Wear-resistant volute based on bionic arrangement and bionic morphology and design method

Abstract

The invention provides a wear-resistant volute based on bionic arrangement and bionic morphology and a design method, a convex hull arrangement area is arranged on the inner wall of a spiral flow channel of the volute, and the convex hull arrangement area is located between the second section and the eighth section of the volute; a plurality of convex hulls are arranged in the convex hull arrangement area, and a center point set of the convex hulls is obtained by unfolding the side face of a cylinder according to the columnar phyllotaxy arrangement rule and mapped to the convex hull arrangement area on the inner wall of the spiral flow channel of the volute. According to the bionic arrangement mode, the surface space utilization efficiency is improved, and the direct impact frequency of solid particles on the wall surface is remarkably reduced by orderly guiding fluid and particle movement. The bionic morphology of the convex hulls is combined with the phyllotaxy arrangement rule, the flow field structure of a near-wall area can be effectively changed, particle impact energy is weakened, and therefore the abrasion rate of the wall face of the volute is greatly reduced in long-term operation, and the service life of equipment is prolonged.

Description

Technical Field

[0001] This invention relates to the field of fluid machinery, specifically to a wear-resistant volute and its design method based on biomimetic arrangement and morphology in the field of pump biomimetic surfaces. Background Technology

[0002] In the field of hydraulic engineering, the volute, as the core component of a solid-liquid two-phase centrifugal pump, operates in a complex environment of high-speed solid-containing fluids for extended periods, continuously enduring the combined effects of particle erosion, cavitation damage, and alternating loads. This leads to easy peeling of the surface material, gradual efficiency degradation, and serious impacts on long-term operational stability. Traditional improvement methods often rely on homogeneous material reinforcement or symmetrical geometry optimization, which can improve wear resistance to some extent, but often exacerbate the uneven distribution of the internal flow field and are difficult to effectively suppress local stress concentration caused by particle impacts. Summary of the Invention

[0003] To address the shortcomings of existing technologies, this invention provides a wear-resistant volute and its design method based on biomimetic arrangement and morphology. While maintaining flow continuity, this invention can reduce the effective impact probability and kinetic energy of particles, suppress stress concentration and wear propagation in the near-wall region. The leaf sequence arrangement is not a simple "dense arrangement". Its quasi-uniform angular distribution still exhibits low repetition and low resonance point characteristics under curved surface mapping.

[0004] The present invention achieves the above-mentioned technical objectives through the following technical means.

[0005] A wear-resistant volute based on biomimetic arrangement and biomimetic morphology has a convex hull arrangement area on the inner wall of the spiral flow channel of the volute, which is located between the second and eighth cross sections of the volute. The convex hull arrangement area has several convex hulls, and the set of center points of the several convex hulls is obtained by unfolding the side of the cylinder according to the columnar leaf sequence arrangement law and mapped onto the convex hull arrangement area on the inner wall of the spiral flow channel of the volute.

[0006] In the above scheme, the columnar leaf sequence arrangement rule is as follows: Let the divergence angle of the kth leaf sequence point satisfy α=k·φ, where k=1,2,…, and φ is taken as the leaf sequence divergence angle of 137.508°; let m=α / 2π represent the circle number of the point;

[0007] When α ≤ 2π, the expanded coordinates of the k-th leaf sequence point in the Cartesian coordinate system are x k = R0·α,y k = k·h; When α > 2π, its expanded coordinates are x k = R0·α−2π·m·R0,y k= k·h, where R0 is the radius of the phyllotaxis base circle of the Van Iterson cylindrical phyllotaxis model, and h is the phyllotaxis coefficient, that is, the spacing between adjacent phyllotaxis points along the height direction of the generatrix of the parent cylinder.

[0008] In the above solution, k is a positive integer and is numbered sequentially along the flow direction starting from the second section.

[0009] In the above solution, increasing the number of phyllotaxis points or reducing h is used to increase the convex hull arrangement density in the area with larger wear.

[0010] In the above solution, the width of the convex hull arrangement area is the width b of the spiral flow channel of the volute, satisfying k·h < b.

[0011] In the above solution, the length of the convex hull arrangement area along the direction of the spiral flow channel is L, and 2πR0 < L.

[0012] In the above solution, the width b of the spiral flow channel is 20 - 40 mm; the base circle diameter D3 of the volute is 150 - 300 mm; the thickness of the volute housing is 4 - 8 mm.

[0013] In the above solution, the convex hull arrangement area is obtained by importing the phyllotaxis point set obtained by unfolding the cylindrical side surface into 3D modeling software and mapping it to the inner wall of the spiral flow channel of the volute through local transformation.

[0014] In the above solution, the convex hull is a spherical convex hull.

[0015] The design method of the wear-resistant volute based on bionic arrangement and bionic morphology includes the following steps:

[0016] Step 1. Determine the basic structure parameters of the volute

[0017] Select the thickness of the volute, the width of the spiral flow channel and the base circle diameter; a convex hull arrangement area is set on the internal spiral flow channel of the volute, and the convex hull arrangement area covers the range from the second section to the eighth section of the volute;

[0018] Step 2. Design the convex hull arrangement area

[0019] The convex hull arrangement area is arranged along the wall surface of the spiral flow channel and satisfies the relational expression k·h < b, where k is a parameter; the length of each convex hull arrangement area is related to the parameter k and the phyllotaxis base circle radius R0;

[0020] Step 3. Determine the shape and size of the convex hull

[0021] The convex hull adopts a spherical convex hull with a convex height of 1 - 2 mm;

[0022] Step 4. Distribute the convex hulls according to the phyllotaxis arrangement rule

[0023] Within each convex hull arrangement region, the distribution density of the convex hulls is determined according to the columnar phyllotaxis arrangement pattern; the phyllotaxis arrangement unfolds along the lateral side as follows:

[0024] Suppose we expand the k-th phyllotaxis point of the phyllotaxis arrangement, then the relevant parameters of the k-th phyllotaxis point are as follows:

[0025] α = k·φ(4)

[0026] (5)

[0027] Where α is the total angle between the kth leaf sequence point and the initial leaf sequence point, φ is the leaf sequence divergence angle, the leaf sequence divergence angle is selected as the golden section angle of 137.508°, and m is the number of circles in the leaf sequence arrangement where the kth leaf sequence point is located.

[0028] If α ≤ 2π, then the coordinates of the k-th leaf sequence point in the Cartesian coordinate system after expansion are as follows:

[0029] (6)

[0030] Where, x k Let y be the x-coordinate of the k-th leaf sequence point after expansion, and y be the x-coordinate of the leaf sequence point after expansion. k Let R0 be the coordinate of the kth leaf sequence point on the y-axis after unfolding, R0 be the radius of the leaf sequence base circle of Van Iterson's cylindrical leaf sequence model, h be the leaf sequence coefficient of the leaf sequence arrangement, and h be the distance between the kth leaf sequence point and the (k+1)th leaf sequence point along the height of the generatrix of the parent cylindrical model.

[0031] If α > 2π, then the coordinates of the k-th leaf sequence point in Cartesian coordinates after expansion are as follows:

[0032] (7)

[0033] The number of leaf sequence points in the convex hull array is determined by the wear degree of the volute. For a larger wear area, the number of leaf sequence points is increased and h is decreased.

[0034] Step 5: Generate leaf sequence diagram using Python programming.

[0035] By importing the mathematical formulas of foliation theory into Python, a foliation pattern diagram is generated, and the Cartesian coordinates of the foliation points are found.

[0036] Step 6: Adjust the convex bulge density according to the degree of wear.

[0037] For the wear-prone areas inside the volute, the distribution density of the convex hull in the corresponding areas is increased by increasing the number of sequence points and decreasing the sequence coefficient h, thereby enhancing its wear resistance.

[0038] Step 7: Apply the convex hulls to the volute according to the leaf sequence arrangement.

[0039] The leaf sequence point coordinates obtained in steps six and seven are imported into 3D software to generate a convex hull; the convex hull is then distributed on the inner wall of the volute according to the leaf sequence arrangement rule.

[0040] Beneficial effects of this invention:

[0041] 1. Based on the mathematical laws of leaf sequence arrangement following the golden ratio and Fibonacci sequence - maintaining uniform distribution after unfolding along the curved surface, this invention couples this characteristic with the biomimetic convex hull morphology to construct a convex hull region with leaf sequence arrangement characteristics, significantly improving space utilization efficiency, and effectively reducing the impact frequency and impact kinetic energy of solid particles by optimizing the surface flow field distribution.

[0042] 2. Unlike conventional anti-wear coatings, random bumps, or regular grid arrays, this invention employs a columnar leaf sequence design method based on a synergistic approach of golden ratio angle, surface mapping, and distribution density. This design method, while maintaining flow continuity, reduces the effective impact probability and kinetic energy of particles, suppressing stress concentration and wear propagation in the near-wall region. The leaf sequence arrangement in this invention is not a simple "dense arrangement"; its quasi-uniform angular distribution still exhibits low repetition and low resonance point characteristics under surface mapping. The convex hull unit, through local structural reinforcement, utilizes a stress dispersion mechanism to reduce damage caused by a single impact. After arranging this biomimetic coupling structure in the volute wear zone, the stress concentration effect caused by solid particle impact is significantly alleviated, and the surface wear is significantly reduced.

[0043] 3. This invention incorporates a regularly arranged array of convex hulls in the critical, easily worn areas of the spiral flow channel of the volute. The arrangement of this array draws inspiration from the Van Iterson mathematical model in phyllotaxis theory in botany, adapting its planar unfolded form to the curved surface of the volute. This biomimetic arrangement not only improves surface space utilization efficiency but also significantly reduces the frequency of direct impacts between solid particles and the wall surface by orderly guiding fluid and particle movement. The biomimetic morphology of the convex hulls, combined with the phyllotaxis arrangement, effectively alters the flow field structure near the wall, weakening particle impact energy and thus significantly reducing the wear rate of the volute wall surface during long-term operation, extending the equipment's service life. Attached Figure Description

[0044] Figure 1 The leaves are arranged in a columnar inflorescence, typical of natural plants.

[0045] Figure 2 Cylindrical coordinates for leaf order theory;

[0046] Figure 3 This is a diagram showing the distribution of columnar leaf arrangement.

[0047] Figure 4 This is a diagram of the volute structure according to an embodiment of the present invention;

[0048] Figure 5 for Figure 4 A left-view diagram;

[0049] Figure 6 This is a cross-sectional view of the volute casing according to an embodiment of the present invention;

[0050] Figure 7 This is a diagram showing the thickness of the volute shell in an embodiment of the present invention;

[0051] Figure 8 This is a schematic diagram of the volute cross-section position according to an embodiment of the present invention;

[0052] Figure 9 This invention provides a Python-generated diagram of leaf arrangement patterns.

[0053] Figure 10 This is a three-dimensional model diagram of an embodiment of the present invention. Detailed Implementation

[0054] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0055] Inspired by the efficient wear-resistant structures found in biological evolution, this invention proposes an innovative design strategy that integrates the theory of phyllotaxis spatial arrangement with biomimetic convex hull morphology. In plant morphology, leaves, petals, and fruits, such as sunflower seed discs and pine cone scales, follow the Fibonacci sequence, as... Figure 1 As shown, efficient filling and mechanical distribution within a limited space are achieved; and micro-convex hull morphology is commonly found in organisms that survive in harsh environments such as deserts, such as the skin texture of tamarisk and the scale morphology of lizards. This type of convex hull structure has the advantages of small size, strong adaptability and excellent erosion resistance.

[0056] This invention proposes an alternate phyllotaxis, characterized by leaves arranged in an alternating spiral along the main stem. This arrangement is influenced by two key parameters: the divergence angle (the angle between the petiole of adjacent leaves and the projection of the main stem) and the axial distance between the centers of adjacent leaves along the main stem. To optimize photosynthetic efficiency, the phyllotaxis of most plants in nature conforms to the Fibonacci sequence. Specifically, the spiral growth of plant leaves on the main stem can be represented by a fraction, where the numerator is the total number of turns of the phyllotaxis around the main stem, and the denominator is the total number of leaves growing in that spiral. This defines the average number of turns each leaf occupies in the spiral structure, and this ratio tends towards the golden ratio limit. In alternate phyllotaxis, this principle ensures the optimal spatial distribution of leaves, providing theoretical support for the optimization and control of biomimetic phyllotaxis arrangement in solid-liquid two-phase flow in this project.

[0057] (1)

[0058] The growth of plant leaves follows the interval ratio of the Fibonacci sequence, resulting in a specific arrangement of adjacent leaves along the circumference of the stem. The angle between adjacent leaves along the stem circumference remains constant. Further statistical analysis has led to the conclusion that this angle is the golden divergence angle of the phyllotaxis, mathematically expressed as:

[0059] (2)

[0060] In practical applications, by analyzing the phyllotaxis distribution pattern, a mathematical model of cylindrical phyllotaxis pattern can be established, which can be described by the following formula in cylindrical coordinates:

[0061] (3)

[0062] in n and h are respectively the cylinder radius of the foliation configuration, the ordinal number of the foliation element, the spiral angle (divergence angle) between consecutive foliation points, and the foliation coefficient in foliation theory. h is an important parameter for measuring the density of foliation elements, such as... Figure 2 As shown.

[0063] Based on the aforementioned biological observations, this invention applies the spatial regularity of leaf arrangement to constrain the distribution pattern of the biomimetic convex hull on the volute surface, achieving optimized configuration of the surface structure through biomimetic principles. This design not only improves hydrodynamic performance but also promises to significantly enhance the erosion resistance and lifespan of the volute in solid-containing fluids, providing a new technical path for the stable and efficient operation of centrifugal pumps under harsh conditions.

[0064] Unfold Van Iterson's cylindrical leaf sequence model into a plane as follows: Figure 3 As shown, the spatial distribution of the convex hull shape constrained by the leaf sequence arrangement law is applied to the spiral flow channel wall structure, providing a wear-resistant volute that combines biomimetic arrangement and biomimetic morphology to improve the wear resistance of traditional centrifugal pump volutes.

[0065] To achieve the above objectives, the present invention includes the following steps:

[0066] Step 1: Determine the basic structural parameters of the volute.

[0067] The selected volute thickness is 4 - 8 mm, the width of the spiral flow channel b = 20 - 40 mm, and the base circle diameter D3 = 150 - 300 mm; a convex hull arrangement area is provided on the spiral flow channel inside the volute, and this area covers the range from the second cross-section to the eighth cross-section of the volute.

[0068] Step Two: Design the convex hull arrangement area

[0069] The convex hull arrangement area is arranged along the wall surface of the spiral flow channel, and its width is all b, and it satisfies the relational expression k·h < b, where k is a parameter; the length of each area is related to the parameter k and the phyllotaxis base circle radius R0;

[0070] Step Three: Determine the shape and size of the convex hull

[0071] The convex hull adopts a spherical convex hull, and the protruding height is controlled within 1 - 2 mm;

[0072] Step Four: Distribute the convex hulls according to the phyllotaxis arrangement rule

[0073] Within each convex hull arrangement area, the distribution density of the convex hulls is determined according to the columnar phyllotaxis arrangement rule; the phyllotaxis arrangement along the side expansion method is as follows:

[0074] Assume that the k-th phyllotaxis point of the phyllotaxis arrangement is expanded, then the relevant parameters of the k-th phyllotaxis point will be as shown in the formula:

[0075] α = k·φ (4)

[0076] (5)

[0077] Among them, α is the total angle between the k-th phyllotaxis point and the initial phyllotaxis point, φ is the phyllotaxis divergence angle, and the phyllotaxis divergence angle of this invention selects the golden section angle 137.508°, m is the number of circles in the phyllotaxis arrangement where the k-th phyllotaxis point is located;

[0078] If α ≤ 2π, the coordinates of the expanded k-th phyllotaxis point in the Cartesian coordinate system are as follows:

[0079] (6)

[0080] Among them, x k is the coordinate of the expanded k-th phyllotaxis point on the x-axis, y k is the coordinate of the expanded k-th phyllotaxis point on the y-axis, R0 is the phyllotaxis base circle radius of the Van Iterson's cylindrical phyllotaxis model, h is the phyllotaxis coefficient of the phyllotaxis arrangement, and it is the distance between the k-th phyllotaxis point and the (k + 1)-th phyllotaxis point along the generatrix height of the parent cylindrical model;

[0081] If α > 2π, the coordinates of the expanded k-th phyllotaxis point in the Cartesian coordinate system are as follows:

[0082] (7)

[0083] The number of leaf sequence points in the convex hull array of each part is determined by the wear degree of the volute. For the larger wear area, the number of leaf sequence points is increased and h is decreased to improve the convex hull density.

[0084] Step 5: Generate leaf sequence diagram using Python programming.

[0085] By importing the mathematical formulas of foliation theory into Python, a foliation pattern diagram is generated, and the Cartesian coordinates of the foliation points are found.

[0086] Step 6: Adjust the convex bulge density according to the degree of wear.

[0087] For the wear-prone areas inside the volute, the distribution density of the convex hull in the corresponding areas is increased by increasing the number of sequence points and decreasing the sequence coefficient h, thereby enhancing its wear resistance.

[0088] Step 7: Apply the convex hulls to the volute according to the leaf sequence arrangement.

[0089] By obtaining reasonable leaf sequence point coordinates through steps six and seven, the coordinates are imported into 3D software to generate a convex hull; then, through a series of functions such as transformation, the convex hull is distributed on the inner wall of the volute according to the leaf sequence arrangement rule.

[0090] An embodiment of the present invention provides a wear-resistant volute based on biomimetic arrangement and biomimetic morphology, and the specific implementation steps are as follows:

[0091] Step 1: Based on the initial volute structure, determine the volute base circle diameter D3 = 232 mm and the volute spiral flow channel width b = 35 mm. A specific example is shown below. Figure 4-6 As shown.

[0092] Step two: Using the columnar phyllotaxis arrangement method in biology, the golden ratio angle of 137.508° is selected as the phyllotaxis divergence angle, and the columnar phyllotaxis is spread out along the side into a planar arrangement pattern. Step 3: Based on the wear area distribution of the spiral flow channel in the volute, define the convex hull arrangement area as a specific section of the inner wall of the flow channel. The convex hull arrangement area is arranged along the spiral flow channel wall, and its width is 2πR0, satisfying the relationship 2πR0≤b. The width of this area is less than or equal to the flow channel width b=35mm, where R0=5mm. A specific example is shown below. Figure 4-5 As shown. Step 4, the unfolded length L along the flow channel needs to satisfy the condition of k·h < L. The actual unfolded length corresponds to the second cross-section to the eighth cross-section of the volute. Here, k = 400, L = 770 mm, and h = 1.9 are selected. However, some phyllotaxis points exceed the specified area. Actually, the number of phyllotaxis points k = 383 is taken. Specific examples are as Figure 6 shown. Step 5, create a spherical convex hull at each phyllotaxis point position, and set the protrusion height to 1.5 mm to form a bionic morphology. The wall thickness of the volute is 5 mm. Specific examples are as Figure 7 shown in Fig. 8. Step 6, by importing the phyllotaxis theory mathematical formula into Python 3.12, program to generate a phyllotaxis arrangement pattern diagram, and find the Cartesian coordinates of the phyllotaxis points. Specific examples are as Figure 9 shown.

[0093] Step 7, apply the spherical convex hulls distributed according to the phyllotaxis arrangement pattern to the volute. Through the phyllotaxis point coordinates obtained in Step 6, import the coordinates into the 3D modeling software NX 12.01 to generate spherical convex hulls. Then, through a series of functions such as transformation, distribute the spherical convex hulls on the inner wall of the volute according to the phyllotaxis arrangement pattern. Finally, the generated wear-resistant volute 3D model integrates the bionic arrangement and bionic morphology features. Specific examples are as Figure 10 shown.

[0094] In the description of this specification, the description referring to terms such as "one embodiment", "some embodiments", "example", "specific example", or "some examples" means that the specific features, structures, materials, or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in a suitable manner in any one or more embodiments or examples.

[0095] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limitations to the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention without departing from the principles and purposes of the present invention.

Claims

1. A wear-resistant volute based on biomimetic arrangement and biomimetic morphology, characterized in that, There is a convex hull arrangement area provided on the inner wall of the spiral flow channel of the volute, and the convex hull arrangement area is located between the second cross-section and the eighth cross-section of the volute; Several convex hulls are arranged in the convex hull arrangement area, and the set of the center points of the several convex hulls is obtained by unfolding the side surface of a cylinder according to the columnar phyllotaxis arrangement rule and mapped to the convex hull arrangement area on the inner wall of the spiral flow channel of the volute.

2. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The columnar phyllotaxis arrangement rule is as follows: Let the divergence angle of the k-th phyllotaxis point satisfy α = k·φ, where k = 1, 2, …, and φ takes the phyllotaxis divergence angle of 137.508°; Let m = α / 2π represent the number of circles where this point is located; When α ≤ 2π, the expanded coordinates of the k-th leaf sequence point in the Cartesian coordinate system are x k = R0·α,y k = k·h; when α > 2π, its expanded coordinates are x k = R0·α−2π·m·R0,y k = k·h, where R0 is the radius of the leaf sequence base circle of the Van Iterson cylindrical leaf sequence model, and h is the leaf sequence coefficient, i.e., the distance between adjacent leaf sequence points along the height direction of the parent cylinder generatrix.

3. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 2, characterized in that, k is a positive integer and is sequentially numbered along the flow direction starting from the second cross-section.

4. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 2, characterized in that, Increasing the number of phyllotaxis points or reducing h is used to increase the convex hull arrangement density in the area with greater wear.

5. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The width of the convex hull arrangement area is the width b of the spiral flow channel of the volute, satisfying k·h < b.

6. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The length of the convex hull arrangement area along the direction of the spiral flow channel is L, and 2πR0 < L.

7. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The width b of the spiral flow channel is 20 - 40 mm; the base circle diameter D3 of the volute is 150 - 300 mm; the thickness of the volute housing is 4 - 8 mm.

8. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The convex hull arrangement area is obtained by importing the phyllotaxis point set obtained by unfolding the side surface of a cylinder into three-dimensional modeling software and mapping it to the inner wall of the spiral flow channel of the volute through local transformation.

9. The wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to claim 1, characterized in that, The convex hull is a spherical convex hull.

10. The design method for a wear-resistant volute based on biomimetic arrangement and biomimetic morphology according to any one of claims 1 to 8, characterized in that, It includes the following steps: Step 1: Determine the basic structure parameters of the volute Select the thickness of the volute, the width of the spiral flow channel, and the base circle diameter; a convex hull arrangement area is provided on the internal spiral flow channel of the volute, and the convex hull arrangement area covers the range from the second cross-section to the eighth cross-section of the volute; Step 2: Design the convex hull arrangement area The convex hull arrangement area is arranged along the wall surface of the spiral flow channel and satisfies the relationship k·h < b, where k is a parameter; the length of each convex hull arrangement area is related to the parameter k and the phyllotaxis base circle radius R0; Step 3: Determine the shape and size of the convex hull The convex hull adopts a spherical convex hull with a convex height of 1 - 2 mm; Step 4: Distribute the convex hulls according to the phyllotaxis arrangement rule Within each convex hull arrangement area, the distribution density of the convex hulls is determined according to the columnar phyllotaxis arrangement rule; the phyllotaxis arrangement along the side unfolding method is as follows: Assume that the k-th phyllotaxis point of the phyllotaxis arrangement is unfolded, then the relevant parameters of the k-th phyllotaxis point are as shown in the formula: α = k·φ (4) （5） where α is the total angle between the k-th phyllotaxis point and the initial phyllotaxis point, φ is the phyllotaxis divergence angle, the phyllotaxis divergence angle selects the golden section angle of 137.508°, and m is the number of circles where the k-th phyllotaxis point is located in the phyllotaxis arrangement; If α ≤ 2π, the coordinates of the unfolded k-th phyllotaxis point in the Cartesian coordinate system are as follows: （6） Where, x k Let y be the x-coordinate of the k-th leaf sequence point after expansion, and y be the x-coordinate of the leaf sequence point after expansion. k Let R0 be the coordinate of the kth leaf sequence point on the y-axis after unfolding, R0 be the radius of the leaf sequence base circle of Van Iterson's cylindrical leaf sequence model, h be the leaf sequence coefficient of the leaf sequence arrangement, and h be the distance between the kth leaf sequence point and the (k+1)th leaf sequence point along the height of the generatrix of the parent cylindrical model. If α > 2π, the coordinates of the unfolded k-th phyllotaxis point in the Cartesian coordinate system are as follows: （7） where the number of phyllotaxis points of the convex hull array is determined by the wear degree of the volute. For the larger part of the worn area, increasing the number of phyllotaxis points and reducing h are selected; Step 5: Generate a phyllotaxis diagram through Python programming By importing the phyllotaxis theory mathematical formula into Python, programming to generate a phyllotaxis arrangement rule diagram and find the Cartesian coordinates of the phyllotaxis points; Step 6: Adjust the convex hull density according to the wear degree For the wear-prone areas inside the volute, the distribution density of the convex hull in the corresponding areas is increased by increasing the number of sequence points and decreasing the sequence coefficient h, thereby enhancing its wear resistance. Step 7: Apply the convex hulls to the volute according to the leaf sequence arrangement. The leaf sequence point coordinates obtained in steps six and seven are imported into 3D software to generate a convex hull; the convex hull is then distributed on the inner wall of the volute according to the leaf sequence arrangement rule.

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