An analysis method for round billet spray cooling water flow density and heat exchange coefficient based on nozzle layout

Abstract

The application provides a nozzle layout-based analysis method for round billet spray cooling water flow density and heat exchange coefficient; including data fitting on the water flow density distribution data of the homogenized nozzle; three-dimensional spherical grid division processing is performed on the nozzle according to the spray angle; the grid characteristic parameters of each spray grid are calculated; the spray water quantity of each spray deviation angle is calculated according to the grid characteristic parameters; the area of each spray grid on the surface of the casting billet is calculated; the area of the sphere surrounded by the projection points of the connecting line between each set point on the equal water flow density sphere is obtained; the casting billet cylindrical water flow density is calculated based on the area of the sphere surrounded by the projection points, the spray water quantity of each spray deviation angle and the area of each spray grid on the surface of the casting billet; and finally the heat exchange coefficient of the casting billet surface is calculated. The results of the water flow density and heat exchange coefficient analysis provide boundary condition basis for the fine round billet three-dimensional temperature field mathematical model.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of continuous steel casting, and particularly relates to a method for analyzing the water flow density and heat exchange coefficient of round billet spray cooling based on nozzle layout. BACKGROUND

[0002] Continuous casting is a short name of continuous steel casting, and its process is that under a continuous operation state, molten steel continuously transfers and releases heat, and gradually solidifies into a specific shape of a casting billet. Continuous steel casting has the advantages of few processes, short flow, high metal yield, low energy consumption and high automation degree.

[0003] Round billet continuous casting machine is a near-net shape continuous casting technology, and is mainly used for the production of seamless steel pipe billets. Since the round billets formed by direct casting are used for perforation rolling, the steel plant can significantly reduce energy consumption and improve metal yield.

[0004] Compared with slab or bloom, the round billet does not have the problem of corner and corner crack, but the complex heat exchange boundary conditions in the round billet continuous casting process still have an impact on the circumferential uniform cooling of the casting billet. Among various heat exchange boundary conditions, the crystallizer cooling heat exchange and the radiation heat exchange in the secondary cooling zone are generally considered to be conducive to the circumferential uniform cooling of the casting billet. Although the contact heat exchange of a single group of guide rollers belongs to a circumferential non-uniform heat exchange condition, the overall heat exchange effect of the guide rollers can be approximately regarded as uniform due to the alternate arrangement of the guide rollers in the drawing direction. The spray cooling of the nozzle is easy to cause uneven cooling due to the difference in the number of outer circle nozzles and the distance between the nozzle and the casting billet. If the casting billet is unevenly cooled, surface longitudinal cracks and ovality defects may be caused. Therefore, it is of great significance to study the influence of nozzle layout on the cooling of round billets for the fine study of heat exchange boundary conditions and the improvement of the quality of casting billets. In view of the problems in the related art, no effective solution has been proposed so far. SUMMARY

[0005] The purpose of the present application is to provide a method for analyzing the water flow density and heat exchange coefficient of round billet spray cooling based on nozzle layout.

[0006] The present application is realized by the following technical solutions:

[0007] The present application relates to a method for analyzing the water flow density and heat exchange coefficient of round billet spray cooling based on nozzle layout, comprising the following steps:

[0008] Step S1, uniformization processing and data fitting are performed on the nozzle water flow density distribution data; and then three-dimensional spherical grid division processing is performed on the nozzles according to the spray angle, to obtain a plurality of spray grids;

[0009] Step S2: Based on the data fitting results of the nozzle water flow density distribution data, calculate the grid characteristic parameters of each spray grid; then calculate the spray water volume for each spray deviation angle according to the grid characteristic parameters.

[0010] Step S3: Obtain the coordinates of the boundary nodes in the cross-section of the center of the circle at any position along the billet pulling direction. Based on the connection network structure between the nozzle, boundary nodes and several set points on the billet, calculate the area of ​​each spray grid on the surface of the billet.

[0011] Step S4: Obtain the area of ​​the sphere enclosed by the projection points of the lines connecting each set point onto the sphere with equal water flow density; calculate the water flow density on the cylindrical surface of the billet based on the area of ​​the sphere enclosed by the projection points, the spray water volume of each spray deviation angle, and the area of ​​each spray grid on the surface of the billet.

[0012] Step S5: Calculate the heat transfer coefficient of the billet surface based on the water flow density on the billet cylindrical surface and in combination with the temperature of the billet surface and the temperature of the sprayed water.

[0013] Preferably, in step S1, the process of homogenizing and fitting the nozzle water flow density distribution data is as follows: The process of homogenizing the nozzle water flow density distribution data based on the nozzle grid position division principle, and then fitting the homogenized nozzle water flow density distribution data, specifically includes:

[0014] (1) Obtain nozzle water flow density distribution data through water flow density testing;

[0015] (2) Set the grid position directly below the nozzle as the reference point, and adjust the water flow density of the nozzles at other positions to complete the uniform distribution of water flow in the nozzles.

[0016] (3) The nozzle flow density after uniform distribution treatment is fitted based on the normal distribution function to obtain the nozzle fitting function.

[0017] Preferably, in step S1, the process of dividing the nozzle into three-dimensional spherical meshes according to the spray angle to obtain several spray meshes specifically involves: based on the spray angle of the nozzle, dividing the nozzle into three-dimensional spherical meshes according to a preset angle each time, and terminating at the termination angle.

[0018] Preferably, in step S2, the process of calculating the grid characteristic parameters of each spray grid based on the data fitting results of the nozzle water flow density distribution data specifically involves:

[0019] (1) Obtain the spray length of each spray grid on a preset plane on one side of the spray;

[0020] (2) Based on the data fitting results of the nozzle water flow density distribution data and the spray height of the nozzle, calculate the water flow density of each spray grid on the predefined horizontal axis and the area of ​​the spray grid on the nozzle spray sphere.

[0021] (3) Based on the water flow density of the spray grid on the predefined horizontal axis, the area of ​​the spray grid on the spray sphere of the nozzle, and the spray length of each spray grid on the preset plane, calculate the comprehensive dimensionless water flow density of each spray grid.

[0022] Preferably, in step S2, the process of calculating the spray water volume for each spray deviation angle based on the grid characteristic parameters specifically involves:

[0023] (1) Calculate the water volume weight of each spray grid based on the comprehensive dimensionless water flow density of each spray grid;

[0024] (2) Calculate the spray water volume for each spray deviation angle based on the water volume weight of each spray grid, the spray area, and the area of ​​each spray grid on the spray sphere of the nozzle.

[0025] Preferably, in step S3, the process of obtaining the coordinates of the boundary nodes within the cross-section of the center of a circle at any position along the throwing direction specifically involves:

[0026] (1) Construct a three-dimensional coordinate system with the center of the meniscus of the billet as the origin, and establish the relationship between the center and the boundary nodes at any position along the throwing direction;

[0027] (2) Obtain the angle between the line connecting the boundary node and the center of the circle and the line connecting the nozzle and the center of the circle, the geometric radius of the blank and the coordinates of the center of the circle, and calculate the coordinates of the boundary node by using the correlation law between the center of the circle and the boundary node at any position along the throwing direction.

[0028] Preferably, in step S3, the process of calculating the area of ​​each spray grid on the surface of the billet based on the connection network structure between the nozzle, boundary nodes, and several set points on the billet is specifically as follows:

[0029] (1) The boundary nodes rotate in two directions along the circumference of the blank and extend forward and backward along the pulling direction to form an arc surface with several set points as boundaries.

[0030] (2) Obtain the intersection point, boundary node, and line length between the line connecting the set point and the center of the billet and the sphere with equal water flow density, as well as the line length between the set point and the nozzle, and calculate the area of ​​each spray grid on the surface of the billet.

[0031] Preferably, in step S4, the process of obtaining the area of ​​the sphere enclosed by the projection points of the lines connecting the various set points on the sphere with equal flow density specifically involves: obtaining the length of the arc formed by connecting the lines connecting the various set points between the projection points of the sphere with equal flow density, and calculating the area of ​​the sphere enclosed by the projection points of the arc formed by the boundaries of several set points on the sphere with equal flow density.

[0032] Preferably, in step S4, the process of calculating the water flow density on the cylindrical surface of the cast billet based on the area of ​​the sphere enclosed by the projection points, the spray water volume at each spray deviation angle, and the area of ​​each spray grid on the surface of the cast billet is as follows:

[0033] (1) Calculate the water flow density of a single nozzle at the boundary node based on the area of ​​the sphere enclosed by the projection points, the spray water volume at each spray deviation angle, and the area of ​​each spray grid on the surface of the billet.

[0034] (2) The water flow density on the billet column surface is calculated based on the water flow density of a single nozzle at the boundary node.

[0035] Preferably, in step S5, the formula for calculating the heat transfer coefficient of the billet surface is:

[0036]

[0037] In formula (1),

[0038] H[z][r] is the spray heat transfer coefficient of the grid with z as the drawing coordinate and r as the circumferential coordinate;

[0039] Vs[z][r] represents the water flow density of the grid with z as the throwing coordinate and r as the circumferential coordinate;

[0040] Ts[z][r] represents the surface temperature of the billet in the grid with z as the drawing coordinate and r as the circumferential coordinate.

[0041] Tanh is the hyperbolic tangent function;

[0042] Tw represents the temperature of the spray water.

[0043] The principle of the analytical method involved in this invention is as follows: data fitting is performed on the nozzle water flow density distribution data after homogenization; the nozzles are divided into three-dimensional spherical meshes according to the spray angle; the mesh characteristic parameters of each spray mesh are calculated; the spray water volume at each spray deviation angle is calculated based on the mesh characteristic parameters; the area of ​​each spray mesh on the surface of the billet is calculated; the area of ​​the sphere enclosed by the projection points of the lines connecting each set point on the sphere of equal water flow density is obtained; based on the area of ​​the sphere enclosed by the projection points, the spray water volume at each spray deviation angle, and the area of ​​each spray mesh on the surface of the billet, the water flow density on the cylindrical surface of the billet is calculated; finally, the heat transfer coefficient of the billet surface is calculated.

[0044] The present invention has the following advantages:

[0045] This invention addresses the secondary cooling process in continuous casting of round billets by establishing a mathematical model for the water distribution on a spherical grid of nozzles. Using this model and test data from the original nozzles, the amount of water sprayed at different locations and areas under nozzle spray is calculated. A mathematical model for the water distribution on a cylindrical grid of the billet is also established. This model converts the nozzle water volume data into water flow density data on the cylindrical surface of the billet, providing a data foundation for calculating the spray heat transfer coefficient. Finally, the spray heat transfer coefficient at each node on the cylindrical surface of the billet is calculated using the water flow density, providing a boundary condition basis for a refined three-dimensional temperature field mathematical model of the round billet. Attached Figure Description

[0046] Figure 1 This is a flowchart of an analysis method for the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout, which is involved in this invention.

[0047] Figure 2 This is a diagram showing the water flow density distribution of a pure water nozzle;

[0048] Figure 3 This is a diagram showing the water flow density distribution of the air-water nozzle;

[0049] Figure 4 This is a comparison chart of the raw data and the fitted curve of the pure water nozzle;

[0050] Figure 5 This is a comparison chart of the raw data and fitted curves of the air-water nozzle;

[0051] Figure 6 This is a schematic diagram of the two-dimensional division of the spray grid;

[0052] Figure 7 This is a schematic diagram of the three-dimensional division of the spray grid;

[0053] Figure 8 This is a schematic diagram of the calculation results of the water volume weight of the spray grid;

[0054] Figure 9 This is a schematic diagram of the cross-section of a round billet;

[0055] Figure 10 This is a schematic diagram for calculating water flow density;

[0056] Figure 11 This is a top view for calculating water flow density;

[0057] Figure 12 This is a side view for calculating water flow density;

[0058] Figure 13 This is a diagram showing the heat transfer coefficient of the pure water nozzle casting surface;

[0059] Figure 14 This is a diagram showing the heat transfer coefficient of the gas-water nozzle casting surface. Detailed Implementation

[0060] The present invention will now be described in detail with reference to specific embodiments. It should be noted that the following embodiments are merely further illustrations of the present invention, but the scope of protection of the present invention is not limited to the following embodiments.

[0061] Example

[0062] This embodiment relates to an analysis method for the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout, see [link to relevant documentation]. Figure 1 As shown, it includes the following steps:

[0063] Step S1: The nozzle water flow density distribution data is homogenized and fitted; then, the nozzle is divided into three-dimensional spherical meshes according to the spray angle to obtain several spray meshes.

[0064] Step S2: Based on the data fitting results of the nozzle water flow density distribution data, calculate the grid characteristic parameters of each spray grid; then calculate the spray water volume for each spray deviation angle according to the grid characteristic parameters.

[0065] Step S3: Obtain the coordinates of the boundary nodes in the cross-section of the center of the circle at any position along the billet pulling direction. Based on the connection network structure between the nozzle, boundary nodes and several set points on the billet, calculate the area of ​​each spray grid on the surface of the billet.

[0066] Step S4: Obtain the area of ​​the sphere enclosed by the projection points of the lines connecting each set point onto the sphere with equal water flow density; calculate the water flow density on the cylindrical surface of the billet based on the area of ​​the sphere enclosed by the projection points, the spray water volume of each spray deviation angle, and the area of ​​each spray grid on the surface of the billet.

[0067] Step S5: Calculate the heat transfer coefficient of the billet surface based on the water flow density on the billet cylindrical surface and in combination with the temperature of the billet surface and the temperature of the sprayed water.

[0068] The specific analytical method involved in this embodiment is as follows:

[0069] 1. Mathematical model of heat transfer during round billet spray cooling

[0070] 1.1 Calculation of the heat transfer coefficient of the billet surface

[0071] The heat transfer characteristics of atomized water droplets sprayed onto the surface of high-temperature steel by an air-water nozzle were analyzed, and the surface heat transfer coefficient was obtained as a function of the surface temperature of the cast billet and the water flow density:

[0072]

[0073] In formula (1):

[0074] H[z][r] represents the spray heat transfer coefficient of the grid with z as the throwing coordinate and r as the circumferential coordinate, W / (㎡·℃);

[0075] Vs[z][r] represents the water flow density of the grid with coordinate z in the throwing direction and coordinate r in the circumferential direction, in kg / (m³). 2 ·s);

[0076] Ts[z][r] represents the surface temperature of the billet in °C for the grid with z as the drawing coordinate and r as the circumferential coordinate.

[0077] Tanh() represents the hyperbolic tangent function;

[0078] Tw represents the temperature of the spray water, in °C;

[0079] As can be seen from formula (1), the corresponding heat transfer coefficient can be obtained by calculating the water flow density at the boundary.

[0080] 1.2 Calculation of Nozzle Water Flow Density

[0081] The model numbers and parameters of pure water cone nozzles and air-water cone nozzles were obtained from a nozzle testing center, as shown in Table 1.

[0082] Table 1

[0083] Parameter Pure water nozzle Air-water nozzle Model 1 / 4PZ1577QZ5 HPZ1.8-65QZ2 Air pressure [MPa] 0 0.2 Water pressure [MPa] 0.3 0.2 Gas flow [Nm 3 / h]]]> 0 7.30 Water flow W [L / min] 11.42 1.80 Spray height H [mm] 150 150 Spray angle [°] 77.8 66.5

[0084] See Figure 2 and Figure 3 As shown, the water flow density distribution was obtained in the water flow density test of the pure water nozzle and the air-water nozzle.

[0085] The grid position directly below the nozzle was set to 0, and the other grid positions were increased sequentially to homogenize the water flow density distribution of the pure water nozzle and the air-water nozzle. The resulting water flow density distribution is shown in Table 2.

[0086] Table 2

[0087] Position (mm) Pure water nozzle Air-water nozzle Position (mm) Pure water nozzle Air-water nozzle 0 100 100 375 0 0 25 95.5 93.5 400 0 0 50 86 65.5 425 0 0 75 74.5 37.5 450 0 0 100 58 9.5 475 0.5 0.5 125 36.5 3.0 500 0 0 150 77.8 1.0 525 0.5 0.5 175 4 0 550 0.5 0.5 200 0.5 0 575 0 0 225 0.5 0.5 600 0 0 250 0 0 625 0 0 275 0 0 650 0 0 300 0.5 0.5 675 0 0 325 0.5 0.5 700 0 0 350 0 0 725 0 0

[0088] The data above is fitted according to the normal distribution function to obtain the fitting function of the pure water nozzle. The calculation formula is shown in formula (2):

[0089]

[0090] In formula (2):

[0091] fv(x) represents the water flow density at a vertical distance x from the center of the pure water nozzle, which is dimensionless;

[0092] x represents the vertical distance from the center of the nozzle, in mm.

[0093] A comparison chart of the raw data and fitted curves for the pure water nozzle can be found here. Figure 4 As shown.

[0094] The fitting function for the air-water nozzle is shown in formula (3):

[0095]

[0096] In formula (3):

[0097] fn(x) represents the water flow density at a vertical distance x from the center of the air-water nozzle, which is dimensionless;

[0098] x represents the vertical distance from the center of the nozzle, in mm.

[0099] A comparison chart of the raw data and fitted curves for the pure water nozzle can be found here. Figure 5 As shown.

[0100] like Figure 6 As shown, the conical nozzle is divided into three-dimensional spherical meshes at 1° intervals according to the spray angle, and the division ends when the angle is η; the spray length MN of each spray mesh on the plane can be obtained on one side of the spray, as shown in the figure below, and its calculation is shown in formula (4):

[0101] Lmn(ε)=H·(tan(ε+0.5)-tan(ε-0.5)) (4)

[0102] In formula (4):

[0103] ε represents the angle between the line connecting the center point of the ε-th spray grid and the nozzle and the center line of the nozzle, in °;

[0104] H represents the distance between the nozzle and the test surface during the water flow density test, in meters (m).

[0105] Lmn(ε) represents the spray length MN, m of the ε-th spray grid.

[0106] The calculation of the water flow density on the x-axis for each spray grid is shown in formula (5):

[0107]

[0108] In formula (5):

[0109] ε represents the angle between the line connecting the center point of the ε-th spray grid and the nozzle and the center line of the nozzle, in °;

[0110] H represents the minimum distance between the nozzle and the billet, in meters (m).

[0111] fv() represents a function that calculates the dimensionless water flow density of a pure water nozzle based on the vertical distance from the nozzle center;

[0112] fn() represents a function that calculates the dimensionless water flow density of the corresponding air-water nozzle based on the vertical distance from the nozzle center;

[0113] W(ε) represents the dimensionless water flow density of the ε-th spray grid.

[0114] The area of ​​each spray grid on the spray sphere of the conical nozzle is calculated as shown in formula (6):

[0115]

[0116] In formula (6):

[0117] ε represents the angle between the line connecting the center point of the ε-th spray grid and the nozzle and the center line of the nozzle, in °;

[0118] H represents the minimum distance between the nozzle and the billet, in meters (m).

[0119] A(ε) represents the spray area of ​​the ε-th spray grid, in square meters.

[0120] The shape of the spray grid after division, such as Figure 7 As shown.

[0121] Therefore, the comprehensive dimensionless flow density of each spray grid is calculated as shown in formula (7):

[0122] Wfd(ε)=Lmn(ε)·W(ε)·A(ε) (7)

[0123] In formula (7):

[0124] ε represents the angle between the line connecting the center point of the ε-th spray grid and the nozzle and the center line of the nozzle, in °;

[0125] Lmn(ε) represents the spray length MN, m of the ε-th spray grid;

[0126] W(ε) represents the dimensionless water flow density of the ε-th spray grid;

[0127] A(ε) represents the spray area of ​​the ε-th spray grid, in square meters;

[0128] Wfd(ε) represents the overall dimensionless flow density of the ε-th spray grid.

[0129] The calculation of the water volume weight for each spray grid is shown in formula (8):

[0130]

[0131] In formula (8):

[0132] ε represents the angle between the line connecting the center point of the ε-th spray grid and the nozzle and the center line of the nozzle, in °;

[0133] Wfd(ε) refers to the overall dimensionless flow density of the ε-th spray grid;

[0134] η represents the maximum spray angle of the nozzle, in degrees;

[0135] Wfw(ε) refers to the water flow density weight of the ε-th spray grid, which is dimensionless;

[0136] The calculated water weight of the spray grid is shown below. Figure 8 As shown.

[0137] In a spray sphere formed by a total spray water volume of W and a spray height of H, the spray water volume SPw for a spray area of ​​Sr and a spray deviation angle of e is calculated as shown in formula (9):

[0138]

[0139] In formula (9):

[0140] Sr represents the area of ​​a specific region, expressed in square meters;

[0141] e represents the angle between the line connecting the center point of the specific area and the nozzle and the center line of the nozzle, in degrees;

[0142] W represents the total water flow rate of the nozzle, in L / min;

[0143] Wfw(e) represents the water flow density weight of the e-th spray grid, which is dimensionless;

[0144] A(e) represents the area of ​​the e-th spray grid, in square meters;

[0145] SPw(Sr,e) represents the spray water volume (L / min) in a specific area with an area of ​​Sr and a spray angle of e.

[0146] 1.3 Calculation of water flow density on the surface of the cast billet

[0147] A three-dimensional coordinate system is established with the center of the meniscus of the billet as the origin. The influence of the casting machine radius on the coordinate system is ignored. The horizontal direction of the origin is the X direction, the vertical direction is the Y direction, and the direction of billet pulling is the Z direction.

[0148] like Figure 9 As shown, the water spray volume of the nozzle is set to m, the number of nozzles around the billet is n, the geometric radius of the billet is r, the minimum distance between the nozzle and the surface of the billet is R, and the maximum spray angle of the nozzle is γ.

[0149] Let the coordinates of the center M of any position along the throwing direction be (0,0,Zm), the coordinates of the boundary node A in the cross section of the center be (Xa,Ya,Za), and the coordinates of the nozzle N where node A is located be (Xn,Yn,Zn). Since the nozzle N is directly above the center M, its calculation is shown in formula (10):

[0150]

[0151] In formula (10):

[0152] Xn represents the x-coordinate of nozzle N, in meters (m).

[0153] Yn represents the y-coordinate of nozzle N, in meters;

[0154] Zn represents the z-axis coordinate of nozzle N, in meters;

[0155] r represents the radius of the circular billet cross-section, in meters (m).

[0156] R represents the minimum distance between the nozzle and the surface of the billet, in meters (m).

[0157] The area of ​​the sphere with equal water flow density formed by taking the coordinates of nozzle N as the center, R as the radius, and the nozzle spray angle γ as the radiation angle is Sn, and its calculation is shown in formula (11):

[0158] Sn=2πR 2 ·(1-cos(γ / 2)) (11)

[0159] In formula (11):

[0160] R represents the minimum distance between the nozzle and the surface of the cast billet, in meters (m).

[0161] γ represents the maximum spray angle of the nozzle, in °.

[0162] The angle between the line connecting node A and center M and the line connecting nozzle N and center M is β. The maximum value of β is βm. Each nozzle row has a total of n nozzles. The range of β is (0, βs), as shown in formula (12):

[0163]

[0164] In formula (12):

[0165] n represents the number of nozzles in each nozzle row, which is dimensionless;

[0166] βs represents the research range of the spray angle on one side of the nozzle, in °.

[0167]

[0168] In equation (13):

[0169] r represents the radius of the circular billet cross-section, in meters (m).

[0170] R represents the minimum distance between the nozzle and the surface of the cast billet, in meters (m).

[0171] βm represents the maximum angle at which the nozzle sprays onto the billet on one side, in °.

[0172] Since node A and the center M are in the same cross-section, therefore:

[0173]

[0174] In equation (14):

[0175] Xa represents the x-coordinate of node A, in meters (m).

[0176] Ya represents the y-coordinate of node A, in meters (m).

[0177] Za represents the z-coordinate of node A, in meters (m).

[0178] r represents the radius of the circular billet cross-section, in meters (m).

[0179] β represents the angle, in degrees, between the line connecting node A and center M and the line connecting nozzle N and center M;

[0180] Z represents the z-axis coordinate of the center M of the circle, in meters.

[0181] A cylindrical mesh is formed with node A as the center point. This mesh is formed by rotating the line connecting the center point and node A circumferentially by dθ / 2 degrees clockwise and counterclockwise, respectively, and extending it by dl / 2 degrees in the positive and negative directions along the casting direction, respectively, and projecting it onto the surface of the cast billet. This mesh is an arc surface with points A1, A2, A3, and A4 as its boundaries, and the area of ​​this arc surface is defined as Sa. Figure 10 As shown, then:

[0182] Sa=dθ·r·dl (15)

[0183] In equation (15):

[0184] dθ represents the circumferential grid step size, in degrees;

[0185] r represents the radius of the cast billet, in meters;

[0186] dl represents the step size in the pulling direction, in meters;

[0187] Sa represents the area of ​​the cylindrical grid, in square meters;

[0188] Let the coordinates of A1, A2, A3, and A4 be (Xa1, Ya1, Za1), (Xa2, Ya2, Za2), (Xa3, Ya3, Za3), and (Xa4, Ya4, Za4) respectively. Then we can obtain:

[0189]

[0190]

[0191] In equations (16)-(19):

[0192] dθ represents the circumferential grid step size, in degrees;

[0193] β represents the angle, in degrees, between the line connecting node A and center M and the line connecting nozzle N and center M;

[0194] r represents the radius of the cast billet, in meters;

[0195] dl represents the step size in the pulling direction, in meters;

[0196] Xa1, Ya1, and Za1 represent the x-axis, y-axis, and z-axis coordinates of point A1, in meters (m).

[0197] Xa2, Ya2, and Za2 represent the x-axis, y-axis, and z-axis coordinates of point A2, in meters (m).

[0198] Xa3, Ya3, and Za3 represent the x-axis, y-axis, and z-axis coordinates of point A3, in meters (m).

[0199] Xa4, Ya4, and Za4 represent the x-axis, y-axis, and z-axis coordinates of point A4, in meters (m).

[0200] Za represents the z-axis coordinate of node A, in meters.

[0201] Let the lines connecting points A, A1, A2, A3, and A4 to the center of the circle intersect the sphere Sn at points B, B1, B2, B3, and B4, respectively, with corresponding coordinates (Xb, Yb, Zb), (Xb1, Yb1, Zb1), (Xb2, Yb2, Zb2), (Xb3, Yb3, Zb3), and (Xb4, Yb4, Zb4). Let the length of the line connecting point A and nozzle N be Lan, the length of the line connecting point A1 and nozzle N be La1n, the length of the line connecting point A2 and nozzle N be La2n, the length of the line connecting point A3 and nozzle N be La3n, and the length of the line connecting point A4 and nozzle N be La4n. We can calculate:

[0202]

[0203] In equation (20):

[0204] Lan, La1n, La2n, La3n, and La4n represent the lengths of the lines connecting points A, A1, A2, A3, and A4 to nozzle N, in meters (m).

[0205] Xa, Ya, and Za represent the x-axis, y-axis, and z-axis coordinates of node A, in meters (m).

[0206] Xn, Yn, and Zn represent the x-axis, y-axis, and z-axis coordinates of nozzle N, in meters (m).

[0207] Xa1, Ya1, and Za1 represent the x-axis, y-axis, and z-axis coordinates of point A1, in meters (m).

[0208] Xa2, Ya2, and Za2 represent the x-axis, y-axis, and z-axis coordinates of point A2, in meters (m).

[0209] Xa3, Ya3, and Za3 represent the x-axis, y-axis, and z-axis coordinates of point A3, in meters (m).

[0210] Xa4, Ya4, and Za4 represent the x-axis, y-axis, and z-axis coordinates of point A4, in meters (m).

[0211] The length of the line connecting points A1 and A2 is La1a2, the length of the line connecting points A1 and A3 is La1a3, the length of the line connecting points A2 and A4 is La2a4, the length of the line connecting points A3 and A4 is La3a4, and the length of the line connecting points A1 and A4 is La1a4. Calculate:

[0212]

[0213] In equation (21):

[0214] La1a2, La1a3, La2a4, La3a4, and La1a4 represent the lengths (in meters) of the lines connecting points A1 and A2, A1 and A3, A2 and A4, A3 and A4, and A1 and A4, respectively.

[0215] dθ represents the circumferential grid step size, in degrees;

[0216] dl represents the step size in the pulling direction, in meters;

[0217] r represents the radius of the cast billet, in meters.

[0218] The formula for calculating the area GS of each cylindrical grid on the surface of the cast billet is as follows:

[0219] GS=La1a2·La1a3 (22)

[0220] In equation (22):

[0221] La1a2 and La1a3 represent the lengths of the lines connecting points A1 and A2, and A1 and A3, in meters (m).

[0222] GS represents the area of ​​the cylindrical grid, in square meters (m²).

[0223] The angle between lines La1n and La2n is α[La1n,La2n], the angle between lines La1n and La3n is α[La1n,La3n], the angle between lines La2n and La4n is α[La2n,La4n], the angle between lines La3n and La4n is α[La3n,La4n], the angle between lines La1n and La4n is α[La1n,La4n], and the angle between lines Lmn and Lan is α[Lmn,Lan]. Then:

[0224]

[0225] In equation (23):

[0226] La1a2, La1a3, La1a4, La3a4, and La2a4 represent the lengths (in meters) of the lines connecting points A1 and A2, A1 and A3, A1 and A4, A3 and A4, and A2 and A4, respectively.

[0227] La1n, La2n, La3n, La4n, and Lan represent the lengths of the lines connecting points A1, A2, A3, A4, and A to nozzle N, in meters (m).

[0228] Lmn represents the length of the line connecting the center M and the nozzle N, in meters;

[0229] α[La1n,La2n], α[La1n,La3n], α[La2n,La4n], α[La3n,La4n], α[La1n,La4n], α[Lmn,Lan] represent the angles between lines La1n and La2n, La1n and La3n, La2n and La4n, La3n and La4n, La1n and La4n, Lmn and Lan, in °;

[0230] r represents the radius of the cast billet, in meters;

[0231] R represents the minimum distance between the nozzle and the surface of the billet, in meters (m).

[0232] The projection of line La1a2 onto the spherical surface Sn of the spray system is an arc formed by connecting points B1 and B2, with a length of Ab1b2. The projection of line La1a3 onto the spherical surface Sn is an arc formed by connecting points B1 and B3, with a length of Ab1b3. The projection of line La2a4 onto the spherical surface Sn is an arc formed by connecting points B2 and B4, with a length of Ab2b4. The projection of line La3a4 onto the spherical surface Sn is an arc formed by connecting points B3 and B4, with a length of Ab3b4. Then:

[0233]

[0234] In equation (24):

[0235] α[La1n,La2n], α[La1n,La3n], α[La2n,La4n], α[La3n,La4n], α[La1n,La4n] represent the angles between lines La1n and La2n, La1n and La3n, La2n and La4n, La3n and La4n, and La1n and La4n, in °;

[0236] R represents the minimum distance between the nozzle and the surface of the billet, in meters (m).

[0237] Ab1b2, Ab1b3, Ab2b4, Ab3b4, and Ab1b4 represent the lengths (in meters) of the projected arcs of points B1 and B2, B1 and B3, B2 and B4, B3 and B4, and B1 and B4 on the spray sphere Sn.

[0238] The projection of the node-representing surface Sa onto the spray sphere Sn is a sphere Sb bounded by points B1, B2, B3, and B4. This sphere can be considered as being formed by splicing triangle E (composed of arcs B1B2, B1B4, and B2B4) and triangle F (composed of arcs B1B3, B1B4, and B3B4). Therefore, the formulas for calculating the areas of triangle E (Se), triangle F (Sf), and Sb are as follows:

[0239]

[0240] In equations (25)-(27):

[0241] Ab1b2, Ab1b3, Ab2b4, Ab3b4, and Ab1b4 represent the lengths (in meters) of the projected arcs of points B1 and B2, B1 and B3, B2 and B4, B3 and B4, and B1 and B4 on the spray sphere Sn.

[0242] SEa represents half the sum of the three sides of triangle E, in meters;

[0243] SFa represents half the sum of the three sides of triangle F, in meters (m).

[0244] Se represents the area of ​​triangle E, in square meters;

[0245] Sf represents the area of ​​triangle F, in square meters;

[0246] Sb represents the projected area of ​​the cylindrical grid Sa on the spray sphere, in square meters.

[0247] The formula for calculating the projected area St of Sb on the standard spray sphere of the nozzle is:

[0248]

[0249] In equation (28):

[0250] Sb represents the projected area of ​​the cylindrical mesh Sa on the spray sphere, in square meters;

[0251] St represents the projected area of ​​Sb on the standard spray sphere of the nozzle, in square meters;

[0252] R represents the minimum distance between the nozzle and the surface of the billet, in meters (m).

[0253] H represents the distance between the nozzle and the test surface during the water flow density test, in meters (m).

[0254] Then the water flow density Ws(β) of a single nozzle at node A is:

[0255]

[0256] In equation (29):

[0257] β represents the angle, in degrees, between the line connecting node A and center M and the line connecting nozzle N and center M;

[0258] βm represents the maximum angle at which the nozzle sprays onto the billet on one side, in °;

[0259] GS represents the area of ​​the cylindrical grid, in square meters;

[0260] St represents the projected area of ​​Sb on the standard spray sphere of the nozzle, in square meters;

[0261] α[Lmn,Lan] represents the angle between lines Lmn and Lan, in degrees;

[0262] SPw(St,β=α[Lmn,Lan]) represents the spray water volume in a specific area with an area of ​​St and a spray angle of α[Lmn,Lan], in L / min;

[0263] Ws(β) represents the water flow density of the cylindrical mesh under a single nozzle when the angle between the line connecting node A and center M and the line connecting nozzle N and center M is β, in kg / (m³). 2 ·s).

[0264] Because the water spraying areas of multiple nozzles on the billet overlap, the water flow density Wm(β) of the multiple nozzles at node A is:

[0265]

[0266] In equation (30):

[0267] β represents the angle, in degrees, between the line connecting node A and center M and the line connecting nozzle N and center M;

[0268] βm represents the maximum angle at which the nozzle sprays onto the billet on one side, in °;

[0269] βs represents the research range of the spray angle on one side of the nozzle, in °;

[0270] Ws(β) represents the water flow density of the cylindrical mesh under a single nozzle when the angle between the line connecting node A and center M and the line connecting nozzle N and center M is β, in kg / (m³). 2 ·s);

[0271] Wm(β) represents the water flow density of the cylindrical mesh under multiple nozzles when the angle between the line connecting node A and center M and the line connecting nozzle N and center M is β, in kg / (m²). 2 ·s).

[0272] Top and side views for water flow density calculation are as follows: Figure 11 and Figure 12 As shown.

[0273] 2. Analysis of Calculation Results

[0274] Geometric parameters, nozzle parameters, and physical property parameters are shown in Table 3:

[0275] Table 3

[0276]

[0277]

[0278] The heat transfer coefficient distributions of the pure water nozzle and the air-water nozzle were calculated, as shown in the figure. Figure 13 and Figure 14 As shown.

[0279] In summary, this invention addresses the secondary cooling process of round billet continuous casting by establishing a mathematical model for the water distribution of a spherical mesh nozzle. Using this model and test data from the original nozzles, the spray water volume at different locations and areas under nozzle spray is calculated. A mathematical model for the water distribution of a cylindrical mesh on the billet is also established. This model converts the nozzle water volume data into water flow density data on the billet's cylindrical surface, providing a data foundation for calculating the spray heat transfer coefficient. Finally, the spray heat transfer coefficient at each node on the billet's cylindrical surface is calculated using the water flow density, providing the boundary conditions for a refined three-dimensional temperature field mathematical model of the round billet.

[0280] The specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various modifications or variations within the scope of the claims, which do not affect the essence of the present invention.

Claims

1. A method for analyzing the flow density and heat transfer coefficient of spray cooling water for round billets based on nozzle layout, characterized in that, Includes the following steps: Step S1: The nozzle water flow density distribution data is homogenized and fitted; then, the nozzle is divided into three-dimensional spherical meshes according to the spray angle to obtain several spray meshes. Step S2: Based on the data fitting results of the nozzle water flow density distribution data, calculate the grid characteristic parameters of each spray grid; then calculate the spray water volume for each spray deviation angle according to the grid characteristic parameters. Step S3: Obtain the coordinates of the boundary nodes in the cross-section of the center of the circle at any position along the billet pulling direction. Based on the connection network structure between the nozzle, boundary nodes and several set points on the billet, calculate the area of ​​each spray grid on the surface of the billet. Step S4: Obtain the area of ​​the sphere enclosed by the projection points of the lines connecting each set point onto the sphere with equal water flow density; calculate the water flow density on the cylindrical surface of the billet based on the area of ​​the sphere enclosed by the projection points, the spray water volume of each spray deviation angle, and the area of ​​each spray grid on the surface of the billet. Step S5: Calculate the heat transfer coefficient of the billet surface based on the water flow density on the billet cylindrical surface and in combination with the temperature of the billet surface and the temperature of the sprayed water. In step S2, the process of calculating the grid characteristic parameters of each spray grid based on the data fitting results of the nozzle water flow density distribution data is as follows: (1) Obtain the spray length of each spray grid on a preset plane on one side of the spray; (2) Based on the data fitting results of the nozzle water flow density distribution data and the spray height of the nozzle, calculate the water flow density of each spray grid on the predefined horizontal axis and the area of ​​the spray grid on the nozzle spray sphere; (3) Based on the water flow density of the spray grid on the predefined horizontal axis, the area of ​​the spray grid on the spray sphere of the nozzle, and the spray length of each spray grid on the preset plane, calculate the comprehensive dimensionless water flow density of each spray grid. The process of calculating the spray water volume for each spray deviation angle based on the grid characteristic parameters is as follows: (1) Calculate the water volume weight of each spray grid based on the comprehensive dimensionless water flow density of each spray grid; (2) Calculate the spray water volume for each spray deviation angle based on the water volume weight of each spray grid, the spray area, and the area of ​​each spray grid on the spray sphere of the nozzle; In step S3, the process of calculating the area of ​​each spray grid on the surface of the billet based on the connection network structure between the nozzle, boundary nodes, and several set points on the billet is as follows: (1) The boundary nodes rotate in two directions along the circumference of the round blank and extend forward and backward along the pulling direction to form an arc surface with several set points as boundaries; (2) Obtain the intersection point, boundary node, and line length between the line connecting the set point and the center of the billet and the sphere with equal water flow density, as well as the line length between the set point and the nozzle, and calculate the area of ​​each spray grid on the surface of the billet.

2. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S1, the process of homogenizing and fitting the nozzle water flow density distribution data is specifically as follows: The process involves homogenizing the nozzle water flow density distribution data based on the nozzle grid position division principle, and then fitting the homogenized nozzle water flow density distribution data. (1) Obtain nozzle water flow density distribution data through water flow density testing; (2) Set the grid position directly below the nozzle as the reference point, and adjust the water flow density of the nozzles at other positions to complete the uniform distribution of water flow in the nozzles. (3) The nozzle flow density after uniform distribution treatment is fitted based on the normal distribution function to obtain the fitting function of the nozzle.

3. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S1, the process of dividing the nozzle into three-dimensional spherical meshes according to the spray angle to obtain several spray meshes is as follows: based on the spray angle of the nozzle, the three-dimensional spherical mesh is divided according to a preset angle each time, and the process ends at the termination angle.

4. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S3, the process of obtaining the coordinates of the boundary nodes within the cross-section of the center of a circle at any position along the throwing direction is specifically as follows: (1) Construct a three-dimensional coordinate system with the center of the meniscus of the billet as the origin, and establish the relationship between the center and the boundary nodes at any position along the throwing direction; (2) Obtain the angle between the line connecting the boundary node and the center of the circle and the line connecting the nozzle and the center of the circle, the geometric radius of the blank and the coordinates of the center of the circle, and calculate the coordinates of the boundary node by using the correlation law between the center of the circle and the boundary node at any position along the throwing direction.

5. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S4, the process of obtaining the area of ​​the sphere enclosed by the projection points of the lines connecting the various set points on the sphere with equal flow density is specifically as follows: obtain the length of the arc formed by connecting the lines connecting the various set points between the projection points of the sphere with equal flow density, and calculate the area of ​​the sphere enclosed by the projection points of the arc formed by the boundaries of several set points on the sphere with equal flow density.

6. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S4, the process of calculating the water flow density on the cylindrical surface of the billet based on the area of ​​the sphere enclosed by the projection points, the spray water volume at each spray deviation angle, and the area of ​​each spray grid on the billet surface is as follows: (1) Calculate the water flow density of a single nozzle at the boundary node based on the area of ​​the sphere enclosed by the projection points, the spray water volume at each spray deviation angle, and the area of ​​each spray grid on the surface of the billet. (2) The water flow density on the billet column surface is calculated based on the water flow density of a single nozzle at the boundary node.

7. The method for analyzing the flow density and heat transfer coefficient of round billet spray cooling water based on nozzle layout as described in claim 1, characterized in that, In step S5, the formula for calculating the heat transfer coefficient of the billet surface is shown in formula (1): ， In formula (1), H [ z ][ r [The coordinates for the billet pulling are] z Circumferential coordinates are r The spray heat transfer coefficient of the grid; Vs [ z ][ r [The coordinates for the billet pulling are] z Circumferential coordinates are r The water flow density of the grid; Ts [ z ][ r [The coordinates for the billet pulling are] z Circumferential coordinates are r The surface temperature of the grid-like billet; Tanh It is the hyperbolic tangent function; Tw The temperature of the spray water.

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