Standardized Description of the Generation Principle and Process of the Surface of Archimedes Spiral Wind Blade

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This paper refines, expands and standardizes the existing qualitative description of the generation principle of the surface of Archimedes Spiral Wind Blade, and attempts to describe the blade’s surface using an infinite number of 3D Archimedes Spiral curves. The study provides theoretical support for exploring more manufacturing methods for ASWB beyond the realm of 3D printing.

Abstract

Currently, there are three fundamental theoretical issues that need to be addressed in the research field of the Archimedes Spiral Wind Blade (ASWB): (1) the existing description of the generation principle of the ASWB’s surface is qualitative, which needs to be formally described using mathematical tools; (2) in the published literature, no studies were found that attempt to fully describe the surface of the ASWB using mathematical tools; (3) in the published literature, no clear definition of the ASWB can be found. The first and second of the above three problems are relatively easy to solve, whereas the ultimate solution to the third problem requires long-term communication and discussion among researchers from various countries. Therefore, this paper focuses primarily on the first of the above problems, namely, to describe the intermediate process of transforming an irregular planar figure into the 3D surface of an ASWB in a standardized manner using mathematical tools including a polar coordinate system, planar and spatial Cartesian coordinate systems, the curve equation of an Archimedes spiral, differential equations, and so on. For the second problem, this paper proposes an alternative approach: namely, conjecturing that the ASWB surface can be approximated by an infinite number of 3D Archimedes spirals, and an example with a finite number of such spirals is provided. For the third problem, this paper can currently only suggest that the precise definition of the ASWB should be based on a standardized description of the generation principle of the ASWB surface, or, alternatively, on an accurate geometric description of the shape of the ASWB surface. The scientific contribution of this paper lies in proposing, for the first time from a purely geometric perspective (independent of aerodynamics), three fundamental theoretical problems concerning ASWB, along with preliminary alternative ideas toward their solution.

1. Introduction

The Archimedes Spiral Wind Blade (ASWB) is a new type of wind blade invented by Dutchman Rinus MIEREMET in 2006. The appearance of a typical ASWB is shown in Figure 1. The rotor composed of three ASWBs surrounding the rotating axis is called an Archimedes Spiral Wind Rotor (ASWR). The appearance of a typical ASWR is shown in Figure 2. A wind turbine using three ASWBs (i.e., one ASWR) is called an Archimedes Spiral Wind Turbine (ASWT). The appearance of a typical ASWT is shown in Figure 3. The ASWT is a small type of wind turbine with a history of no more than 20 years, and has a high aesthetic value, since its rotor, the ASWR, looks like a rose.

Researchers from various countries have proved through computer simulation, wind tunnel tests, field tracking, and other methods that the ASWT is not only of appreciative value, but also has excellent aerodynamic performance compared with traditional small wind turbines, such as having a low cut-in wind speed, high tip speed ratio, and high wind energy utilization coefficient, etc. In addition, it also has environmental protection properties such as low noise and no harm to birds.

In many books and papers on wind turbines, according to the relationship between the wind blade axis and the wind direction, wind turbines can be divided into two categories: horizontal axis wind turbines and vertical axis wind turbines, for example, as described in references [1,2,3,4,5]. According to Figure 1, Figure 2 and Figure 3, since the rotation axis of an ASWB or ASWR is parallel to the ground (also parallel to the wind direction), the ASWT belongs to the horizontal axis wind turbine category.

The rotor is an important component of a wind turbine. At present, the rotor of a mainstream horizontal axis wind turbine is generally composed of 2 or 3 wind blades and a hub [1,3,5,6,7,8]. In terms of the number of blades, the ASWT is consistent with mainstream wind turbines, using three ASWB blades. These three ASWB blades are evenly arranged around the shaft and connected to the shaft to form an ASWR.

When the wind turbine is working, the aerodynamic force acting on the blades can be decomposed into two components: the component perpendicular to the wind direction is called lift and the component parallel to the wind direction is called drag [4,5,6,7,9,10]. According to the working principle of the blades, wind turbines can be divided into two categories, namely lift-type and drag-type. Turbines that mainly rely on lift to work are called lift-type turbines, and turbines that mainly rely on drag to work are called drag-type turbines [5,6,7,9,10,11,12,13,14,15]. The literature shows that the ASWT is fully dependent on the drag force [16,17]. Therefore, the ASWT is recognized as a drag-type wind turbine by researchers from various countries [18].

The tip-speed ratio is the ratio of the linear velocity of the wind blade tip to the wind speed [1,3,5,6,9]. Generally, the tip-speed ratio of a drag-type wind turbine is less than 1, while the tip-speed ratio of a lift-type wind turbine is greater than 1 [3,5,19]. Although the ASWT is a drag-type wind turbine, numerous experiments and simulations have found that its tip-speed ratio can exceed 1, and the highest tip-speed ratio reported in the literature reaches 4.0. Therefore, in terms of tip-speed ratio, ASWT differs significantly from traditional drag-type wind turbines [20].

The power coefficient is a key indicator of wind turbine performance, representing the proportion of input wind energy converted into rotor kinetic energy or the efficiency of converting wind energy into electrical energy [1,4,5,6,9]. Most studies, through experiments and simulations, have shown that the maximum power coefficient achieved by an ASWT is usually around 26%, with a corresponding tip-speed ratio of 2.5 [21,22,23,24].

Low cut-in speed is another advantage of the ASWT. Simulations and experiments have demonstrated that an ASWT can operate at a minimum wind speed of as low as 2.5 m/s [25,26,27].

The yaw system is an essential windward steering mechanism for horizontal-axis wind turbines, enabling the rotor to rotate around a vertical axis to face the wind head-on. According to traditional views, a horizontal-axis wind turbine must be equipped with a yaw system, while a vertical-axis wind turbine does not require one [1,2,3,5,9]. However, although the ASWT is a horizontal-axis wind turbine, the unique shape of its rotor (ASWR) allows it to automatically face the wind, exhibiting self-yawing capability. Consequently, the ASWT does not require an additional yaw system, marking a significant distinction from traditional horizontal-axis wind turbines [16].

In summary, compared with mainstream wind turbines, the characteristics of the ASWT are as follows:

(1)

Although an ASWT belongs to horizontal-axis wind turbines, it has a self-yawing capability and does not need to be equipped with an additional yaw system.

(2)

An ASWT belongs to a drag-type wind turbine.

(3)

Although the tip-speed ratio of traditional drag-type turbines is typically less than 1, the tip-speed ratio of an ASWT, being a drag-type turbine, can be greater than 1, resulting in a higher power coefficient of ASWT compared with traditional drag-type turbines.

(4)

In addition to conventional evaluation criteria, the ASWT offers other advantages compared to traditional wind turbines: (a) low noise during operation; (b) environmentally friendly, posing no harm to birds; (c) capable of operating in urban areas or building complexes; not picky about the working environment.

Since its invention in 2006, the ASWB has had a development history of less than 20 years, which is far shorter than that of traditional wind blades. Therefore, the research on fundamental theories of the ASWB is not enough, leaving many problems unsolved to date. For example, the generation principle of the ASWB surface is currently available in only one paper, where the description is brief and purely qualitative [28].

The earliest academic research on the ASWT is conducted by Timmer and Toet at the Delft University of Technology, and the corresponding study was published as a technical report written in Dutch [29]. Since the full text of this report is unavailable, it cannot be determined whether a standardized description of the generation principle of the ASWB surface is included therein, whether an accurate definition of ASWB is provided, or whether the modeling method for an ASWB is introduced.

The first academic paper on ASWBs was published in 2012 and written in Korean [21]. The first academic paper on an ASWB written in English was published in 2013 [20]. Other important early academic papers on ASWBs were published in 2014 [16,30], 2015 [17], 2016 [24], and 2019 [18].

It is noteworthy that all the above-mentioned papers directly present the final 3D model of an ASWR (composed of three ASWBs). In these papers, no accurate definition of an ASWB can be found, nor any detailed modeling process, nor any relevant explanation of the generation principle of the ASWB surface, nor any mathematical equation describing the ASWB surface. Similarly, in more recent publications, such as [23,26,31,32], no studies addressing the aforementioned issues can be identified.

Given that most authors of the early publications on ASWBs have maintained close collaborations with the inventor of the ASWB, and nearly all the published literature focuses exclusively on the aerodynamic performance of the ASWB without defining the ASWB, it seems that an accurate definition of an ASWB appears to be absent from the published literature.

Since an accurate definition of an ASWB should be based on a standardized and rigorous description of the generation principle of the ASWB surface, this paper will, in the following sections, conduct its investigation based on the original description provided by the inventor [28].

To comprehensively use mathematical formulas, geometric models, and scientific language to describe the generation principle and process of surface of the new wind blade, the ASWB, in a standardized and academic manner, this paper conducts the following work:

(1)

Section 1 introduces the development history and basic characteristics of the ASWB, and differences between ASWBs and traditional wind blades.

(2)

In Section 2, the descriptions of the generation principle and process of the ASWB surface in the existing published literature are summarized. In fact, in the published literature, only the inventor of the ASWB has given a brief and qualitative description of the generation principle of the ASWB surface, which is the most original and only description available. Therefore, Section 2 quotes the original description of the generation principle of the ASWB given by the inventor and then discusses it.

(3)

In Section 3, the original description is improved and rewritten. Rigorous mathematical tools and geometric models are used, including a planar polar coordinate system, a planar and spatial Cartesian coordinate system, the curve equation of the Archimedes spiral, the derivation of differential equations and their numerical solutions, etc., to provide a standardized, academic, and scientific description of the generation basis, i.e., an irregular closed plane figure, of the ASWB surface.

(4)

In Section 4, based on the standardized description of the generation basis of the ASWB surface proposed in Section 3 and combined with the spatial Cartesian coordinate system, a standardized, academic, and detailed explanation of the generation process of the ASWB surface is given.

(5)

In Section 5, the factors that affect the shape of the ASWB surface and the characteristics of the ASWB surface are revealed; the existing modeling process for ASWBs is introduced; and a conjecture on describing the ASWB surface with equations is proposed.

(6)

In Section 6, the work of this paper is concluded and outlooks for further research are presented.

2. Related Work

Regarding the generation principle of the ASWB surface, the inventor provided a brief qualitative description in his article. The original text is quoted as follows [28]:

“The Archimedes rotor blade is a flat surface elongated to give it depth and, therefore it’s shape perceived to have volume. From a sheet of paper one can obtain the spatial form of an Archimedes rotor blade by turning and simultaneously stretching out a cutout of the plane between a circle with radius R and a flat spiral”.

[28]

The quoted text above can be roughly translated as follows: “The Archimedes rotor blade is an elongated plane shape. After it is elongated along the depth direction (i.e., the direction perpendicular to the plane), the shape has volume. On a piece of paper, by cutting out a figure between an arc with a radius R and a plane spiral curve, and then rotating and simultaneously stretching the figure, the spatial shape of the Archimedes rotor blade is obtained.”

Alongside the textual description, the inventor provided two schematic diagrams, as shown in Figure 4 and Figure 5 [28]. In Figure 4, the shaded area represents a closed plane figure, which is roughly located between a circular arc and a segment of a spiral curve. Obviously, this plane figure is the basis for generating the ASWB surface, and it must be first cut out along its boundary and then deformed to form the ASWB surface shown in Figure 5.

As discussed in the introduction section, all the published literature on ASWBs (whether from the early papers or those published in recent years) focuses exclusively on the aerodynamic performance of ASWBs. And it seems that an accurate definition of ASWBs appears to be absent from the published literature.

As discussed in the Introduction, studying the generation principle of the ASWB surface helps to find its precise definition. Therefore, Section 3 and Section 4 in this paper focus on the basis for generating the ASWB surface and the generation principle of the ASWB surface, respectively.

3. Description of Basis for Generating the ASWB Surface

In this section, based on the original description of the generation principle of the ASWB surface quoted in Section 2, a standardized and academic description of the basis for generating the ASWB surface (i.e., the plane figure in Figure 4) is provided. The work includes the following:

(1)

Using symbols, coordinate systems, and formulas to standardize the definition of all geometric elements on the shaded closed irregular plane figure, which is the planar basis for generating the ASWB surface, shown in Figure 4, including points, straight lines, and curves.

(2)

Finding curve equations for all curves that constitute the boundary of the irregular plane figure in Figure 4.

(3)

Determination of the coordinates of each intersection and tangent point on the boundary of the irregular plane figure in Figure 4.

(4)

Finding all line equations for all straight lines that constitute the boundary of the irregular plane figure in Figure 4.

3.1. Standardized Definition of Geometric Elements of the Irregular Plane Figure

In the original description of ASWB surface generation, no clear mathematical tools and geometric models are used to define the closed shaded plane figure in Figure 4, and the corresponding text description is also brief and qualitative. Therefore, the first task of this section is to use mathematical and geometric tools to define geometric elements of the irregular closed shaded plane figure in Figure 4, including all points, lines and curves.

Based on the original description and Figure 4 provided by the inventor, a plane Cartesian coordinate system xOy is established, and then the arc, the Archimedes spiral, and the straight-line segments in Figure 4 are redrawn, and all geometric elements are annotated, as shown in Figure 6.

All geometric elements in Figure 6 are described as follows:

(1)

Point O is the origin of the plane Cartesian coordinate system xOy, the horizontal axis is the x-axis, and the vertical axis is the y-axis.

(2)

Draw a complete circle, with O as the center and R the radius, denoted as circle O.

(3)

Draw a straight-line segment from center O that coincides with the y-axis, in the positive y-direction, intersecting circle O at point P, denoted as OP, with the color red. The length of OP equals the radius R.

(4)

In the plane xOy, draw an in-plane Archimedes spiral that starts at O and ends at P, denoted by AS (Archimedes spiral), which is colored green.

(5)

Draw a straight-line segment parallel to the y-axis, tangent to the spiral AS at point T, and intersecting the circle O at point Q. Define this straight-line segment as QT, colored red.

(6)

Points P and Q divide the complete circle O into two arcs: the red arc on the right, denoted by ARC, and the black arc on the left, denoted by ARC1.

(7)

The tangent point T divides the green curve AS into two parts: the first part is denoted as TO, and the second part is denoted as TP.

(8)

Obviously, the shaded area in Figure 6 (i.e., the shaded area in Figure 4) is an irregular closed plane figure with its boundary consisting of four parts: the red straight-line OP, the red circular arc ARC, the red straight-line QT, and the green curve TO connected head-to-tail in sequence.

In this paper, the irregular closed plane figure in Figure 6 is represented by OPQTO. Obviously, OPQTO serves as the planar basis for generating the ASWB surface.

At this point, this section has completed the standardized definition of the irregular plane figure OPQTO which is the basis for generating the ASWB surface.

Obviously, after completing the definition of the plane figure OPQTO, the next steps, according to Figure 6, should be as follows: (1) determine the equations for each curve constituting the boundary of OPQTO, including curve TO and curve ARC (see Section 3.2.1 and Section 3.2.2); (2) determine the coordinates for each turning point on the boundary of OPQTO, especially the point Q and T (see Section 3.3.1 and Section 3.3.2); (3) determine the equations for each straight-line segment constituting the boundary of OPQTO, including OP and QT (see Section 3.4).

3.2. Equations of the Boundary Curves of Plane Figure OPQTO

The curves forming the boundary of the plane figure OPQTO include curve TO and curve ARC. Note that curve TO is also part of the Archimedes spiral AS, and curve ARC is also part of circle O. Therefore, determining the equation of curve TO is equivalent to determining the equation of curve AS, and determining the equation of curve ARC is equivalent to determining the equation of the circle O. Since none of the existing published research, including the original description (see Section 2), has used mathematical tools to strictly define curve AS and circle O in Figure 6 (also Figure 4), this section consists of two sub-sections: Section 3.2.1 determines the equation of curve AS, i.e., the equation of curve TO; Section 3.2.2 determines the equation of curve ARC, i.e., the equation of circle O.

3.2.1. Equation of Curve TO (Archimedes Spiral AS)

Since AS in Figure 6 is a special Archimedes spiral lying in the xOy plane, to determine the equation of AS, the general-form definition of a two-dimensional Archimedes spiral is first given:

In the plane xOy, when a point P moves along the rotating ray OP at a constant speed v while OP rotates around the point O at a constant angular velocity ω, the trajectory of the point P is called the “Archimedes spiral”. As shown in Figure 7:

In Figure 7, with time t as the independent variable, let the point P depart from the point O at time t = 0, and when t = 0, the instantaneous velocity of the point P is along the positive direction of the y-axis; let the direction of rotation of the ray OP be clockwise, and the magnitude of the rotational speed is ω (ω > 0); the angle between the ray OP and the y-axis is denoted by θ and the length of the OP is denoted by r.

Then, with the y-axis as the polar axis, the coordinates of point P under the polar coordinate system, i.e., the equation of the Archimedes spiral, is given as follows:

$$r=b\times \theta , -2\pi \le \theta \le 0$$

(1)

In the above equation, r = vt, and b is a constant to be determined; in order to be consistent with the original description (see Figure 4), this paper assumes that the ray OP rotates clockwise; thus, taking into account that counterclockwise is generally the positive direction, after time t, the angle between the ray OP and the y-axis is as follows: θ = −ωt.

For the convenience of describing Figure 4, Figure 6 and Figure 7, it is assumed that when the angle between the ray OP and the y-axis is exactly θ = −2π, the length of OP is exactly R. Substituting this assumption into Equation (1), the following equation is obtained:

$$R=b\times (-2\pi )$$

(2)

From Equation (2), we get the expression of b:

$$b=-{\displaystyle \frac{R}{2\pi }}$$

(3)

Then, the Archimedes spiral equation in the polar coordinate system, i.e., the coordinate of point P(r, θ) is as follows:

$$r=-{\displaystyle \frac{R}{2\pi }}\theta , -2\pi \le \theta \le 0$$

(4)

By converting the polar coordinate P(r, θ) to Cartesian coordinate P(x, y), we obtain the following:

$$\left\{\begin{array}{l}x=r\sin (\theta )=b\theta \sin (\theta )=-{\displaystyle \frac{R}{2\pi }}\theta \sin (\theta )\\ y=r\cos (\theta )=b\theta \cos (\theta )=-{\displaystyle \frac{R}{2\pi }}\theta \cos (\theta )\end{array}\right.$$

(5)

That is, in Figure 7, the coordinates of point P in the Cartesian coordinate system xOy, i.e., the equation of the Archimedes spiral AS, are as follows:

$$\left\{\begin{array}{l}x=-{\displaystyle \frac{R}{2\pi }}\theta \sin (\theta )\\ y=-{\displaystyle \frac{R}{2\pi }}\theta \cos (\theta )\end{array}\right.$$

(6)

In summary, the equation of the boundary curve TO of the plane figure OPQTO, i.e., the equation of the Archimedes spiral AS (in Figure 6 and Figure 7), is given by Equation (6).

3.2.2. Equation of Curve ARC (Circle O)

As shown in Figure 6 and Figure 7, since the radius of the circle O is R, the equation of the circle O is given as follows:

$$x^2+y^2=R^2$$

(7)

While the curve ARC which forms part of the boundary of plane figure OPQTO is also part of the circle O, the equation of curve ARC is given by Equation (7).

3.3. Solving the Coordinates of Key Points on the Boundary of Plane Figure OPQTO

After determining the equations of the curve TO (i.e., AS) and the curve ARC (i.e., circle O), the next step should be to determine the coordinates of the points T and Q, which are key points on the boundary of plane figure OPQTO, using the geometric relationship between curves and straight lines (see Figure 6). No study has ever explicitly given the equations of the straight-lines OP and QT or the relationship between the straight-line QT and the curve AS (perhaps the inventor and many researchers consider this to be self-evident, but this ambiguity precisely contributes to the weak theoretical foundation of ASWBs). Therefore, this paper must first clearly state the following: (1) in Figure 6, the straight lines OP and QT are both parallel to the y-axis; and (2) the straight-line QT is tangent to curve AS at the point T. On this basis, the coordinates of point T can be found accurately.

Therefore, this section consists of 2 sub-sections: Section 3.3.1, according to the geometric relation that the curve AS is tangent to QT at the point T, establishes a differential equation, by solving this equation the coordinates of the point T can be obtained; Section 3.3.2, based on the coordinates of point T and in conjunction with the equation of the curve ARC, obtains the coordinates of point Q.

3.3.1. Finding Coordinates of Point T by Establishing Differential Equations

According to Figure 6, the coordinates of point T are denoted as (x_T, y_T) and the angle between the line OT and the y-axis is denoted as θ_T. Notice that QT is parallel to the y-axis and tangent to AS at point T, as shown in Figure 8:

According to Equation (6), the differentiations of x and y on the curve AS, dx and dy, with θ as the parameter, are given as follows:

$$\left\{\begin{array}{l}dx=-{\displaystyle \frac{R}{2\pi }}d[\theta \sin (\theta )]=-{\displaystyle \frac{R}{2\pi }}[\sin (\theta )+\theta \cos (\theta )]d\theta \\ dy=-{\displaystyle \frac{R}{2\pi }}d[\theta \cos (\theta )]=-{\displaystyle \frac{R}{2\pi }}[\cos (\theta )-\theta \sin (\theta )]d\theta \end{array}\right.$$

(8)

Notice that QT is parallel to the y-axis and tangent to AS at point T. Thus, at point T, it is obvious that dx/dy = 0, that is,

$${\displaystyle \frac{dx}{dy}}{|}_{\theta ={\theta }_T}={\displaystyle \frac{\sin (\theta )+\theta \cos (\theta )}{\cos (\theta )-\theta \sin (\theta )}}{|}_{\theta ={\theta }_T}=0$$

(9)

Collating Equation (9) yields the following equation:

$$\sin (\theta )+\theta \cos (\theta ){|}_{\theta ={\theta }_T}=0$$

(10)

Since Equation (10) has no analytical solution, using numerical methods, we obtain the following: θ_T ≈ −2.028758(rad), noting that θ_T is in radians.

Further, substituting the numerical solution of θ_T into Equation (6) yields the following:

$$\left\{\begin{array}{l}{x}_T=-{\displaystyle \frac{R}{2\pi }}{\theta }_T\sin ({\theta }_T)\approx 0.2896R\\ y_T=-{\displaystyle \frac{R}{2\pi }}{\theta }_T\cos ({\theta }_T)\approx -0.1428\mathrm{R}\end{array}\right.$$

(11)

That is, line QT is tangent to the Archimedes spiral AS at point T, and the coordinates of point T are (x_T, y_T) = (0.2896R, −0.1428R).

3.3.2. Finding Coordinates of Point Q

According to Figure 6 and Figure 8, assume that the coordinates of point Q are Q(x_Q, y_Q). Since line QT is parallel to y-axis, it is obvious that x_Q = x_T = 0.2896R.

According to Figure 6, the vertical coordinate of point Q is less than 0, and point Q is on the circumference of circle O. Based on the equation of circle O, Equation (7), the vertical coordinate of point Q, y_Q, is obtained as follows:

$$y_Q=-\sqrt{R^2-x_Q^2}=-\sqrt{R^2-x_T^2}=-\sqrt{R^2-{(0.2896R)}^2}\approx -0.9571R$$

(12)

To sum up: the straight-line QT intersects the circle O at point Q, and the coordinates of the intersection point Q are (x_Q, y_Q) = (0.2896R, −0.9571R).

3.4. Determination of the Equations of All Straight Lines

(1)

Obviously, the equation of the straight-line OP is as follows: x = 0, 0 ≤ y ≤ R; (see Figure 6);

(2)

Given that the coordinates of point Q are Q(0.2896R, −0.9571R), the coordinates of point T are T(0.2896R, −0.1428R), and the straight line QT is parallel to y-axis, the equation of the straight line QT is as follows: x = 0.2896R, −0.9571R ≤ y ≤ −0.1428R.

3.5. Summary of the Standardized Description of Plane Figure OPQTO

Based on the work in Section 3.1, Section 3.2, Section 3.3 and Section 3.4, for the plane figure OPQTO in Figure 6, assuming that the radius of circle O is R, we obtain the following complete, detailed, and standardized description and conclusion:

(1)

The basis for the generation of the ASWB surface is an irregular closed plane figure OPQTO, which is composed of the shaded area and its boundaries in Figure 6 (i.e., Figure 4).

(2)

The boundary of the irregular plane figure OPQTO in Figure 6 is formed by connecting the straight-line OP, the curve ARC (a circular arc of radius R), the straight-line QT, and the curve TO (a part of the Archimedes spiral) in a clockwise direction.

(3)

The coordinates of point O are O(0, 0).

(4)

The coordinates of point P are P(0, R).

(5)

The equation of the straight-line OP is as follows: x = 0, 0 ≤ y ≤ R.

(6)

The arc ARC is represented by the equation of circle O, i.e., Equation (7).

(7)

The coordinates of point Q are Q(0.2896R, −0.9571R).

(8)

The coordinates of point T are T(0.2896R, −0.1428R).

(9)

The equation of the straight-line QT is as follows: x = 0.2896R, −0.9571R ≤ y ≤ −0.1428R.

(10)

The curve TO is represented by the equation of Archimedes spiral AS, i.e., Equation (6).

So far, based on the radius R of the circle O, under the plane Cartesian coordinate system and the plane polar coordinate system, by utilizing the geometric relationship between the lines and curves, combined with differential equations and their numerical solutions, we have found the expressions of all geometric elements on the boundary of the irregular plane figure OPQTO (see Figure 6), including all straight lines, curves, tangent points, and intersection points, providing a comprehensive and standardized description of the irregular plane figure OPQTO, i.e., the basis for the generation of the ASWB surface.

4. Standardized Generation Principle and Process of the ASWB Surface

Regarding the generation principle of the ASWB surface, the original description quoted in Section 2 provides a brief, qualitative description with two images. However, this description is not standardized, and by reading it the reader can only have a vague understanding. Therefore, based on the original description in Section 2 and the work in Section 3, this section presents a standardized description of the generation principle of ASWB surface. The principle is firstly given as follows:

The basis for generating the ASWB surface is an irregular closed plane figure OPQTO (see Figure 6), which has been provided with a standardized description in Section 3.5. The ASWB surface is then generated by transformation of the plane figure OPQTO. This transformation is roughly divided into three stages: (1) establishing the 3D Cartesian coordinate system (see Section 4.1); (2) entire rotation of the plane figure (see Section 4.2); (3) deformation of the plane figure (see Section 4.3). And each stage consists of several steps.

Then, the corresponding generation process of the ASWB surface is described in Section 4.1, Section 4.2 and Section 4.3, respectively, in detail.

4.1. Stage 1: Establishing 3D Cartesian Coordinate System

Step 1.1: Starting from the origin O of the plane coordinate system xOy in Figure 6, establish a new coordinate axis OZ perpendicular to the xOy plane (see Figure 9).

Step 1.2: Starting from point O, establish a new coordinate axis OX, and make it coincide with the x-axis of the coordinate system xOy in Figure 6 (see Figure 9).

Step 1.3: Starting from point O, establish a new coordinate axis OY, and make it coincide with the y-axis of the coordinate system xOy in Figure 6 (see Figure 9).

Step 1.4: Based on steps 1.1, 1.2, and 1.3, a 3D Cartesian coordinate system XYZ is established with its origin point O (see Figure 9).

Step 1.5: Place and display all geometric elements in Figure 6, including the plane figure OPQTO, into the XOY plane of the 3D coordinate system XYZ (see Figure 9).

4.2. Stage 2: Entire Rotation of the Plane Figure OPQTO

Step 2.1: Cut the plane figure OPQTO along its boundary to make it independent of other geometric elements (see Figure 9).

Step 2.2: Using the OY axis of the 3D coordinate system XYZ as the rotation axis, rotate the entire plane figure OPQTO counterclockwise. There is no specific requirement for the rotation angle, as long as it is greater than 90° and less than 180° (see Figure 9).

Step 2.3: After the entire rotation of the plane figure OPQTO, the points P, Q, and T on its boundary are denoted by P’, Q’, and T’, respectively. Therefore, after the rotation, the plane figure OPQTO is denoted by OP’Q’T’O (see Figure 9).

Step 2.4: After the entire rotation of the plane figure OPQTO, the plane arc ARC on its boundary is denoted by ARC’ as shown in Figure 9.

In the above four steps of this stage, it is important to note the following:

During the rotation of the plane figure OPQTO, the relative position and distance between any two points inside it do not change, and thus after rotation, the figure OP’Q’T’O is still a plane figure, as shown in Figure 9.

After the entire rotation, point P’ coincides with point P, point Q’ does not coincide with point Q, and point T’ does not coincide with point T, as shown in Figure 9.

This stage requires the plane figure OPQTO to be rotated as a whole, and the rotation angle to be greater than 90° and less than 180°. The reason is that if this stage (see Figure 9) is not included, it is difficult for the human brain to intuitively visualize how the plane figure OPQTO in Figure 6 can be directly morphed into an ASWB surface (see Figure 5 or Figure 10).

4.3. Stage 3: Deformation of the Plane Figure OP’Q’T’O

This stage involves twisting and stretching, which ultimately transforms the plane figure OP’Q’T’O into a 3D surface, i.e., the ASWB surface.

Step 3.1: For all points within the plane figure OP’Q’T’O, first make point O fixed, and then allow only point Q’ to move actively. Except for points O and Q’, all other points in the plane figure OP’Q’T’O move passively, driven by point Q’ (see Figure 9).

Step 3.2: Point Q’ serves as the traction point, and its motion is complex, which can be decomposed into three types of simple motions: (1) counterclockwise rotation around the OZ axis; (2) movement along the positive direction of the OZ axis; (3) movement along the positive direction of the OX axis (see Figure 9).

Step 3.3: Under the traction of the complex motion of point Q’, the plane figure OP’Q’T’O undergoes two types of motion: (1) the rotation of the plane figure OP’Q’T’O as a whole around point O; and (2) the deformation of the plane figure OP’Q’T’O itself, that is, the relative motion between any two points within it, ultimately causing the plane figure to become a 3D curved surface (see Figure 9).

Step 3.4: Point Q’ in Figure 9 eventually moves to a point on the OZ axis, which is denoted by Q″, but its specific location is variable.

Step 3.5: Point P’ in Figure 9 eventually moves to point P″, and the location of point P″ is variable (see Figure 10). However, it is certain that in the existing literature, OP″ is generally a straight line lying in the plane YOZ. Therefore, the angle <P″OZ> between OP″ and the OZ axis is variable (see Figure 10). Many of the literature sources suggest that when angle <P″OZ> is 60°, the efficiency of the ASWB blade is highest.

Step 3.6: Point T’ in Figure 9 eventually moves to point T″, but the position of point T″ is uncertain.

Step 3.7: The plane curve OT’ in Figure 9 is transformed into the 3D complex curve OT″ in Figure 10, and the shape of OT″ is uncertain.

Step 3.8: The plane straight line T’Q’ in Figure 9 is transformed into the 3D complex curve T″Q″ in Figure 10, and the shape of T″Q″ is uncertain.

Step 3.9: After deformation, the plane arc ARC’ in Figure 9 becomes the 3D curve ARC″ in Figure 10. It is particularly important to note that the projection of ARC″ on the XOY plane is exactly the Archimedes spiral AS in the xOy plane in Figure 6. This projection relationship is obvious in Figure 10.

In summary, through steps 3.1–3.9, the plane figure OP’Q’T’O (in Figure 9) is transformed into the 3D surface OP″Q″T″O (in Figure 10), colored yellow, which is the surface of ASWB.

5. Discussion

Section 4 describes the entire process of transforming the plane figure OP’Q’T’O (see Figure 9) into the ASWB surface (see Figure 10). There are some influencing factors in this process which ultimately lead to the fact that the ASWB surface has an infinite number of shapes. The influencing factors can be classified into two categories, uncertain and deterministic, which are explained below, respectively.

5.1. Uncertain Factors Influencing ASWB Surface Shape

The uncertain factors can be summarized as follows:

(1)

Point Q’ in Figure 9 eventually falls to point Q″ in Figure 10. Point Q″ is on the OZ axis, but the coordinates of point Q″ are variable.

(2)

Point P’ in Figure 9 eventually falls to point P″ in Figure 10. Point P″ lies within the YOZ plane, but the angle <P″OZ> is variable, meaning that the coordinates of point P″ are variable.

(3)

Point T’ in Figure 9 eventually falls to point T″ in Figure 10, but the coordinates of point T″ are uncertain.

(4)

Since the coordinates of the above points are uncertain, the number of shapes of the ASWB surface is infinite.

5.2. Deterministic Factors Influencing the ASWB Surface Shape

The deterministic factors can be summarized as follows:

(1)

Point O does not move before and after the deformation of the plane figure OP’Q’T’O.

(2)

After the deformation, the plane arc ARC’ in Figure 9 becomes the 3D curve ARC″ in Figure 10, and the projection of ARC″ on the XOY plane is exactly the Archimedes spiral AS on the original xOy plane in Figure 6. This projection relationship (see Figure 10) is believed to be the consensus of all relevant researchers, including the inventor of the ASWB, but this consensus has not been formally stated to date.

(3)

According to the previous conclusion, since the projection of ARC″ on the XOY plane is exactly the Archimedes spiral AS on the original xOy plane (see Figure 6), the 3D curve ARC″ is a 3D Archimedes spiral (see Figure 10).

5.3. Characteristics of the ASWB Surface

Based on the above summary of uncertain and deterministic factors influencing the shape of the ASWB surface, the following characteristics of the ASWB surface are proposed:

(1)

During the entire process of transforming the plane figure OPQTO into the ASWB surface, there are some uncertainties, thus the final shape of the generated ASWB surface is uncertain. Therefore, the first feature of the ASWB surface is the following: the ASWB surface is a general term for a class of surfaces composed of an infinite number of similar surfaces.

(2)

The second feature of the ASWB surface is the following: the boundary of the ASWB surface consists of one straight line and three 3D curves.

(3)

The curve ARC″ (see Figure 10) is a 3D Archimedes spiral, and ARC″ is also a boundary curve of the ASWB surface. Therefore, the third important feature of the ASWB surface is the following: among the curves that constitute the boundary of the ASWB surface, one of them must be a 3D Archimedes spiral.

5.4. Current Modeling Approaches for ASWBs

Since nearly all published papers directly present the final ASWR model (where one ASWR consists of three ASWBs), we briefly introduce the currently widely adopted modeling process for ASWBs using SolidWorks 2020 SP5 as an example, as detailed video tutorials on the modeling process are widely available online.

In SolidWorks, the 3D model of the ASWB’s surface is generated based on three boundary curves, as shown in Figure 11.

The first boundary curve consists of two straight line segments, OA and AB; the second boundary curve consists of a single straight-line segment, OC; the third boundary curve consists of a single 3D helical curve BC with continuously varying pitch. However, it is worth noting that its projection onto the XOY plane can be easily proven to be a mathematically strict 2D Archimedes spiral; therefore, the proof process is not presented here. (see Figure 4, Figure 7, Figure 9 and Figure 10).

The ASWB surface is generated in SolidWorks using the “Boundary Surface” command by successively clicking on the first, second, and third boundary curves, as shown in Figure 12.

Using the “thicken” command in SolidWorks and entering the desired thickness value, the ASWB surface can be converted into a solid model of an ASWB, as shown in Figure 1.

Using the “circular pattern” command in SolidWorks, the ASWR model can be obtained, as shown in Figure 2.

It should be noted that some well-known geometric features of ASWBs (or ASWRs) are precisely defined by the aforementioned software operation workflow. For example, the length of the ASWB (or ASWR) is just the length of AC; the radius R of the ASWR is just the length of OB; the opening angle of the ASWR is just the angle <BAC>; and the aspect ratio is just the ratio of lengths of AC to OB.

Although these geometric features are closely related to the aerodynamic performance of the ASWB, the primary objective of this study is to ultimately explore new manufacturing methods for the ASWB. Therefore, the relationship between these geometric features and the aerodynamic performance of the ASWB is not further discussed in this paper.

5.5. Conjectures on the Equation of the ASWB Surface

As noted in Section 5.3, one of the boundary curves of the ASWB surface is a 3D Archimedes spiral. Consequently, we are naturally led to further conjecture that there may exist additional 3D Archimedes spirals on the ASWB surface.

Due to the author’s insufficient mathematical background, we were unable to derive a single surface equation capable of fully describing the ASWB surface. Consequently, we propose a conjecture that is more suitable for engineering applications: namely, that the ASWB surface may be approximated by an arbitrarily large number of 3D Archimedes spirals. After extensive experimentation and parameter tuning, we present a simple and illustrative example based on eight 3D Archimedes spirals.

As shown in Figure 13, we display these eight curves simultaneously in the same coordinate system and provide four different viewpoints.

Due to intellectual property considerations, this paper can only provide the parametric equations for three of these curves.

The parametric equations for the first curve are given by Equation (13), those for the fourth curve by Equation (14), and those for the eighth curve by Equation (15).

$$\left.\begin{array}{c}x={\displaystyle \frac{150}{2\pi }}\theta \cos \left(\theta \right)\\ y={\displaystyle \frac{150}{2\pi }}\theta \sin \left(\theta \right)\\ z={\displaystyle \frac{150}{2\pi }}\theta \end{array}\right\}0\le \theta \le 2\pi$$

(13)

$$\left.\begin{array}{c}x={\displaystyle \frac{90}{2\pi }}\theta \cos \left(\theta \right)\\ y={\displaystyle \frac{90}{2\pi }}\theta \sin \left(\theta \right)\\ z=-{\displaystyle \frac{60·0.42-150}{2\pi }}\theta +60\end{array}\right\}0\le \theta \le 2\pi$$

(14)

$$\left.\begin{array}{c}x={\displaystyle \frac{10}{2\pi }}\theta \cos \left(\theta \right)\\ y={\displaystyle \frac{10}{2\pi }}\theta \sin \left(\theta \right)\\ z=-{\displaystyle \frac{140\bullet 0.42-150}{2\pi }}\theta +140\end{array}\right\}0\le \theta \le 2\pi$$

(15)

In the above equations, θ is the parameter, and its definition has already been provided in Section 3.2.1 (see Figure 7). The numerical values 150, 90, and 10 appearing in the equations represent different specific values of R. The definition of R is given in Figure 7 (note that this R corresponds to the distance between points O and B in Figure 11).

In the above equations, x, y, and z are all functions of parameter θ. The forms of x and y are governed by the definition of the 2D Archimedes spiral (see Figure 7), whereas z remains unrestricted. Although z is expressed as a linear function of θ here, alternative forms for z may offer better aerodynamic performance. For example, expressing z as z = f (θ²), or z = f (e^θ), and so on.

The other numerical values appearing in the above equations are derived from experimentation and practical experience; therefore, they are not elaborated upon further.

According to the parametric equations, in SolidWorks, eight curves were generated, and a surface was subsequently generated using the “Boundary Surface” command based on these curves. As shown in Figure 14.

By observing the two surfaces shown in Figure 12 and Figure 14, we find that they are very similar. In fact, through extensive experimentation (including inserting additional curves between the first and eight curves) and performing extensive parameter tuning, we have made the shapes of the two surfaces extremely close to each other. Nevertheless, quantitatively computing their similarity using numerical methods is currently not meaningful, for reasons to be detailed in Section 5.6.

5.6. On the Definition of ASWB

Since none of the published literature had identified what truly constitutes an ASWB surface, nor has any paper demonstrated that the surface generated using the method described in Section 5.4 is indeed the genuine ASWB surface, there is currently no necessity to quantitatively compare the two surfaces shown in Figure 12 and Figure 14.

In other words, the published literature does not provide an accurate definition of an ASWB.

Obviously, a precise definition of the ASWB should be based on a standardized description of the generation principle of the ASWB surface, or on an exact mathematical description of its shape. Only pure mathematical language (independent of any commercial computer-aided design software) can accurately define the ASWB.

However, two observations currently hold: (1) existing descriptions of the generation principle of the ASWB surface remain vague; (2) no mathematical equation has yet been identified that comprehensively describes the ASWB surface.

Based on these observations, this paper hereby suggests the following: to date, a clear and exact definition of ASWB appears to be absent from the published literature.

5.7. Demonstration of an Alternative Manufacturing Method for ASWB Based on Parametric Equations

In Section 5.5, we proposed a conjecture that the surface of the ASWB can be approximated by an infinite number of 3D Archimedes spirals. This conjecture naturally suggests a novel and alternative manufacturing approach for ASWBs: by using a finite number of 3D Archimedes spirals as the structural skeleton and then covering this skeleton with a lightweight and compliant skin, the ASWB blade can be fabricated in a manner similar to making a kite.

Therefore, based on the conjecture proposed in Section 5.5, we demonstrate this kite-like manufacturing scheme as follows:

Step 1: Using the parametric Equations (13)–(15), three 3D Archimedes spirals are plotted; specifically, the first, fourth, and eighth curves shown in Figure 13;

Step 2: In SolidWorks, these curves are converted into solid bodies, generating three spiral-shaped skeletons. The centerline of each skeleton corresponds exactly to curve 1, curve 4, and curve 8 shown in Figure 13, respectively. These skeletons are named Skeleton_1, Skeleton_4, and Skeleton_8, as shown in Figure 15.

Step 3: A complete skeleton of one ASWR is assembled based on three pieces of Skeleton_1, three pieces of Skeleton_4, three pieces of Skeleton_8, one rotating shaft, and three straight connecting rods, as shown in Figure 16.

Step 4: Fabricate all the parts involved in step 3 individually using 3D printing technology. Then, assemble these parts into a complete ASWR skeleton. Subsequently, cover the skeleton with a soft plastic film, thereby completing the fabrication of the physical ASWR prototype, as illustrated in Figure 17.

A preliminary principal validation test was performed on the physical ASWR prototype shown in Figure 17. Only qualitative conclusions can be obtained at this stage: compared to the ASWR produced entirely via 3D printing, this kite-like ASWR exhibits three clear advantages: (1) lower manufacturing cost; (2) lighter overall weight; (3) superior ease of rotation under identical wind speeds.

Naturally, while the proposed manufacturing method demonstrated clear advantages, it has limitations. For example,

(1)

The fabrication of the spiral-shaped skeleton still relies on 3D printing technology, and the size of the skeleton is constrained by the dimensions of the 3D printer. Therefore, producing a large-scale ASWR is currently not feasible;

(2)

The cost of fabricating a spiral-shaped skeleton using 3D printing technology remains high;

(3)

The shape of the ASWB based on the spiral-shaped skeletons is an approximation of the genuine ASWB shape, resulting in non-optimal aerodynamic performance.

Nevertheless, despite the trade-off in blade shape precision, the proposed method holds promise for several breakthroughs in the future:

(1)

The difficulties associated with fabricating complex curved thin-walled blades are bypassed by redirecting the key manufacturing focus to the production of spiral-shaped skeletons;

(2)

Alternative technologies beyond 3D printing, such as 3D freeform tube bending technology, can offer low-cost and high-efficiency manufacturing of large-scale spiral-shaped skeletons, thereby making low-cost and high-efficiency production of large-sized ASWBs or ASWRs feasible;

(3)

By further selecting or developing materials for both the skeleton and the covering skin, the realization of lightweight, high-strength ASWBs becomes practical.

6. Conclusions

The ASWB is a novel wind blade with promising performance and significant academic research value. However, compared to traditional wind turbine blades, the development history of the ASWB is far too short. As a result, many researchers have not yet become aware of the ASWB, leading to the current existence of three fundamental, yet unresolved theoretical issues in its research field:

(1)

The existing description of the generation principle of the ASWB surface is qualitative;

(2)

In the published literature, no mathematical equations were found that attempt to fully describe the surface of the ASWB;

(3)

A clear and exact definition of the ASWB appears to be absent from the published literature.

For the first problem, based on the original qualitative description given by the inventor, this paper uses geometric and mathematical tools to provide a standardized and academic explanation of the generation principle of the ASWB surface.

For the second problem, this paper proposes a conjecture: the ASWB surface may be approximated by an arbitrarily large number of 3D Archimedes spirals, along with a simple and illustrative example based on eight 3D Archimedes spirals and corresponding parametric equations.

For the third problem, this paper can currently only suggest that the precise definition of the ASWB should be based on a standardized description of the generation principle of the ASWB surface, or, alternatively, on an accurate geometric description of the shape of the ASWB surface. Only pure mathematical language (independent of any commercial CAD software) can accurately define the ASWB.

We believe that ultimately arriving at a widely accepted definition of the ASWB will require long-term collaborative efforts from more researchers across various countries. The work presented makes a modest and preliminary contribution toward that objective.