A digital design method for sleeve valve based on flow characteristic surface

Abstract

The present invention discloses a digital design method for a sleeve valve based on a flow characteristic surface, specifically, by designing the area and shape of a throttling window to change the flow capacity and flow characteristics of the sleeve valve, a combined valve core principle model with a flow characteristic surface function can be constructed while ensuring the flow regulation accuracy, thereby obtaining a valve flow characteristic surface map represented by a sleeve valve, and according to the characteristic surface map and the target flow characteristic curve of the designed valve, a specific point set is selected on the flow characteristic surface, a fitting flow characteristic curve is drawn, and design information is extracted to quickly design the throttling window shape and profile. The method is based on digital design, avoids repeated theoretical formulas and experimental verification processes, and improves the problem of low numerical simulation accuracy of sleeve valves with irregular throttling windows, saving time and saving manpower and material resources.

Description

Technical Field

[0001] The invention relates to a digital design of a sleeve valve, in particular to a digital design method of a sleeve valve based on a flow characteristic surface. Background Art

[0002] As a key component in the industrial pipeline system, valves regulate and control parameters such as flow, pressure, and temperature. They are widely used in aerospace, military, nuclear energy, electricity and other fields, and are called the "throat" of the industrial system. As the key motion unit of the valve, the valve core regulates and controls the flow, pressure and other parameters of the medium, and directly determines the key performance indicators such as the valve's adjustment accuracy and adjustable ratio. With the continuous emergence of extreme working conditions such as large flow and high pressure difference in process industries and hydraulic systems, my country's valves are currently often faced with technical difficulties such as low adjustment accuracy and small adjustable ratio. This makes the key valves in many major projects in my country heavily dependent on imports. Once a "neck-stuck" incident occurs in this field, many major engineering projects in my country will face the risk of paralysis, posing a major threat to national security, social stability and economic development.

[0003] Sleeve valves have good stability and can affect the flow state in the valve by changing the valve core structure and the shape of the throttling window. They are a widely used type of regulating valve. According to the design requirements of specific flow characteristic curves, the design process of traditional sleeve valve cores is complicated, and the designed sleeve valve has low regulation accuracy. Therefore, a high-precision and fast sleeve valve core design method is urgently needed. Summary of the invention

[0004] The purpose of the present invention is to provide a digital design method for a sleeve valve based on a flow characteristic surface. Based on digital design, a characteristic curve is quickly selected from the flow characteristic surface according to the flow demand to obtain the shape of the throttling window, which facilitates the accurate and rapid design of the sleeve valve. The method is based on digital design, avoids repeated theoretical formulas and experimental verification processes, saves design time, and saves manpower and material resources.

[0005] The technical solution adopted by the present invention is:

[0006] The present invention provides a digital design method for a sleeve valve based on a flow characteristic surface, and the steps are as follows:

[0007] S1: Based on the original sleeve valve design file, the valve solid model is constructed using 3D modeling software;

[0008] S2: extracting a valve internal flow channel model according to the valve entity model; and then discretizing the valve internal flow channel model into an internal flow channel mesh model by using meshing software;

[0009] S3: Using computational fluid dynamics software, according to the original sleeve valve design file, in the internal flow channel grid model, a pressure inlet boundary condition is set for the fluid inlet surface, a pressure outlet boundary condition is set for the fluid outlet surface, and a no-slip wall boundary condition is set for the valve inner wall surface, and fluid properties are assigned to the entire space occupied by the internal flow channel grid model to obtain an internal fluid grid model;

[0010] S4: using the internal fluid grid model and computational fluid dynamics software to obtain the flow conditions in the valve under different throttling windows, and using the flow rate as a basis to obtain the flow value in the valve, thereby calculating the corresponding flow coefficient;

[0011] S5: According to the flow coefficients corresponding to different throttling windows calculated in step S4, flow characteristic curves under different throttling window areas and aspect ratios are obtained through numerical simulation, and then all the flow characteristic curves are fitted into a flow characteristic surface of the sleeve valve; the x-axis of the flow characteristic surface is the valve core opening, the y-axis is the rotation angle of the outer sleeve, and the z-axis is the magnitude of the flow coefficient;

[0012] S6: According to the target flow characteristic curve, a fitting flow characteristic curve is selected on the flow characteristic surface, the flow coefficient, valve core opening and outer sleeve rotation angle corresponding to several points on the fitting flow characteristic curve are extracted, and the shape and area of ​​the target sleeve throttling window are calculated according to the equal area principle;

[0013] S7: According to the shape and area of ​​the target sleeve throttling window and the original sleeve valve design file, a sleeve valve that meets the target flow characteristic curve is designed.

[0014] Preferably, in step S1, the original sleeve valve design file includes composition information and operating condition information of the sleeve valve, the composition information includes the structure, size and material properties of the sleeve valve, and the operating condition information includes the physical properties of the incoming fluid, the valve inlet and outlet, and the boundary conditions at the inlet.

[0015] Preferably, in step S1, the method for establishing the coordinate system in the process of constructing the valve entity model is as follows:

[0016] The origin of the global rectangular coordinate system is located at the center of the bottom surface of the sleeve, the XOZ plane coincides with the longitudinal section of the valve; the Z axis coincides with the axis of the sleeve, and its positive direction points to the top of the sleeve; the Y axis points to the valve outlet; the valve longitudinal section is the plane determined by the line connecting the center points of the valve inlet surface and the outlet surface and the axis of the sleeve.

[0017] Preferably, the 3D modeling software is one of Solidworks, Creo, Inventor, UG / NX, CATIA, ANSYS Workbench Design Modeler and ANSYS Workbench Space Claim.

[0018] Preferably, the meshing software is one of ICEM CFD, HyperMesh, TGrid, PointWise, ANSA, GridPro, ANSYS Workbench Mesh, and Fluent Meshing.

[0019] Preferably, the computational fluid dynamics software is one of ANSYS Fluent, ANSYS CFX, STAR-CD, STAR-CCM, NUMECA, and OpenFOAM.

[0020] Preferably, the throttling window of the sleeve valve is rectangular in shape, with a width of b and a height of h. There are 4 throttling windows on the outer sleeve, and the maximum area of ​​the throttling window is 4bh; the number and shape of the throttling windows on the inner sleeve are exactly the same as the throttling windows of the outer sleeve; all throttling windows are symmetrically arranged about the longitudinal section of the valve; the inner sleeve of the sleeve valve is fixed, and the outer sleeve can rotate freely around the sleeve axis. The length of the rectangular throttling window can be controlled by the rotation angle, thereby changing the area of ​​the throttling window, and then changing the flow characteristics of the valve.

[0021] Furthermore, the flow coefficients corresponding to different throttling windows are obtained by the following method:

[0022] The valve plug height and the aspect ratio of the rectangular throttling window are taken as variables, the valve flow rate is taken as the dependent variable, n sleeve rotation angles are selected as variables, and the radius of the outer sleeve is R; the complete overlap of the throttling windows of the inner and outer sleeves is taken as the starting state, and the complete offset of the throttling windows of the inner and outer sleeves is taken as the end state; from the starting state to the end state, the outer sleeve valve rotates every (b / nR) angle, and the sleeve valve is calculated once according to steps S2 to S4, and a total of n groups of flow coefficients corresponding to different throttling windows are obtained.

[0023] Preferably, the step S5 is specifically as follows:

[0024] According to the mechanism of the effect of the characteristic parameters of the throttling window on the flow characteristics, the flow characteristic curve is imported into the professional mathematical optimization software, and all the flow characteristic curves are nonlinearly fitted by the interpolation method and the multi-data point integrated processing method to obtain a smooth surface, thereby realizing the acquisition of the flow characteristic surface.

[0025] Furthermore, the professional mathematical optimization software is one of MATLAB, Origin, and 1STOPT.

[0026] Preferably, step S6 is specifically as follows:

[0027] According to the characteristics of the target flow characteristic curve, select k points under the opening of the fitting flow characteristic curve, extract the valve core height, sleeve rotation angle and flow coefficient of the k points on the fitting flow characteristic curve; calculate the flow area corresponding to the k points a1 = 4bhx1y1, a2 = 4bhx2y2...a k =4bhx k y k , using the valve core opening and the throttling window area as the horizontal and vertical axes respectively, connect the k points on the coordinate axis through a smooth curve to make a two-dimensional rectangular area curve of the flow area-valve core opening;

[0028] According to the points under k openings on the fitting flow characteristic curve, a flow area rectangle under k openings is made; the opening interval is [x l-1 ,x l ] is approximately a rectangular segment with an opening range of [x l-1 ,x l The area of ​​] is a l -a l-1 , x0=0, a0=0, the value range of l is specified as (1, k); connect the midpoints of the top of the area rectangle to make a smooth curve, which is the shape of the sleeve valve.

[0029] The beneficial effects of the present invention are:

[0030] 1) The present invention can quickly design a sleeve valve structure model by creating a flow characteristic surface, extracting an arbitrary flow curve, and calculating the throttling window area and shape of the sleeve.

[0031] 2) The sleeve valve design of the present invention is based on digital design technology, which avoids repeated theoretical formula calculations and experimental verification processes in the design process, greatly saves manpower and material resources, and improves economic benefits. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is a cross-sectional view of a double-layer sleeve combined valve in an embodiment;

[0033] Figure 2 yes Figure 1 A schematic diagram of the structure of the middle sleeve;

[0034] Figure 3 yes Figure 2 A cross-sectional view of

[0035] Figure 4is the target flow characteristic curve in the embodiment;

[0036] Figure 5 is the flow characteristic surface and the selected fitting flow characteristic curve in the embodiment;

[0037] Figure 6 is a rectangular area curve diagram of two-dimensional flow area-valve core opening in the embodiment;

[0038] Figure 7 is the shape and area of ​​the target sleeve throttling window in the embodiment;

[0039] In the figure: 1 is the valve stem, 2 is the valve cover, 3 is the valve core, 4 is the inner sleeve, 5 is the outer sleeve, and 6 is the valve body. DETAILED DESCRIPTION

[0040] The present invention is further described and illustrated below in conjunction with the accompanying drawings and specific embodiments. The technical features of each embodiment of the present invention can be combined accordingly without conflicting with each other.

[0041] Example

[0042] This embodiment provides a digital design method for a sleeve valve based on a flow characteristic surface. The design method has the following steps:

[0043] S1: Based on the original sleeve valve design file, the valve solid model is constructed using 3D modeling software.

[0044] In actual application, the original sleeve valve design file needs to include the composition information and use condition information of the sleeve valve. The composition information includes the structure, size and material properties of the sleeve valve, and the use condition information includes the physical properties of the fluid entering, the valve inlet and outlet, and the boundary conditions at the inlet. The 3D modeling software can be one of Solidworks, Creo, Inventor, UG / NX, CATIA, ANSYS Workbench DesignModeler or ANSYS Workbench SpaceClaim.

[0045] In this embodiment, the 3D modeling software is Solidworks, and the original 3D structure model of the double-layer sleeve valve is as follows: Figure 1 As shown. Figure 1 It can be seen that the original model of the three-dimensional structure includes the valve stem 1, the valve cover 2, the valve core 3, the inner sleeve 4, the outer sleeve 5 and the valve body 6. The origin of the global rectangular coordinate system is located at the center of the sleeve bottom surface, and the XOZ plane coincides with the valve longitudinal section. The Z axis coincides with the sleeve axis, and its positive direction points to the top of the sleeve. The Y axis points to the valve outlet. The valve longitudinal section is the plane determined by the line connecting the center points of the valve inlet surface and the outlet surface and the sleeve axis, as shown in the attached figure. Figure 2 and 3 shown.

[0046] S2: Extract the valve internal flow channel model based on the valve solid model obtained in step S1, and then discretize the valve internal flow channel model into an internal flow channel mesh model using meshing software.

[0047] In practical applications, the meshing software can be one of ICEM CFD, HyperMesh, TGrid, PointWise, ANSA, GridPro, ANSYS Workbench Mesh, and Fluent Meshing.

[0048] In this embodiment, the meshing software adopts Fluent Meshing.

[0049] S3: Import the internal flow channel grid model obtained in step S2 into computational fluid dynamics software. Using the computational fluid dynamics software, according to the original sleeve valve design file, in the internal flow channel grid model, set pressure inlet boundary conditions for the fluid inlet surface, set pressure outlet boundary conditions for the fluid outlet surface, set no-slip wall boundary conditions for the valve inner wall surface, and assign fluid properties to the entire space occupied by the internal flow channel grid model, so as to obtain an internal fluid grid model with the same fluid properties as the original sleeve valve design file in step S1.

[0050] In practical applications, the computational fluid dynamics software can be one of ANSYS Fluent, ANSYS CFX, STAR-CD, STAR-CCM, NUMECA, and OpenFOAM.

[0051] In this embodiment, the computational fluid dynamics software is ANSYS Fluent; the boundary conditions are set to pressure inlet and pressure outlet, the inlet pressure is 10 MPa, and the outlet pressure is 0 MPa; the standard k-ε turbulence model is adopted, and the fluid medium is water at room temperature.

[0052] S4: Using the internal fluid grid model and computational fluid dynamics software, the flow conditions in the valve under different throttling windows are obtained. Based on the flow rate, the flow value in the valve is obtained, and the corresponding flow coefficient is calculated.

[0053] S5: According to the flow coefficients corresponding to different throttling windows calculated in step S4, the flow characteristic curves under different throttling window areas and aspect ratios are obtained through numerical simulation, and then all the flow characteristic curves are fitted into the flow characteristic surface of the sleeve valve. The flow characteristic surface includes x, y, and z axes, wherein the x axis is the valve core opening (L / Lmax), the y axis is the angle of rotation of the outer sleeve (T / Tmax), and the z axis is the size of the flow coefficient.

[0054] The sleeve of this embodiment is a hollow column, the throttling window is a rectangle, the width is b, the height is h, there are 4 throttling windows on the outer sleeve, and the maximum area of ​​the throttling window is 4bh. Select n sleeve rotation angles as variables, and each time the outer sleeve valve rotates (b / nR), the calculation of the sleeve valve is completed according to S2~S4. The details are as follows:

[0055] First, when the throttling windows of the inner and outer sleeves overlap, by establishing a sliding grid, adjusting the outer sleeve to change the length of the rectangular throttling window, the numerical simulation obtains the first flow characteristic curve that changes with the valve sleeve height when the rectangular length is maximum; the outer sleeve rotates every time (b / nR), repeating the above operation to obtain the second flow characteristic curve, the third flow characteristic curve, the fourth flow characteristic curve... the nth flow characteristic curve, until the valve is completely closed and in a blocked state; a total of n curves about the relationship between the sleeve rotation angle and the valve core height and the flow are obtained. Figure 5 As shown in FIG. 1 , according to the mechanism of action of the characteristic parameters of the throttling window on the flow characteristics, the n curves obtained by the above numerical simulation are imported into the mathematical software through professional mathematical optimization software. The interpolation method and the method of multi-data point integration processing are used to quickly fit the curves nonlinearly to obtain a smooth surface, which is the flow characteristic surface of the combined valve core. The acquisition of the flow characteristic surface is achieved as shown in FIG. Figure 5 shown.

[0056] In practical applications, the mathematical optimization software can be one of MATLAB, Origin, and 1STOPT.

[0057] In this embodiment, the mathematical optimization software is MATLAB, the rectangular window width is 30 mm, the height is 45 mm, the sleeve radius is 90 mm, and the throttling window area is 5400 mm 3 , n is 20. Each time the sleeve rotates 1 / 60, 20 flow characteristic curves are obtained, which are fitted into a flow characteristic surface through MATLAB.

[0058] S6: According to the target flow characteristic curve, such as Figure 4 As shown, a fitting flow characteristic curve is selected on the flow characteristic surface, and several points of the fitting flow characteristic curve are extracted. Each point contains design information such as flow coefficient, valve core opening and outer sleeve rotation angle. According to the equal area principle, the shape and area of ​​the target sleeve throttling window are calculated.

[0059] In practical applications, according to the target flow characteristic curve characteristics of the designed valve, several points n1 (n1>10, the valve core opening is from 0% to 100%) on the target flow characteristic curve are selected, and corresponding points are selected on the flow characteristic surface according to the corresponding principle based on the several points, and the flow characteristic surface is fitted to obtain the fitted flow characteristic curve.

[0060] In this example, the corresponding points under the opening of 5%, 10%, 15%, 20%, 25%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100% are selected to draw the fitting flow characteristic curve, such as Figure 6 shown.

[0061] S7: According to the shape and area of ​​the target sleeve throttling window and the original sleeve valve design file, a sleeve valve that meets the target flow characteristic curve is designed.

[0062] In practical applications, k points on the fitting flow characteristic curve are selected and the coordinates (x k ,y k ,z k ) Extract the valve core height, sleeve rotation angle and flow coefficient information of each point; calculate the flow area at the k point as a1 = 4bhx1y1, a2 = 4bhx2y2...a k =4bhx k y k . Set the opening interval to [x l-1 ,x l ] is approximately a rectangular segment with an opening range of [x l-1 ,x l The area of ​​] is a l -a l-1 , x0=0, a0=0, l’s value range is (1, k); connect the top midpoints of the rectangles of each area interval to make a smooth curve, which is the shape of the valve sleeve. Then, according to the composition information and structure of the sleeve valve in the original sleeve valve design file, the required sleeve valve can be designed. Finally, the rectangular area under each opening interval, as well as the final throttling window shape and profile are as follows Figure 7 shown.

[0063] In this example, the corresponding points under the opening of 5%, 10%, 15%, 20%, 25%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100% are selected, and the flow area under each opening is a1 = 840mm 3 , a2=1270mm 3 ,a3=2350mm 3 …a 13 =5400mm 3 .

[0064] The present invention changes the flow capacity and flow characteristics of the sleeve valve by designing the throttling window area and shape. Under the condition of ensuring the flow regulation accuracy, a combined valve core principle model with a flow characteristic surface function can be constructed, thereby obtaining a valve flow characteristic surface map represented by a double-layer sleeve type combined valve core. According to the characteristic surface map and the target flow characteristic curve of the designed valve, a specific point set is selected on the flow characteristic surface, a fitting flow characteristic curve is drawn, and the design information is extracted to quickly design the throttling window shape and profile. The method is based on digital design, avoids repeated theoretical formulas and experimental verification processes, and improves the problem of low numerical simulation accuracy of sleeve valves with irregular throttling windows, saving time and manpower and material resources.

[0065] It should be clear that the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

Claims

1. A digital design method for a sleeve valve based on a flow characteristic surface, characterized in that: Here are the steps: S1: Based on the original sleeve valve design file, the valve solid model is constructed using 3D modeling software; S2: extracting a valve internal flow channel model according to the valve entity model; and then discretizing the valve internal flow channel model into an internal flow channel mesh model by using meshing software; S3: Using computational fluid dynamics software, according to the original sleeve valve design file, in the internal flow channel grid model, a pressure inlet boundary condition is set for the fluid inlet surface, a pressure outlet boundary condition is set for the fluid outlet surface, and a no-slip wall boundary condition is set for the valve inner wall surface, and fluid properties are assigned to the entire space occupied by the internal flow channel grid model to obtain an internal fluid grid model; S4: using the internal fluid grid model and computational fluid dynamics software to obtain the flow conditions in the valve under different throttling windows, and using the flow rate as a basis to obtain the flow value in the valve, thereby calculating the corresponding flow coefficient; S5: According to the flow coefficients corresponding to different throttling windows calculated in step S4, flow characteristic curves under different throttling window areas and aspect ratios are obtained through numerical simulation, and then all the flow characteristic curves are fitted into a flow characteristic surface of the sleeve valve; the x-axis of the flow characteristic surface is the valve core opening, the y-axis is the rotation angle of the outer sleeve, and the z-axis is the magnitude of the flow coefficient; S6: According to the target flow characteristic curve, a fitting flow characteristic curve is selected on the flow characteristic surface, the flow coefficient, valve core opening and outer sleeve rotation angle corresponding to several points on the fitting flow characteristic curve are extracted, and the shape and area of ​​the target sleeve throttling window are calculated according to the equal area principle; S7: According to the shape and area of ​​the target sleeve throttling window and the original sleeve valve design file, a sleeve valve that meets the target flow characteristic curve is designed.

2. The digital design method for sleeve valve according to claim 1, characterized in that: In step S1, the original sleeve valve design file includes the composition information and operating condition information of the sleeve valve, the composition information includes the structure, size and material properties of the sleeve valve, and the operating condition information includes the physical properties of the incoming fluid, the valve inlet and outlet, and the boundary conditions at the inlet.

3. The digital design method for sleeve valve according to claim 1, characterized in that: In step S1, the method for establishing the coordinate system in the process of constructing the valve entity model is as follows: The origin of the global rectangular coordinate system is located at the center of the bottom surface of the sleeve, the XOZ plane coincides with the longitudinal section of the valve; the Z axis coincides with the axis of the sleeve, and its positive direction points to the top of the sleeve; the Y axis points to the valve outlet; the valve longitudinal section is the plane determined by the line connecting the center points of the valve inlet surface and the outlet surface and the axis of the sleeve.

4. The digital design method for sleeve valve according to claim 1, characterized in that: The three-dimensional modeling software is one of Solidworks, Creo, Inventor, UG / NX, CATIA, ANSYS Workbench DesignModeler and ANSYS Workbench SpaceClaim.

5. The digital design method for sleeve valve according to claim 1, characterized in that: The meshing software is one of ICEM CFD, HyperMesh, TGrid, PointWise, ANSA, GridPro, ANSYS Workbench Mesh, and FluentMeshing.

6. The digital design method for sleeve valve according to claim 1, characterized in that: The computational fluid dynamics software is one of ANSYS Fluent, ANSYS CFX, STAR-CD, STAR-CCM, NUMECA, and OpenFOAM.

7. The digital design method for sleeve valve according to claim 1, characterized in that: The throttling window of the sleeve valve is rectangular in shape, with a width of b and a height of h. There are 4 throttling windows on the outer sleeve, and the maximum area of ​​the throttling window is 4bh; the number and shape of the throttling windows on the inner sleeve are exactly the same as the throttling windows of the outer sleeve; all throttling windows are symmetrically arranged about the longitudinal section of the valve; the inner sleeve of the sleeve valve is fixed, and the outer sleeve can rotate freely around the sleeve axis. The length of the rectangular throttling window can be controlled by the rotation angle, thereby changing the area of ​​the throttling window, and then changing the flow characteristics of the valve.

8. The digital design method for sleeve valve according to claim 7, characterized in that: The corresponding flow coefficient under different throttling windows is obtained by the following method: The valve plug height and the aspect ratio of the rectangular throttling window are taken as variables, the valve flow rate is taken as the dependent variable, n sleeve rotation angles are selected as variables, and the radius of the outer sleeve is R; the complete overlap of the throttling windows of the inner and outer sleeves is taken as the starting state, and the complete offset of the throttling windows of the inner and outer sleeves is taken as the end state; from the starting state to the end state, the outer sleeve valve rotates every (b / nR) angle, and the sleeve valve is calculated once according to steps S2 to S4, and a total of n groups of flow coefficients corresponding to different throttling windows are obtained.

9. The digital design method for sleeve valve according to claim 1, characterized in that: The step S5 is specifically as follows: According to the mechanism of the effect of the characteristic parameters of the throttling window on the flow characteristics, the flow characteristic curve is imported into the professional mathematical optimization software, and all the flow characteristic curves are nonlinearly fitted by the interpolation method and the multi-data point integrated processing method to obtain a smooth surface, thereby realizing the acquisition of the flow characteristic surface.

10. The digital design method for sleeve valve according to claim 9, characterized in that: The professional mathematical optimization software is one of MATLAB, Origin, and 1STOPT.

11. The digital design method for sleeve valve according to claim 1, characterized in that: The step S6 is specifically as follows: According to the characteristics of the target flow characteristic curve, select k points under the opening of the fitting flow characteristic curve, extract the valve core height, sleeve rotation angle and flow coefficient of the k points on the fitting flow characteristic curve; calculate the flow area corresponding to the k points a1 = 4bhx1y1, a2 = 4bhx2y2...a k =4bhx k y k , using the valve core opening and the throttling window area as the horizontal and vertical axes respectively, connect the k points on the coordinate axis through a smooth curve to make a two-dimensional rectangular area curve of the flow area-valve core opening; According to the points under k openings on the fitting flow characteristic curve, a flow area rectangle under k openings is made; the opening interval is [x l-1 ,x l ] is approximately a rectangular segment with an opening range of [x l-1 ,x l The area of ​​] is a l -a l-1 , x0=0, a0=0, the value range of l is specified as (1, k); connect the midpoints of the top of the area rectangle to make a smooth curve, which is the shape of the sleeve valve.

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