Method for fitting working face of water bucket by using catenary equation

Abstract

The invention relates to the field of water turbines, and discloses a method for fitting a working face of a water bucket by utilizing a catenary equation, which comprises the following steps of: establishing the catenary equation of a corresponding fitting curve and solving equation parameters according to the characteristics of a section line of the water bucket of an impulse water turbine and geometric parameters of section line control points; the method can be used for curved surface modeling, modeling, hydraulic optimization design and CFD (computational fluid dynamics) flow analysis of the water bucket of the impulse turbine.

Description

Technical Field

[0001] The invention relates to the field of water turbines, and in particular to a method for fitting a bucket working surface by using a catenary equation. Background Art

[0002] The application of impulse turbines in high-head power stations is relatively common, and the capacity of single units is increasing. The runner of the impulse turbine is the core component that converts the kinetic energy of water into mechanical energy. The energy conversion of the impulse runner is mainly completed on the working surface of the bucket. Therefore, accurate and efficient hydraulic design of the bucket working surface is of great significance for improving the overall efficiency of the turbine and fully completing the energy conversion. The present invention establishes a fitting method for the bucket working surface of the impulse turbine, which meets the needs of bucket modeling, shaping, hydraulic optimization design and CFD flow analysis. Summary of the invention

[0003] The present invention uses the catenary equation to fit the working surface of the impulse turbine bucket, meets the needs of accurate and rapid bucket modeling, shaping, hydraulic optimization design and CFD flow analysis, and improves the hydraulic design efficiency of the impulse turbine runner. The technical solution of the present invention is:

[0004] Step 1: The bucket on the impulse turbine runner includes a working surface and a back surface, wherein the working surface is the main working part of the impulse turbine runner, converting the jet energy into the mechanical energy of the impulse turbine runner, and the working surface is a curved surface with the water inlet edge, the water outlet edge, the water cutting edge and the root transition line as the boundary lines. The present invention uses the catenary equation to establish a series of cross-section lines from the water inlet edge to the water outlet edge for surface fitting of the working surface, and these cross-section lines are divided into two left and right catenary lines, and the connection point of the two catenary lines is the extreme point of the cross-section line, and the key geometric parameter of the cross-section line is the water inlet edge height h i , water edge depth h o , bucket width w, extreme point width w m and the depth d of the extreme point m ;

[0005] Step 2: Establish the reference point and reference plane of the bucket, and use the reference point of the bucket as the origin of the coordinate system, the straight line passing through the reference point of the bucket and parallel to the rotation center line of the runner as the Z axis, the straight line passing through the reference point of the bucket on the reference plane of the bucket and perpendicular to the Z axis as the X axis, and the vector product of the Z axis and the X axis as the Y axis, to establish the body-fitting coordinate system of the bucket;

[0006] Step 3: Each section line of the bucket working surface is a concave curve with equal X value in the bucket body-fitting coordinate system. Each section line of the bucket working surface is divided into left and right parts with the extreme point of the section line as the breaking point;

[0007] Step 4: For the left part of each section line of the bucket working surface, according to the water inlet edge height h of the section line i , the width w of the extreme point m and depth d m The catenary equation with unknowns a1 and y1 can be established And satisfy f(0)=h i ,f(w m )=-d m ;

[0008] in the formula

[0009] a1 and y1 are the unknown coefficients of the catenary equation;

[0010] w m is the width w of the extreme point m ;

[0011] Step 5: According to f(0) = h i and f(w m )=-d m Establish a system of transcendental equations with a1 and y1 as unknowns

[0012]

[0013] In the above equation

[0014] a1, y1, and w m As shown in step 4;

[0015] h i is the height of the water inlet edge of the section line h i ;

[0016] d m is the depth d of the extreme point m ;

[0017] And a1 and y1 can be solved by Newton's downhill method, so as to establish the catenary equation of the left part of each section line;

[0018] Step 6: For the right part of each section line of the bucket working surface, according to the water outlet depth h of the section line o , the width w of the extreme point m and depth d m The catenary fitting equation containing unknowns a2 and y2 can be established And satisfy g(w)=-h o , g(w m )=-d m ;

[0019] in the formula

[0020] a2 and y2 are the unknown coefficients of the catenary equation;

[0021] w m As shown in step 4;

[0022] Step 7: According to g(w)=-h o and g(w m )=-d m Construct a system of transcendental equations with a2 and y2 as unknowns

[0023]

[0024] In the above equation

[0025] w m As shown in step 4;

[0026] d m As shown in step five;

[0027] a2, y2 are as shown in step 6;

[0028] h o is the height of the section line out of the water h o ;

[0029] And a2 and y2 can be solved by Newton's downhill method, thus establishing the catenary equation of the right part of each section line;

[0030] Step 8: The catenary equations on the left and right sides of each section line of the bucket working surface obtained in steps 5 and 7 are combined into a unified curve to complete the catenary fitting of each section line of the bucket working surface;

[0031] Step nine: Use the water cutting blade, each section line established in step eight, and the root transition line as the u-curve, and the water inlet and outlet edges as the v-curve, and complete the fitting of the bucket working surface through two-way lofting surface. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 Schematic diagram of the runner of an impulse turbine.

[0033] Figure 2 Schematic diagram of the water bucket of an impulse turbine.

[0034] Figure 3 This is a schematic diagram of the bucket body-fitting coordinates.

[0035] Figure 4 This is a schematic diagram of the cross-section of the bucket of an impulse turbine.

[0036] In the attached figure:

[0037] 1- Impulse turbine runner; 2- Bucket; 3- Working surface; 4- Back surface; 5- Jet; 6- Water inlet edge; 7- Water outlet edge; 8- Water cutting edge; 9- Root transition line; 10- Section line; 11- Extreme point; 12- Water inlet edge height h i ; 13- Depth of water outlet h o ; 14- bucket width w; 15- extreme point width w m ; 16- Depth d of extreme point m ; 17-reference point; 18-reference plane; 19-origin of coordinate system; 20-rotation centerline; 21-a straight line parallel to the reference point of the water bucket and the rotation centerline of the rotating wheel; 22-a straight line perpendicular to the Z axis and on the reference plane of the water bucket; 23-body-fitting coordinate system.

[0038] The beneficial effects of the present invention are:

[0039] The application of impulse turbines in high-head power stations is relatively common, and the capacity of single units is increasing. The runner of the impulse turbine is the core component that converts the kinetic energy of water into mechanical energy. The energy conversion of the impulse runner is mainly completed on the working surface of the bucket. Therefore, accurate and efficient hydraulic design of the bucket working surface is of great significance for improving the overall efficiency of the turbine and fully completing the energy conversion. The present invention establishes a fitting method for the bucket working surface of the impulse turbine, which meets the needs of bucket modeling, shaping, hydraulic optimization design and CFD flow analysis. DETAILED DESCRIPTION

[0040] The present application is further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present application.

[0041] Step 1: If Figure 1 , 2 As shown, the bucket 2 on the impulse turbine runner 1 includes a working surface 3 and a back surface 4. The impulse turbine runner 1 converts the energy of the jet 5 into the mechanical energy of the impulse turbine runner 1 mainly inside the working surface 3. Therefore, the working surface 3 is the main working part of the bucket 2 on the impulse turbine runner 1. Figure 2 As shown, the working surface 3 of the bucket 2 is a curved surface with the water inlet edge 6, the water outlet edge 7, the water cutting edge 8 and the root transition line 9 as the boundary lines, as shown in FIG. Figure 3 As shown, the present invention uses the catenary equation to establish a series of cross-section lines 10 from the water inlet edge 6 to the water outlet edge 7 to perform surface fitting of the working surface 3. These cross-section lines 10 are divided into two left and right catenary segments. The connection point of the two catenary segments is the extreme point 11 of the cross-section line 10, and the first-order continuous differentiable is at the connection point; the key geometric parameters of the cross-section line 10 are the water inlet edge height h i 12. Depth of water outlet h o13. Bucket width w14. Width w of extreme point 11 m 15 and the depth d of the extreme point 11 m 16;

[0042] Step 2: If Figure 3 As shown, the reference point 17 and the reference plane 18 of the bucket 2 are established, and the reference point 17 of the bucket 2 is used as the origin 19 of the coordinate system, the straight line 21 passing through the reference point 17 of the bucket 2 and parallel to the rotation center line 20 of the runner 1 is used as the Z axis, the straight line 22 passing through the reference point 17 of the bucket 2 and on the reference plane 18 of the bucket 2 and perpendicular to the Z axis is used as the X axis, and the vector product of the Z axis and the X axis is used as the Y axis, and the body-fitting coordinate system 23 of the bucket 2 is established;

[0043] Step 3: If Figure 2 , 4 As shown, each section line 10 of the working surface 3 of the bucket 2 is a concave curve with equal X value in the body-fitting coordinate system 21 of the bucket 2, and each section line 10 of the working surface 3 of the bucket 2 is divided into left and right parts with the extreme point 15 of the section line 10 as the breaking point;

[0044] Step 4: If Figure 4 As shown, for the left side of each section line 10 of the working surface 3 of the bucket 2, according to the water inlet edge height h of the section line 10 i 12. The width w of the extreme point 11 m 15 and depth d m 16 The catenary equation with unknowns a1 and y1 can be established And satisfy f(0)=h i ,f(w m )=-d m ;

[0045] in the formula

[0046] a1 and y1 are the unknown coefficients of the catenary equation;

[0047] w m is the width w of extreme point 11 m 15;

[0048] Step 5: According to f(0) = h i and f(w m )=-d m Establish a system of transcendental equations with a1 and y1 as unknowns

[0049]

[0050] In the above equation

[0051] a1, y1, and w m As shown in step 4;

[0052] h i is the height h of the water inlet edge of section line 10 i 12;

[0053] d m is the depth d of extreme point 11 m 16;

[0054] And a1 and y1 can be solved by Newton's downhill method, so as to establish the catenary equation of the left part of each section line 10;

[0055] Step 6: For the right side of each section line 10 of the working surface 3 of the bucket 2, according to the water outlet depth h of the section line 10 o 13. The width w of extreme point 11 m 15 and depth d m 16 You can establish a catenary fitting equation containing unknowns a2 and y2 And satisfy g(w)=-h o , g(w m )=-d m ;

[0056] in the formula

[0057] a2 and y2 are the unknown coefficients of the catenary equation;

[0058] w m As shown in step 4;

[0059] Step 7: According to g(w)=-h o and g(w m )=-d m Construct a system of transcendental equations with a2 and y2 as unknowns

[0060]

[0061] In the above equation

[0062] w m As shown in step 4;

[0063] d m As shown in step five;

[0064] a2, y2 are as shown in step 6;

[0065] h o is the height h of the water outlet of section line 10 o 13;

[0066] And a2 and y2 can be solved by Newton's downhill method, so as to establish the catenary equation of the right part of each section line 10;

[0067] Step 8: The catenary equations on the left and right sides of each section line 10 of the working surface 3 of the bucket 2 obtained in steps 5 and 7 are combined into a unified curve to complete the catenary fitting of each section line 8 of the working surface 3 of the bucket 2;

[0068] Step 9: If Figure 2 , 3 As shown, the water cutting blade 8, each section line 10 established by step eight, and the root transition line 9 are u-curves, and the water inlet edge 6 and the water outlet edge 7 are v-curves, and the fitting of the bucket working surface is completed by bidirectional lofting surface. Although this specification is described according to the implementation mode, not every implementation mode contains only one independent technical solution. This description is only for the sake of clarity. Those skilled in the art should understand that: taking the specification as a whole, the technical solutions in each embodiment can be combined to form other implementation modes that can be understood by those skilled in the art.

[0069] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for fitting a bucket working surface using a catenary equation, characterized by comprising the following steps: Step 1: The bucket (2) on the impulse turbine runner (1) includes a working surface (3) and a back surface (4), wherein the working surface (3) is the main working part of the impulse turbine runner (1), converting the energy of the jet (5) into the mechanical energy of the impulse turbine runner (1), and the working surface (3) is a surface with the water inlet edge (6), the water outlet edge (7), the water cutting edge (8) and the root transition line (9) as boundary lines. A series of cross-section lines (10) from the water inlet edge (6) to the water outlet edge (7) are established by using the catenary equation to perform surface fitting of the working surface (3). These cross-section lines (10) are divided into two left and right catenary segments, and the connection point of the two catenary segments is the extreme point (11) of the cross-section line (10). The key geometric parameter of the cross-section line (10) is the water inlet edge height h i (12) Depth of water outlet h o (13), bucket width w (14), width w of extreme point (11) m (15) and the depth d of the extreme point (11) m (16); Step 2: Establish the reference point (17) and reference plane (18) of the bucket (2), and use the reference point (17) of the bucket (2) as the origin (19) of the coordinate system, a straight line (21) passing through the reference point (17) of the bucket (2) and parallel to the rotation center line (20) of the runner (1) as the Z axis, a straight line (22) passing through the reference point (17) of the bucket (2) on the reference plane (18) of the bucket (2) and perpendicular to the Z axis as the X axis, and a vector product of the Z axis and the X axis as the Y axis, to establish a body-fitting coordinate system (23) of the bucket (2); Step 3: Each section line (10) of the working surface (3) of the bucket (2) is a concave curve with equal X value in the body-fitting coordinate system (21) of the bucket (2), and each section line (10) of the working surface (3) of the bucket (2) is divided into two left and right parts with the extreme point (15) of the section line (10) as the breaking point; Step 4: For the left side of each section line (10) of the working surface (3) of the bucket (2), according to the height h of the water inlet edge of the section line (10), i (12) The width w of the extreme point (11) m (15) and depth d m (16) Establish the catenary equation with unknowns a1 and y1 And satisfy f(0)=h i ,f(w m )=-d m ; In the formula: a1 and y1 are the unknown coefficients of the catenary equation; w m is the width w of the extreme point (11) m (15); Step 5: According to f(0) = h i and f(w m )=-d m Establish a system of transcendental equations with a1 and y1 as unknowns In the above equation a1, y1, and w m As shown in step 4; h i is the height h of the water inlet edge of the section line (10) i (12); d m is the depth d of the extreme point (11) m (16); And solve a1 and y1 by Newton's downhill method to establish the catenary equation of the left part of each section line (10); Step 6: For the right side of each section line (10) of the working surface (3) of the water bucket (2), according to the water outlet depth h of the section line (10) o (13), the width w of the extreme point (11) m (15) and depth d m (16) Establish the catenary fitting equation containing unknowns a2 and y2 And satisfy g(w)=-h o , g(w m )=-d m ; In the formula: a2 and y2 are the unknown coefficients of the catenary equation; w m As shown in step 4; Step 7: According to g(w)=-h o and g(w m )=-d m Construct a system of transcendental equations with a2 and y2 as unknowns In the above equation: w m As shown in step 4; d m As shown in step five; a2, y2 are as shown in step 6; h o is the height h of the water outlet of the section line (10) o (13); And solve a2 and y2 by Newton's downhill method to establish the catenary equation of the right part of each section line (10); Step 8: The catenary equations on the left and right sides of each cross-sectional line (10) of the working surface (3) of the bucket (2) obtained in step 5 and step 7 are combined into a unified curve to complete the catenary fitting of each cross-sectional line (8) of the working surface (3) of the bucket (2); Step nine: With the water cutting blade (8), each section line (10) established in step eight, and the root transition line (9) as the u-curve, and the water inlet edge (6) and the water outlet edge (7) as the v-curve, the fitting of the bucket working surface is completed by bidirectional lofting surface.

2. The algorithm for fitting the bucket working surface using the catenary equation according to claim 1 is characterized in that: In the step 3, each section line (10) of the working surface (3) of the bucket (2) is a concave curve with equal X values ​​in the body-fitting coordinate system (21) of the bucket (2), and each section line (10) of the working surface (3) of the bucket (2) is divided into left and right parts with the extreme point (15) of the section line (10) as the breaking point.

3. The algorithm for fitting the bucket working surface using the catenary equation according to claim 1 is characterized in that: In the step 4, for the left side portion of each section line (10) of the working surface (3) of the bucket (2), according to the water inlet edge height h of the section line (10), i (12) The width w of the extreme point (11) m (15) and depth d m (16) Establish the catenary equation with unknowns a1 and y1 And satisfy f(0)=h i ,f(w m )=-d m And in step 5, according to f(0)=h i and f(w m )=-d m Establish a system of transcendental equations with a1 and y1 as unknowns And a1 and y1 are solved by Newton's downhill method, so as to establish the catenary equation of the left part of each section line (10).

4. The algorithm for fitting the bucket working surface using the catenary equation according to claim 1 is characterized in that: In step 6, for the right side portion of each section line (10) of the working surface (3) of the bucket (2), according to the water outlet depth h of the section line (10), o (13), the width w of the extreme point (11) m (15) and depth d m (16) Establish the catenary fitting equation containing unknowns a2 and y2 And satisfy g(w)=-h o , g(w m )=-d m And the step 7 according to g(w) = -h o and g(w m )=-d m Construct a system of transcendental equations with a2 and y2 as unknowns And by using Newton's downhill method, a2 and y2 are solved to establish the catenary equation of the right part of each section line (10).

5. The algorithm for fitting the bucket working surface using the catenary equation according to claim 1 is characterized in that: In the step eight, the catenary equations on the left and right sides of each cross-sectional line (10) of the working surface (3) of the bucket (2) obtained in steps five and seven are combined into a unified curve, thereby completing the catenary fitting of each cross-sectional line (8) of the working surface (3) of the bucket (2).

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