CN110000348B - Hyperbolic funnel-shaped crystallizer wide-surface copper plate and preparation method thereof - Google Patents

Hyperbolic funnel-shaped crystallizer wide-surface copper plate and preparation method thereof Download PDF

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CN110000348B
CN110000348B CN201910263790.5A CN201910263790A CN110000348B CN 110000348 B CN110000348 B CN 110000348B CN 201910263790 A CN201910263790 A CN 201910263790A CN 110000348 B CN110000348 B CN 110000348B
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李富帅
韩占光
倪赛珍
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MCC Southern Continuous Casting Technology Engineering Co Ltd
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    • B22—CASTING; POWDER METALLURGY
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    • B22D11/00—Continuous casting of metals, i.e. casting in indefinite lengths
    • B22D11/04—Continuous casting of metals, i.e. casting in indefinite lengths into open-ended moulds
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Abstract

The invention discloses a hyperbolic funnel-shaped crystallizer broad-face copper plate and a preparation method thereof. The middle part of the wide-surface copper plate of the crystallizer is inwards concave to form a funnel-shaped curved surface, the curved surface is formed by the topology of a transverse curve family and a longitudinal curve family, wherein the longitudinal curve of the wide-surface copper plate is distributed in a hyperbolic curve along the height direction of the crystallizer. The hyperbolic longitudinal taper curve enables an air gap between the crystallizer copper plate and the solidified blank shell to be smaller, and heat can be effectively taken away; through the calculation of the thermal coupling model, the stress level of the billet shell in the crystallizer is lower, the forming chance of the casting billet crack is effectively reduced, and the casting billet quality is improved.

Description

Hyperbolic funnel-shaped crystallizer wide-surface copper plate and preparation method thereof
Technical Field
The invention relates to the technical field of continuous casting of thin slabs, in particular to a broad-faced copper plate of a hyperbolic-funnel-shaped crystallizer and a preparation method thereof.
Background
Continuous thin slab casting is a revolutionary leading technology in the iron and steel industry in the world today. The funnel-shaped crystallizer is the core of the thin slab continuous casting technology and is characterized in that a funnel shape with a taper in the vertical direction is adopted between wide-surface copper plates so as to meet the requirements of the insertion of a submerged nozzle, the melting of casting powder and the thickness of a slab.
The curved surface shape of the funnel area is closely related to the strain, the strain rate and the stress level of the plastic deformation area in the high-temperature blank shell, and the surface quality of a casting blank is directly influenced. In a funnel-shaped crystallizer, the transverse curvature of the primary shell decreases from top to bottom as it moves in the funnel region in the withdrawal direction, similar to the straightening process. Because the transverse solidification shrinkage of the blank shell is uneven, the track of a point on the blank shell along the blank drawing direction is not straight, when the point passes through the profile contour line of a rough crystallizer, the curvature and the movement direction are suddenly changed, and larger local stress and strain are generated; meanwhile, creep deformation of the high-temperature blank shell needs a certain time, and due to the high drawing speed, the curvature change rate of the blank shell is inconsistent with the curvature change rate of the inner cavity curve of the crystallizer, so that an air gap is formed, the blank shell is thinned, and cracks are generated when local strain exceeds the critical strain of steel. In the actual casting process, longitudinal cracks often appear on the surface of a casting blank at the joint of the edge of a funnel area and an arc line, and bleed-out is caused in severe cases. Therefore, the reasonable design of the shape of the inner cavity curved surface of the funnel-shaped crystallizer according to the high-temperature mechanical property and the metallurgical solidification characteristics of steel is the key for ensuring the high drawing speed and high quality of the sheet billet continuous casting.
The funnel-shaped curved surface of the crystallizer is formed by the topology of a transverse curve family and a longitudinal curve family. The transverse curve of the crystallizer in the prior art is generally formed by connecting a plurality of sections of straight lines, straight lines and circular arcs, double arcs and ellipses, but the crystallizer in the prior art has the following problems: the transverse curve is designed to be formed by connecting a plurality of sections of straight lines, and the curvature change of the blank shell cannot follow the curvature change of the inner cavity of the crystallizer due to the unsmooth curve, so that a gap of about 0.6mm exists between the blank shell and the wall of the crystallizer at the inflection point of the curve, and cracks are generated on the surface of a casting blank along with the change. The transverse curve is designed to be connected with the circular arc through a straight line, so that the curve is smooth, but the curvature of the curve still has sudden change, and the surface of the blank shell often has longitudinal cracks at high drawing speed. The funnel crystallizer formed by double arcs and ellipses is adopted, the joint of the two arc surfaces of the funnel crystallizer has great influence on metal deformation, and the surface strain of a casting blank reaches the maximum value. The prior art also provides the design requirements and the method of the funnel shape, and provides a transverse curve with continuously changing curvature; and the high-order polynomial is used for carrying out multiple calculations through a computer, so that the crystallizer transverse curve with continuous curvature change and small curvature change rate is obtained. The high temperature mechanical properties of steel indicate that strain to fracture is mainly affected by strain rate, and critical strain value is reduced by the increase or mutation of strain rate. The maximum strain rate of the blank shell in the crystallizer should be controlled in the deformation process, and the maximum strain rate of the blank shell can be minimized only under the condition that the curve curvature change rate of the funnel area is constant.
The curvature change rate of the curve of the funnel area of the thin slab crystallizer in the prior art only considers the stability of curvature change, and the inventor of the application carries out intensive research on the curved surface of the hyperbolic-curve funnel crystallizer on the basis of the curvature change rate and also considers the characteristics that the continuous casting billet shell has larger heat flow near a meniscus and larger solidification shrinkage deformation.
Disclosure of Invention
Based on the problems, the invention aims to provide a hyperbolic funnel-shaped crystallizer broad-surface copper plate to solve the problems that the stress level of an inner blank shell of a crystallizer is high, a casting blank is cracked frequently, the quality of the casting blank is not high and the like in the prior art.
The invention also aims to provide a preparation method of the broad-side copper plate of the hyperbolic funnel-shaped crystallizer.
The above purpose is realized by the following technical scheme:
according to one aspect of the invention, the hyperbolic-funnel-shaped wide-side copper plate of the crystallizer is inwards concave to form a funnel-shaped curved surface in the middle, the curved surface is formed by topological horizontal curves and longitudinal curves, and the longitudinal curves of the wide-side copper plate are distributed in a hyperbolic curve mode along the height direction of the crystallizer.
Preferably, the hyperbola has a large curve gradient in the vicinity of the meniscus and conforms to the curve equation:
Figure BDA0002016115880000021
wherein y represents a coordinate in the direction of drawing; w(y)Representing the projection height of the center point of the y position; w represents the height of the crystallizer; a. b, c and theta are constants and satisfy the following expression:
Figure BDA0002016115880000031
Figure BDA0002016115880000032
Figure BDA0002016115880000033
Figure BDA0002016115880000034
Figure BDA0002016115880000035
wherein h represents the height of the funnel protrusion at the upper opening of the crystallizer, theta is the central angle corresponding to the middle arc part of the transverse curve, and omega is a parameter determined by the contour curve of the blank shell, the temperature gradient at the meniscus and the linear expansion coefficient of the blank shell.
Preferably, the parameter ω is expressed as:
ω＝[α·(dT(y)/dy)·(dW(y)/dy)]/(dL/dy)
wherein α is the linear shrinkage of the blank shell;
Figure BDA0002016115880000036
is the temperature gradient of the blank shell at the meniscus; dL/dy represents the shrinkage of the perimeter of the transverse profile curve of the crystallizer in the direction of drawing; dW(y)And/dy represents the rate of change of the height of the projections of the crystalliser in the direction of the drawing.
Preferably, the transverse curve of the wide-face copper plate is formed by externally cutting two circles with equal diameters.
Preferably, the centers of two circles tangent to each other in the transverse curve are respectively 0 and O1, the starting point of the transverse curve is N, the end point is M, the tangent point is R, and the distance x between the cross-section protrusion height H (x) of the crystallizer copper plate and the center of the copper plate in the width direction satisfies the following piecewise curve equation:
x2+H(x)2＝R02(0≤x≤R0*sin(θ))
(x-xo1)2+(H(x)-yo1)2＝R02 (R0*sin(θ)＜x≤2*R0*sin(θ))
H(x)＝R0-WP(2*R0，sin(θ)＜x≤L/2)
wherein R0 is the radius of the tangent circle, xo1 and yo1 are the coordinates of the O1 point of the center of the tangent circle of the transverse curve, theta is the central angle corresponding to the arc line RN formed by the tangent point R and the starting point N, and Wp is the central protrusion height of the cross section position of the copper plate of the crystallizer; l represents the perimeter of the transverse profile curve of the crystallizer;
the radius R0 of the tangent circle is determined by the following equation:
Figure BDA0002016115880000041
wherein Zm-Zn represents the transverse curve pitch.
According to another aspect of the present invention, there is provided a method for preparing a broad-faced copper plate of the hyperbolic funnel-shaped mold described above, the method comprising: determining a longitudinal curve in the height direction of the crystallizer according to solidification shrinkage and deformation of the blank shell, wherein the longitudinal curve is distributed in a hyperbolic curve mode, and the method specifically comprises the following steps:
assuming that the hyperbola conforms to the curve formula:
Figure BDA0002016115880000042
wherein y represents a coordinate in the direction of drawing; w(y)Representing the projection height of the center point of the y position; w represents the height of the crystallizer; a. b, c and theta are constants;
according to the principle that the shape design of the funnel crystallizer should meet, the expressions of the constants a, b, c and theta are obtained as follows:
Figure BDA0002016115880000043
Figure BDA0002016115880000044
Figure BDA0002016115880000045
Figure BDA0002016115880000046
Figure BDA0002016115880000047
wherein h represents the height of the funnel protrusion at the upper opening of the crystallizer, theta is a central angle corresponding to a curve part of a transverse curve, and omega is a parameter determined by a blank shell contour curve, a temperature gradient at a meniscus and a linear expansion coefficient of the blank shell;
calculating omega to obtain specific values of the constants a, b, c and theta;
the values of the constants a, b, c and theta are substituted into the formula 1 to obtain a hyperbolic longitudinal taper curve.
Preferably, ω is calculated according to the linear shrinkage rate of the blank shell, the temperature gradient of the blank shell at the meniscus, the maximum height in the middle of the funnel-shaped curve of the upper mouth of the crystallizer, and the transverse distance of the transition arc. Further, according to the linear shrinkage rate of the blank shell, the temperature gradient of the blank shell at the position of a meniscus, and the ratio of the change rate of the height of the protrusion of the crystallizer in the casting direction to the shrinkage rate of the perimeter of the transverse profile curve of the crystallizer in the casting direction, omega is calculated.
Preferably, the method further comprises: the method comprises the following steps of determining a transverse curve according to the arrangement requirement of a thin slab nozzle, wherein the transverse curve is formed by externally cutting two circles with equal diameters, and specifically comprises the following steps:
according to the obtained hyperbolic formula, the height W of the central point corresponding to the position of a certain height y is calculated(y)So as to obtain the radius R0 of the tangent circle;
assuming that the centers of the two circles are 0 and O1 respectively, the starting point of the transverse curve is N, the end point is M, the tangent point is R, the height H (x) of the protrusion on the cross section of the copper plate of the crystallizer and the distance x from the center of the copper plate in the width direction satisfy the following piecewise curve equation:
x2+H(x)2＝R02(0≤x≤R0*sin(θ))
(x-xo1)2+(H(x)-yo1)2＝R02 (R0*sin(θ)＜x≤2*R0*sin(θ))
H(x)＝R0-WP(2*R0*sin(θ)＜x≤L/2)
wherein R0 is the radius of the tangent circle, xo1 and yo1 are the coordinates of the O1 point of the center of the tangent circle of the transverse curve, theta is the central angle corresponding to the arc line RN formed by the tangent point R and the starting point N, and Wp is the central protrusion height of the cross section position of the copper plate of the crystallizer; l represents the perimeter of the crystallizer transverse profile curve;
and obtaining the bulge height H (x) of the cross section of the copper plate of the crystallizer according to the piecewise curve equation.
Preferably, the method further comprises: calculating transverse contour lines from the meniscus to different positions of the lower opening of the crystallizer; and topologically obtaining the curved surface graph of the wide-surface copper plate of the crystallizer according to the obtained longitudinal curve family and the transverse curve family of the wide-surface copper plate of the crystallizer.
Preferably, the method further comprises: and processing and preparing a funnel-shaped crystallizer wide-surface copper plate according to the curved surface graph.
Compared with the prior art, the hyperbolic funnel-shaped crystallizer wide-surface copper plate has the following beneficial effects:
according to the characteristics that the continuous casting blank shell has larger heat flow and larger solidification shrinkage deformation near the meniscus, the curvature change rate is large when the continuous casting blank shell is positioned at the meniscus, the curvature is gradually reduced along the casting direction, the curvature at the lower opening of the crystallizer becomes zero, and the longitudinal curve of the wide-surface copper plate of the crystallizer is distributed in a hyperbolic curve along the height direction of the crystallizer, so that the air gap between the copper plate of the crystallizer and the solidified blank shell is smaller, the heat can be efficiently taken away, and the quality of a casting blank is improved.
According to the invention, the transverse curve of the wide-face copper plate of the crystallizer is formed by externally cutting two circles with the same diameter according to the arrangement requirement of a thin slab water gap, so that the curvature change is continuous and stable, and the forming chance of casting blank cracks can be effectively reduced.
According to the curve design of the wide-surface copper plate of the crystallizer, a novel funnel-shaped crystallizer is formed, the novel funnel-shaped crystallizer is called as a hyperbolic curve funnel-shaped crystallizer, and through calculation of a thermal coupling model, the stress level of a blank shell in the crystallizer is lower, so that the forming chance of casting blank cracks is effectively reduced, and the quality of the casting blank is obviously improved.
Drawings
FIG. 1 is a schematic view of a curved surface of a wide-surface copper plate of a crystallizer in an embodiment of the invention;
FIG. 2 is a schematic view showing the configuration of arcs in the transverse curve of the wide-face copper plate of the crystallizer in the embodiment of the invention.
Detailed Description
The technical solution in the embodiment of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiment of the present invention:
the design principle of the shape of the funnel of the crystallizer is as follows:
the funnel shape design of the funnel-shaped crystallizer must satisfy the following three principles.
(1) The requirement of placing the submerged nozzle is met. When the immersion type water gap is inserted to about 200mm, the distance between the wall of the crystallizer and the immersion type water gap is more than 20 mm.
(2) The shrinkage rate of the perimeter of the transverse profile curve of the inner cavity of the crystallizer in the drawing direction is equal to or less than the solidification shrinkage rate of the blank shell.
(3) The curvature of the transverse and longitudinal curves of the funnel area is continuous, and the curvature change rate is small, so that the continuous deformation of the blank shell is ensured, and smaller bending stress is borne.
According to the design principle of the shape of the funnel of the crystallizer, a function W (y) is constructed, which meets the following requirements:
1. assuming that the crystallizer height is w, the function is continuously decreased in the interval from 0 to w. Considering that the newly designed curve is in accordance with the solidification shrinkage rule of the casting blank in the crystallizer, the curve equation is assumed as follows:
Figure BDA0002016115880000071
wherein y represents the coordinate in the direction of drawing, W(y)The height of the projection of the center point at the y position is shown, and w is the height of the crystallizer; a, b, c and theta are constants, boundary conditions which should be met by the formula 1 are determined according to the design principle of the shape of the crystallizer funnel, and then the values of a, b, c and theta are determined, which is specifically as follows.
2. The design principle (2) of the shape of the crystallizer funnel is met, various factors are comprehensively considered, and the proper protrusion height h of the funnel at the upper opening position of the crystallizer is given. Obtain the curveThe first boundary condition of the equation, i.e. when y is 0, W(y)＝h。
3. The shrinkage change rate of the shell at the upper opening position meets the requirement of solidification shrinkage, and the change rate of the perimeter of the contour line of the primary shell determined by the shrinkage change rate in the drawing direction is equivalent to the solidification shrinkage rate at the position. Namely:
Figure BDA0002016115880000072
in equation 2, α is the linear expansion coefficient of the shell at that temperature, and dt/dy is the temperature gradient at that location. The second boundary condition of the equation is found, i.e. when y is 0,
Figure BDA0002016115880000081
in equation 3, ω is determined by the billet contour curve, the temperature gradient at the meniscus, and the billet linear expansion coefficient.
4. The curve is to be reduced to 0 at the position of the lower opening of the crystallizer, and the third boundary condition of the equation is obtained, namely when y is equal to W, W(y)＝0。
5. Taking into account the variation of the surface temperature of the cast strand in the mould, which is obtained by finite element calculations, and which has a small variation of temperature at the position of the lower mouth, approaching 0, the fourth boundary condition of the equation is obtained, i.e. when y is equal to w,
Figure BDA0002016115880000082
and connecting the four boundary condition formulas with the formula 1 to solve each coefficient in the formula.
Figure BDA0002016115880000083
Derivation of equation 1:
Figure BDA0002016115880000084
according to equation 4:
Figure BDA0002016115880000085
Figure BDA0002016115880000086
when y is 0, W(y)H. Substituting equation 1 yields:
Figure BDA0002016115880000087
substituting equation 7 into equation 8 yields:
Figure BDA0002016115880000088
Figure BDA0002016115880000089
when y is equal to W, W (y)0. Substituting equation 1 yields:
Figure BDA0002016115880000091
Figure BDA0002016115880000092
Figure BDA0002016115880000093
Figure BDA0002016115880000094
Figure BDA0002016115880000095
in addition, when y is 0, according to formula 3
Figure BDA0002016115880000096
According to equation 5
Figure BDA0002016115880000097
Figure BDA0002016115880000098
Figure BDA0002016115880000099
The following is a solution for ω:
as shown in FIG. 2, the broad-side copper plate half is described as passing through a line P due to its symmetrytsN, the transverse curve NRM, and the straight line MK. Wherein, point PtsThe mid-point of the wide face of the funnel-shaped crystallizer, the point K is the intersection point of the wide face and the narrow face of the funnel-shaped crystallizer, the transverse curve NRM of the funnel-shaped crystallizer is formed by tangency of two equal circles, for △ OO1q, the theory of triangle Pythagorean is known as follows:
OO12＝Oq2+O1q2equation 14
Namely: (2.R0)2＝(Zm-Zn)2+(2.R0-Wp)2Equation 15
Figure BDA00020161158800000910
Therefore, the transition radius R0 varies with Wp, where Wp is the difference between the longitudinal coordinate of the point M (y-axis) and the longitudinal coordinate of the point N, Zm represents the lateral coordinate of the point M in fig. 2 (x-axis), Zn represents the lateral coordinate of the point N in fig. 2, and Zm-Zn represents the distance between the point M and the point N in the lateral direction in fig. 2, i.e., the distance between the points O and q.
For an arc NR, its length LNRComprises the following steps:
Figure BDA0002016115880000101
in equation 17, Zmn also represents the distance between the M point and the N point in the lateral direction (x axis) in fig. 2, and is substantially the same as Zm — Zn, i.e., the distance of the straight line Oq.
The total length L of the primary shell is as follows:
L＝8.R0.θ+L0equation 18
In equation 18, since the arc NRM is 1/4 of the arc segment of the entire mold cavity perimeter, the primary shell is 8 times the arc length NR, L0The length of the shell except the transition arc of the curve of the funnel-shaped crystallizer; r0 is transition arc radius and is the highest point P of the funneltsA function of height W (y); theta is the central angle corresponding to the arc NR.
Figure BDA0002016115880000102
In equations 19 to 21, z also represents the distance in the lateral direction between the M point and the N point in FIG. 2, and is substantially the same as Zm-Zn and Zmn.
Figure BDA0002016115880000103
Figure BDA0002016115880000111
Order:
Figure BDA0002016115880000112
Figure BDA0002016115880000113
according to the shape design principle (2) of the funnel of the crystallizer: namely, the shrinkage rate of the perimeter of the transverse profile curve of the inner cavity of the crystallizer in the drawing direction is equal to or less than the solidification shrinkage rate of the blank shell.
Figure BDA0002016115880000114
Wherein α is the linear shrinkage of the blank shell;
Figure BDA0002016115880000115
is the temperature gradient of the crust at the meniscus.
Therefore, the derivative of w (y) at y ═ 0 can be found:
Figure BDA0002016115880000116
the design of the transverse curve of the wide-surface copper plate of the crystallizer mainly considers the requirement of arrangement of a water gap of a thin slab and is formed by externally cutting two circles with equal diameters. The height position H (x) of the bulge on the cross section of the copper plate of the crystallizer and the distance x from the center of the copper plate in the width direction satisfy the following piecewise curve equation:
x2+H(x)2＝R02(0≤x≤R0*sin(θ))
(x-xo1)2+(H(x)-yo1)2＝R02(R0*sin(θ)＜x≤2*R0*sin(θ))
H(x)＝R0-Wp (2*R0*sin(θ)＜x≤L/2)
wherein xo1 and yo1 are coordinates of a point O1 of a circle center of the tangent circle of the transverse curve of the lumen in fig. 2, θ is a central angle corresponding to the arc NR, and R0 is a radius of the tangent circle, which is determined by formula 16. And Wp is the height of the central bulge of the cross section position of the copper plate of the crystallizer and is determined by the formula 1.
According to the taper curve obtained in the summary of the invention, a thin slab funnel type mold copper plate is taken as an example below to illustrate the specific implementation mode of the invention. The main parameters of the continuous casting machine using the crystallizer copper plate are as follows: case (2): 1620mm by 90mm thin slab.
For producing thin slabs with 1620mm multiplied by 90mm, the maximum height of the middle of a funnel-shaped curve at the upper opening of the crystallizer is 48.2141mm by comprehensively considering all factors; the transition arc NM lateral distance Z is 440mm, and is substituted into equation 21 to obtain:
Figure BDA0002016115880000121
Ω＝0.58163
the temperature gradient at the meniscus position is obtained by finite element calculations:
Figure BDA0002016115880000122
in the ply method used in the finite element calculation, the thickness of each layer of unit is 0.0016m, 1472 and 1476 are temperature difference, and the unit is ℃, namely the surface temperature difference obtained by calculating the two layers of units.
According to equation 24
Figure BDA0002016115880000123
Wherein the linear shrinkage of the shell is 1.987 × 10-5。
According to equation 13
Figure BDA0002016115880000131
And equation 12
Figure BDA0002016115880000132
Wherein, h is 48.2141,
w＝1200
ω＝0.0854
obtaining by solution: theta 2.4593
a＝266.6973
b＝910087.5
c＝-2601.1
Therefore, the above-obtained value is substituted into equation 1
Figure BDA0002016115880000133
And obtaining a function curve of the height of the middle part of the funnel along the throwing direction as follows:
Figure BDA0002016115880000134
for a certain height, such as a position 100mm from the meniscus, the height W (100) of the center point of the position obtained from equation 25 is 39.5923mm, and is substituted into equation 16:
Figure BDA0002016115880000135
resolving to get R0 ═ 1835.4 (mm);
the cross section horizontal contour line equation at this position is as follows:
x2+H(x)2＝1835.42(0≤x≤157.51)
(x-315.03)2+(H(x)-3667.42)2＝R02(157.51＜x≤315.03)
H(x)＝1795.8(315.03＜x≤810)
in the same way, transverse contour lines from the meniscus to different positions of the lower opening of the crystallizer are made, a coordinate system is unified, and then a 1620mm × 90mm thin slab crystallizer copper plate wide-surface cavity type space curved surface graph is obtained in a topological mode, as shown in fig. 1. And then processing and preparing the broad-face copper plate of the hyperbolic funnel-shaped crystallizer according to the curved surface graph.

Claims (9)

1. A hyperbolic funnel-shaped wide-surface copper plate of a crystallizer is concave inwards in the middle to form a funnel-shaped curved surface, and the curved surface is formed by the topology of a transverse curve family and a longitudinal curve family; the hyperbola has a large curve gradient near the meniscus, according to the curve formula:
Figure FDA0002607256810000011
wherein y represents a coordinate in the direction of drawing; w(y)Representing the projection height of the center point of the y position; w represents the height of the crystallizer; a. b, c and theta are constants and satisfy the following expression:
Figure FDA0002607256810000012
Figure FDA0002607256810000013
Figure FDA0002607256810000014
Figure FDA0002607256810000015
Figure FDA0002607256810000016
wherein h represents the height of the funnel protrusion at the upper opening of the crystallizer, theta is the central angle corresponding to the middle arc part of the transverse curve, and omega is a parameter determined by the contour curve of the blank shell, the temperature gradient at the meniscus and the linear expansion coefficient of the blank shell.
2. The hyperbolic funnel-shaped wide-faced copper plate of a crystallizer as claimed in claim 1, characterized in that parameter ω is represented by the following expression:
ω＝[α·(dT(y)/dy)·(dW(y)/dy)]/(dL/dy)
wherein α is the linear shrinkage of the blank shell;
Figure FDA0002607256810000017
is the temperature gradient of the blank shell at the meniscus; dL/dy represents the shrinkage of the perimeter of the transverse profile curve of the crystallizer in the direction of drawing; dW(y)And/dy represents the rate of change of the height of the projections of the crystalliser in the direction of the drawing.
3. The hyperbolic funnel-shaped wide-faced copper plate of a crystallizer as claimed in any one of claims 1-2, wherein the transverse curve of the wide-faced copper plate is formed by the circumscribed of two circles of equal diameter.
4. The hyperbolic funnel-shaped wide-face copper plate of a crystallizer as claimed in claim 3, wherein the centers of two circles tangent to each other in the transverse curve are O and O1 respectively, the starting point of the transverse curve is N, the end point is M, the tangent point is R, the protrusion height H (x) of the cross section of the copper plate of the crystallizer and the distance x from the center of the copper plate in the width direction satisfy the following piecewise curve equation:
x2+H(x)2＝R02(0≤x≤R0*sin(θ))
(x-xo1)2+(H(x)-yo1)2＝R02(R0*sin(θ)＜x≤2*R0*sin(θ))
H(x)＝R0-WP(2*R0*sin(θ)＜x≤L/2)
wherein R0 is the radius of a tangent circle, xo1 and yo1 are the coordinates of a point O1 of the center of the tangent circle of a transverse curve, theta is the central angle corresponding to the middle arc line part of the transverse curve, the middle arc line part of the transverse curve is an arc line RN formed by the tangent point R and a starting point N, and Wp is the central projection height of the cross section position of the copper plate of the crystallizer; l represents the perimeter of the transverse profile curve of the crystallizer;
the radius R0 of the tangent circle is determined by the following equation:
Figure FDA0002607256810000021
wherein Zm-Zn represents the transverse curve pitch.
5. A preparation method of a hyperbolic funnel-shaped crystallizer broad-face copper plate is characterized by comprising the following steps: determining a longitudinal curve in the height direction of the crystallizer according to solidification shrinkage and deformation of the blank shell, wherein the longitudinal curve is distributed in a hyperbolic curve mode, and the method specifically comprises the following steps:
assuming that the hyperbola conforms to the curve formula:
Figure FDA0002607256810000022
wherein y represents a coordinate in the direction of drawing; w(y)Indicating y position center point convex heightDegree; w represents the height of the crystallizer; a. b, c and theta are constants;
according to the design principle of the funnel-shaped crystallizer, the following expressions of constants a, b, c and theta are obtained:
Figure FDA0002607256810000031
Figure FDA0002607256810000032
Figure FDA0002607256810000033
Figure FDA0002607256810000034
Figure FDA0002607256810000035
wherein h represents the height of the funnel protrusion at the upper opening of the crystallizer, theta is a central angle corresponding to a middle arc part of a transverse curve, and omega is a parameter determined by a blank shell profile curve, a temperature gradient at a meniscus and a linear expansion coefficient of the blank shell;
calculating omega to obtain the values of constants a, b, c and theta;
the values of the constants a, b, c and theta are substituted into the formula 1 to obtain a hyperbolic longitudinal taper curve.
6. The method for preparing the hyperbolic funnel-shaped wide-face copper plate of the crystallizer as claimed in claim 5, wherein ω is calculated according to the linear shrinkage rate of the blank shell, the temperature gradient of the blank shell at the meniscus, the maximum height of the middle of the upper mouth funnel-shaped curve of the crystallizer, and the transverse distance of the transition arc.
7. The method for preparing the hyperbolic funnel-shaped wide-face copper plate of the crystallizer as recited in claim 6, further comprising: the method comprises the following steps of determining a transverse curve according to the arrangement requirement of a thin slab nozzle, wherein the transverse curve is formed by externally cutting two circles with equal diameters, and specifically comprises the following steps:
according to the obtained hyperbolic formula, calculating the projection height W of the central point corresponding to the position of a certain height y(y)So as to obtain the radius R0 of the tangent circle;
assuming that the centers of two circles are O and O1 respectively, the starting point of a transverse curve is N, the end point is M, the tangent point is R, the height H (x) of the bulge on the cross section of the copper plate of the crystallizer and the distance x from the center of the copper plate in the width direction satisfy the following piecewise curve equation:
x2+H(x)2＝R02(0≤x≤R0*sin(θ))
(x-xo1)2+(H(x)-yo1)2＝R02(R0*sin(θ)＜x≤2*R0*sin(θ))
H(x)＝R0-WP(2*R0*sin(θ)＜x≤L/2)
wherein R0 is the radius of a tangent circle, xo1 and yo1 are the coordinates of a point O1 of the center of the tangent circle of a transverse curve, theta is the central angle corresponding to the middle arc line part of the transverse curve, the middle arc line part of the transverse curve is an arc line RN formed by the tangent point R and a starting point N, and Wp is the central projection height of the cross section position of the copper plate of the crystallizer; l represents the perimeter of the transverse profile curve of the crystallizer;
and obtaining the bulge height H (x) of the cross section of the copper plate of the crystallizer according to the piecewise curve equation.
8. The method for preparing the hyperbolic funnel-shaped wide-face copper plate of a crystallizer as claimed in claim 7, wherein the method further comprises:
calculating transverse profile curves from the meniscus to different positions of the lower opening of the crystallizer;
and topologically obtaining a curved surface graph of the wide-surface copper plate of the crystallizer according to the obtained longitudinal curve family and the transverse curve family of the wide-surface copper plate of the crystallizer.
9. The method for preparing the hyperbolic funnel-shaped wide-face copper plate of the crystallizer as recited in claim 8, further comprising: and processing and preparing a funnel-shaped crystallizer wide-surface copper plate according to the curved surface graph.
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