Trailing edge attachment

Abstract

And a trailing edge additional device. The invention describes a wind turbine rotor blade attachment (1) comprising a plurality of serrated teeth (10) arranged to extend outwardly from a trailing edge (20TE) of a rotor blade (20), where the serrated teeth (10) have an elongated shape defined by a base (10B) and two at least partially curved side edges (10S) converging at a tip (10T), and where the serrated teeth (10) are arranged to extend outwardly from the trailing edge (20TE) of the rotor blade (20). The length of the serrated tooth (10) from the base (10B) to the tip (10T) is at least 6 times the width of the serrated tooth (10) at its base (10B); the side edge (10S) of the serrated tooth (10) comprises a convex curved portion; and the shortest distance (1G) between adjacent serrated teeth (10) is at most 1 mm. The invention also describes a wind turbine rotor blade (20) comprising a plurality of such additional devices (1) mounted to the trailing edge (20TE) of the airfoil portion of the rotor blade (20).

Description

Background Technology

[0001] Acoustic noise emissions are a key parameter for countries approving the installation of wind turbines. Modern wind turbines can have very large aerodynamic rotors (with diameters of approximately 160 m or more), and the long rotor blades are a significant source of acoustic noise emissions. Acoustic noise from wind turbines is primarily caused by their rotor blades and is generated when air moving above the pressure and suction sides of the rotor blades recombines beyond the trailing edge.

[0002] The essentially straight trailing edge of the rotor blades will efficiently scatter sound waves to an observer near the wind turbine, who will perceive the scattered sound waves as "trailing edge noise." This trailing edge noise is generated when boundary layer turbulence crosses the straight trailing edge and scatters the sound, causing sound pressure fluctuations that can propagate over long distances and may make it difficult to meet noise specifications at the receiving point.

[0003] Trailing-edge noise can be mitigated to some extent by attaching a serrated appendage along a portion of the outer region of the rotor blade. The established trailing-edge appendage design includes a row of triangular "teeth" designed to reduce the radiation efficiency of noise generated at certain points along the blade span. The serrated teeth of this prior art appendage are triangular in shape and are defined by a base and two straight sides converging at the tip or apex of the triangle. This type of serrated trailing edge reduces radiative emissions through several mechanisms: at low frequencies, the corresponding turbulent vortices are large, and the serrations act as an intermediary between the solid surface of the airfoil and the wake, effectively mixing the acoustic impedance (the ratio of sound pressure to the corresponding volume velocity through the surface; solid surfaces have very high impedance, while fluids such as air have very low impedance) at the airfoil edge and reducing scattering intensity; at higher frequencies, the serrations create an angle of incidence between the convective turbulence and the airfoil edge, which reduces the radiative efficiency of the scattered sound waves; in another mode, the serrations induce destructive interference between the noise scattered at the base of each serration and the noise scattered along its straight sides. This "serrated" addition will first scatter from the base of each triangular tooth, then along the straight edges of each triangular tooth, and finally at the apex or tip of each triangle. The base-to-tip time delay induces some destructive interference in the sound field. However, the noise reduction effect of the serrated trailing edge is known to be limited. Therefore, if additional noise reduction is required to obtain permission to install wind turbines, wind turbine operators may have to reduce the maximum aerodynamic rotor speed of the wind turbines. However, this comes at the cost of a reduction in the annual power generation (AEP) of the wind turbines, and is therefore not a satisfactory solution.

[0004] Therefore, the object of the present invention is to provide an auxiliary device for wind turbine rotor blades with improved noise reduction performance.

[0005] This objective is achieved by the stated wind turbine rotor blade attachment and the stated wind turbine rotor blade. Summary of the Invention

[0006] According to the invention, the rotor blade attachment includes a plurality of serrated teeth arranged to extend outward (i.e., in the downstream direction) from the trailing edge of the rotor blade. The serrated teeth have an elongated shape defined by a base and two side edges converging at a tip.

[0007] The additional device of the present invention is characterized in that: the length of the serrated tooth from its base to its tip is at least 6 (six) times the width of the serrated tooth; the side edge of each serrated tooth exhibits a convex edge curvature; and the distance from the base of one serrated tooth to the base of the adjacent serrated tooth is at most 1 mm.

[0008] The geometry of the serrated teeth can be defined by three points: the two outer points of its base and the point where the side edges converge. These three points also define a smaller "basic" triangle, that is, a triangle with a smaller surface area that is substantially contained within the serrated teeth. In the context of this invention, the raised edges of the serrated teeth of the auxiliary device lie outside the straight edges of the basic triangle, as will become clear from the figures.

[0009] The performance of the attachment should be understood as its performance in the context of acoustic noise emissions, i.e., the effect of the shape and form of the attachment on trailing edge noise. As explained above, trailing edge noise is acoustic noise caused by the behavior of turbulence at and beyond the trailing edge of the rotor blade. The rotor blade attachment of the present invention has been shown to result in a surprisingly significant reduction in this trailing edge acoustic noise. It has been demonstrated (e.g., in wind tunnel testing) that the performance of the wind turbine rotor blade attachment of the present invention surpasses that of equivalent attachments with "triangular" or serrated edges as known from the prior art.

[0010] For any given downstream sawtooth tooth length (measured from the midpoint of the base of the sawtooth to its tip), maintaining a high aspect ratio of at least 6:1 for the add-on effectively reduces the spanwise width of any sawtooth tooth (measured across its base), thereby reducing the likelihood of destructive interference of noise radiated along the edges of the sawtooth. The high aspect ratio of each sawtooth tooth of the add-on ensures that relatively small-scale turbulence is also affected by the acoustic impedance mixing of the add-on, thus making the sawtooth effective at higher frequencies as well. The advantageous shape of the sawtooth teeth of the add-on reduces the angle of incidence of turbulence convection through the edges of the teeth and ensures that turbulence is significantly correlated across the width of the sawtooth.

[0011] The very small gap between the convex edge curvature of the sawtooth teeth and the outer corners of adjacent sawtooth teeth optimizes noise reduction. Appropriate selection of the sawtooth tooth geometry ensures that turbulence scatters acoustic energy as inefficiently as possible, thereby reducing the noise observed at the relevant receiving point. The curvature of the sawtooth edges allows for a balanced combination of the noise reduction mechanisms listed above—namely, the mixing of acoustic impedances, the incident angle of turbulence across the edges, and destructive interference at nearby source locations. This combination of geometric features produces acoustic noise reduction at critical frequencies such as 500 Hz to 1500 Hz.

[0012] The wind turbine rotor blades of the present invention have a root portion and an airfoil portion, and can have a length exceeding 80 m. The rotor blades also include at least one instance of the additional device of the present invention mounted to the trailing edge of the airfoil portion of the rotor blade. The wind turbine typically has three rotor blades, and each rotor blade may be equipped with substantially the same arrangement of one or more additional devices along a portion of its trailing edge.

[0013] Particularly advantageous embodiments and features of the invention are given by the dependent claims, as disclosed in the following description. Features from different claim classes may be combined as appropriate to give other embodiments not described herein.

[0014] In the following description, without limiting the invention in any way, it can be assumed that the attachment includes a mounting strip or band to facilitate the attachment of the attachment to the trailing edge of a wind turbine rotor blade. A row of serrated teeth extends downstream from this mounting strip. The bottom edge of the serrated teeth should be assumed to lie along a continuous line, referred to herein as the baseline. The mounting strip can be used to directly attach the attachment to the trailing edge of the wind turbine rotor blade. Alternatively, the mounting strip can be used to attach the attachment to a separate mounting device, which in turn can be attached to the trailing edge of the rotor blade.

[0015] The attachment of this invention can be manufactured as a single piece comprising a row of serrated teeth and a mounting strip. For example, the attachment can be manufactured using injection molding, laser cutting, vacuum forming, additive manufacturing techniques such as 3D printing, subtractive manufacturing techniques such as milling, stamping, and so on. The attachment can be made of suitable materials such as hard and durable plastics (thermoplastics), reinforced composites, wood, vulcanized materials, metals or metal alloys, or combinations thereof, and may have a surface coating.

[0016] The advantageous performance of the additional device of the present invention is partly due to the specific shape of the serrated teeth. The side edges of the serrated teeth exhibit a convex curvature over at least some length between the base and the tip of the serrated teeth, that is, the serrated teeth are essentially "iron-shaped", having a shape similar to the iron plate of a clothing iron.

[0017] The side edges of the serrated teeth may include curved portions and straight portions, for example, the straight portion beginning at the base and transitioning to the curved portion, which terminates at a tip. The point where the curved portion transitions to the straight portion is preferably chosen such that the serrated tooth has a convex curvature over most of its length. The serration may also include, for example, a recessed portion, having an inflection point in the curvature of the edge between the convex or straight portion and the recessed portion.

[0018] Alternatively, the side edges of the serrated teeth are curved along their entire length. In such embodiments, the side edges of the serrated teeth preferably take the form of a conical arc or an arched line. The conical arc can have the shape of a tangential arched line, i.e., a line that begins at the baseline and is tangent to the side edge is perpendicular to the baseline (the side edge intersects the baseline at a 90° angle).

[0019] Similarly, a conical arc can have the shape of a secant arch, that is, a line that begins at the baseline and is tangent to the side edge at an angle relative to the normal. Intersecting the baseline (the side edge intersects the baseline at an angle of less than 90°). In a particularly preferred embodiment of the invention, the side edge of the arched linear sawtooth tooth intersects the baseline at an angle of up to 4.76° relative to the normal, more preferably at an angle of up to 2° relative to the normal.

[0020] The angle at which the lateral edges of the arched linear sawtooth teeth intersect the baseline can be expressed using terms related to the basic triangle formed by the sawtooth teeth (i.e., the triangle formed between the three outer points of the sawtooth teeth). For example, the angle δ between the major axis of the basic triangle and the hypotenuse (the straight lateral edge) of the basic triangle is expressed as... (1) Where W is the width of the base of the triangle, and L is measured from the midpoint of the base of the triangle to its vertex. The angle θ at which the side edge of the arched linear sawtooth intersects the baseline is preferably less than two-thirds (2 / 3) of the angle δ, more preferably less than one-third (1 / 3) of the angle δ. In the case of a tangential arched line shape, the angle θ is reduced to 0°.

[0021] The overall shape of the serrated teeth of the auxiliary device of the present invention can also be defined according to its aspect ratio. The serrated teeth of the auxiliary device of the present invention have a relatively high aspect ratio, that is, their length is significantly longer than their width. The minimum aspect ratio of the serrated teeth of the auxiliary device of the present invention is 6:1, and can be as large as 12:1. The aspect ratio of the serrated teeth of the auxiliary device of the present invention can be expressed as... (2) In other words, in a preferred embodiment of the invention, the length of the serrated tooth is at least six times and at most twelve times the width of the serrated tooth. Several experiments and simulations have been performed to determine the appropriate dimensions of the serrated teeth for the additional device of the invention. For example, it has been determined that the length of the serrated tooth from its base to its tip should preferably not be less than 30 mm and should not exceed 200 mm. Similarly, it has been determined that the width of the serrated tooth at its base should preferably not be less than 3 mm and should not exceed 33 mm.

[0022] The performance improvement of the additional device of the present invention (i.e., the reduction of acoustic noise caused by the rotor blades) is due to the aforementioned advantageous shape (high aspect ratio and convex curvature along the side edges) and the narrow spacing between adjacent sawtooth teeth of the additional device. Any two adjacent sawtooth teeth are separated by a gap of at most 1 mm. This is the distance between the outer corner of one sawtooth tooth and the outer corner of its neighboring sawtooth tooth. This distance is preferably at most 0.5 mm.

[0023] The serrations can be substantially flat, meaning the thickness of the serrated teeth can be substantially constant over its entire area. In a preferred embodiment of the invention, to increase the aerodynamic efficiency of the attachment and improve its structural strength, the thickness of the serrated teeth can decrease towards the edges, i.e., the thickest portion of the serrated teeth faces the center. In a preferred embodiment of the invention, the thickness of the serrated teeth is at most 80% of its width; for example, a serrated tooth with a width of 50 mm can have a thickness of 40 mm or less.

[0024] The serrated teeth can be symmetrical about their longitudinal axis, for example, this axis being at a right angle and intersecting the baseline at the midpoint of the base. However, alternative shapes of the serrated teeth are possible, for example, the longitudinal axis of the serrated teeth can intersect the baseline at an angle relative to the normal, and / or the longitudinal axis of the serrated teeth can intersect the baseline at points other than the midpoint.

[0025] The attachment of this invention can be attached to the rotor blades without affecting the lift of the airfoil at its operating angle of attack. This can be referred to as the "zero-lift" position of the attachment, as the attachment substantially does not affect the lift of the airfoil at this position. Alternatively, in another preferred embodiment of the invention, the attachment can be attached to the rotor blades at an angle relative to the zero-lift position toward the pressure side of the airfoil, thereby also increasing the lift on the airfoil.

[0026] The additional devices of this invention can be mounted to the rotor blades at any suitable portion of their length. Since acoustic noise has been observed to occur primarily in the downwind direction of the outer rotor blade portions, one or more additional devices are preferably mounted to the trailing edge within the outermost 30% of the rotor blade's spanwise length. These additional devices can be placed adjacent to each other, or spaced apart, depending on the situation. Attached Figure Description

[0027] Other objects and features of the invention will become apparent from the following detailed description taken in conjunction with the accompanying drawings. However, it should be understood that the drawings are designed for illustrative purposes only and are not intended to define limitations of the invention.

[0028] Figure 1 An exemplary embodiment of the additional device of the present invention is shown in place along the trailing edge of a wind turbine rotor blade; Figure 2 – 7 illustrates an exemplary implementation of the additional device of the present invention; Figure 8 A cross-section of an exemplary serrated tooth passing through an additional device of the present invention is shown; Figure 9 and Figure 10 This illustrates a possible attachment mode of the present invention to a wind turbine rotor blade; Figure 11 Showing multiple such Figure 1 The rotor blades shown are from a wind turbine. Figure 12 Showing wind turbine rotor blades with existing technology add-ons.

[0029] In the accompanying drawings, similar reference numerals always refer to similar objects. Objects in the accompanying drawings are not necessarily drawn to scale. Detailed Implementation

[0030] Figure 1 Showing along the trailing edge 20 of the wind turbine rotor blade 20 TE An exemplary embodiment of the additional device 1 of the present invention in a suitable position. Here, a plurality of additional devices 1 are located along the trailing edge 20 of the rotor blade 20. TEInstallation. Each attachment 1 includes a plurality of serrated teeth 10 extending outward from an elongated strip 11 (for attaching the attachment 1 to the rotor blades). Unlike the serrated shape of prior art attachments, the serrated teeth 10 of the attachment 1 of the present invention have a shape that can be described as a “flat iron” shape, resembling the soleplate of a clothing iron, with a high aspect ratio of at least 6:1. The enlarged portion of the figure shows that adjacent serrated teeth 10 are separated by a small gap 1G of no more than 1 mm, preferably no more than 0.5 mm. The combined effect of the high aspect ratio, the convex curvature along the side edges, and the narrow gap 1G between adjacent serrated teeth 10 is a significant reduction in trailing edge noise of the wind turbine. Experiments have shown that the wind turbine noise perceived at the receiving point decreases by more than 3 dB at the relevant frequency.

[0031] Figure 2 – 7 illustrates a possible geometry of the serrated teeth 10, an example of the additional device 1 of the present invention. Each serrated tooth 10 is defined by two sides 10S extending from an outer point of its bottom edge 10B or base and converging at its tip 10T. The base or “root” of the serrated tooth can have any form, such as a straight line extending substantially parallel to the trailing edge of the airfoil. Similarly, the base of the serrated tooth can be curved or angled.

[0032] The serrated teeth 10 extend outward from the long strip 11 or mounting strip, and the bottom edge 10B of the row of serrated teeth 10 sits along a common baseline 11B. The baseline 11B essentially defines one long edge of the mounting strip 11.

[0033] exist Figure 2 In this additional device 1, the serrated teeth 10 have a length L of 100 mm and a width W of 10 mm, such that the aspect ratio of the serrated teeth 10 is 10:1. Figure 3 In the figure, the serrated tooth 10 has a length of 140 mm and a width of 20 mm, such that the aspect ratio of the serrated tooth 10 of this additional device 1 is 7:1. Each figure also indicates a smaller basic isosceles triangle T (dashed line), which is contained within the serrated tooth 10 and is defined by the same three vertices. The raised side edges 10S of the serrated tooth lie outside the straight side edges of the triangle T.

[0034] exist Figure 4 In this embodiment, the serrated tooth 10 has a conical arc shape, which in this case is a secant arch shape. In such an embodiment, the side edge 10S of the serrated tooth 10 intersects the baseline 11B at a non-zero angle relative to the normal 11N, where this angle θ is at most 4.76°. In the embodiment shown here, the angle θ is approximately 1°.

[0035] As explained above, the angle at which the side edge of the arched linear sawtooth tooth 10 intersects the baseline 11B can be expressed using terms related to the geometry of the basic triangle T formed by the sawtooth teeth (i.e., the triangle T formed between the three external points of the sawtooth tooth 10). The angle δ between the major axis of the sawtooth tooth 10 and the hypotenuse (straight side edge) of the basic triangle T is given in equation (1) above. The angle θ at which the side edge of the arched linear sawtooth tooth intersects the baseline is preferably less than 2 / 3 δ, more preferably less than 1 / 3 δ.

[0036] exist Figure 5 In this embodiment, the serrated tooth 10 has a conical arc shape, which in this case is a tangential arch shape. In such an embodiment, the side edge 10S of the serrated tooth 10 intersects the baseline 11B at a right angle; that is, the tangent line tangent to the side edge 10S and originating from the baseline 11B coincides with the normal 11N. Another way to describe this shape is that the angle θ between the side edge 10S and the normal decreases to 0° at the intersection of the side edge and the baseline 11B.

[0037] In the above figure, the serrated teeth 10 of the auxiliary device 1 are basically symmetrical about its major axis 10L, that is, the major axis 10L is at the midpoint 10B of the base 10B. mid It intersects with the base 10B at this point. Figure 1 In – 5, the major axis 10L of the serrated tooth 10 is perpendicular to the baseline 10B. In such a case... Figure 6 In the alternative embodiment shown, if the major axis 10L of the serrated tooth 10 is at its midpoint 10B mid If the offset point intersects with the base 10B, then the sawtooth tooth 10 can be asymmetrical. In cases such as... Figure 7 In another alternative embodiment shown, the serrated teeth 10 may exhibit asymmetry if the major axis 10L of the serrated teeth 10 intersects the baseline 11B at an angle β offset from the normal (indicated by the dashed line). The external protrusion shape of the side edges of the serrated teeth 10 is adjusted accordingly.

[0038] In an alternative embodiment, the adjacent sawtooth teeth 10 of the additional device 1 of the present invention may vary within the ranges described above, with respect to parameters θ, L, W, and β.

[0039] exist Figure 2 – In any of the embodiments shown in 7, it can be assumed that the gap between adjacent sawtooth teeth 10 of the auxiliary device 1 is at most 1 mm.

[0040] Figure 8A cross-section of the serrated tooth 10 through an embodiment of the additional device 1 of the present invention is shown. Here, the thickness of the serrated tooth 10 decreases from its maximum thickness at the center toward its minimum thickness along the side edge 10S. The figure shows the thickness h at a certain point along the length of the serrated tooth 10. x and width w x For example, a serrated tooth 10 with a width of 10 mm at its base can have a maximum thickness of 3 mm at its center, tapering to 0.2 mm at its outer edge 10S. Width w x With height h x This ratio can be substantially constant along the length of the serrated tooth 10. The cross-sectional shape of the serrated tooth 10 can be substantially elliptical, as shown here (the serrated tooth has a substantially uninterrupted curved surface). Alternatively, the cross-sectional shape can be substantially polygonal (the serrated tooth has several flat surfaces that meet at various edges).

[0041] Figure 9 and Figure 10 This illustrates an alternative method of mounting the attachment 1 to the airfoil 20A of the wind turbine rotor blades. Figure 9 In this configuration, the attachment 1 is attached such that it is essentially positioned in a "zero-lift" position 20Z. When installed in this manner, the attachment does not contribute lift to the airfoil 20A. Alternatively, as... Figure 10 As shown, the attachment 1 can be attached to the rotor blade at an angle tilted toward the pressure side of the airfoil 20A. Here, the attachment 1 is tilted at an angle of about 3° relative to the zero lift position 20Z, so that it can contribute to the lift on the airfoil 20A.

[0042] Figure 11 A wind turbine 2 is shown with an aerodynamic rotor comprising three rotor blades 20. Each rotor blade 20 is positioned along its trailing edge 20. TE Arrangements of one or more embodiments of the additional device 1 of the present invention. The additional device 1 is placed in the outer region of the rotor blade 20, as shown here.

[0043] Figure 12 A wind turbine rotor blade with prior art add-on 3 is shown. Here, each serrated tooth 30 is triangular in shape, that is, each serrated tooth 30 has the shape of an isosceles triangle. Adjacent serrated teeth meet directly at the baseline, that is, there is no gap between adjacent teeth. The aspect ratio of the serrated teeth 30 of the prior art add-on 3 is generally no greater than 4:1. With the help of these design aspects, the prior art add-on functions as explained in the description.

[0044] While the invention has been disclosed in the form of preferred embodiments and variations thereof, it will be understood that many additional modifications and variations can be made thereto without departing from the scope of the invention. For example, one or more additional devices can be incorporated during the manufacture of the rotor blade. Of course, any existing wind turbine rotor blade can be upgraded by replacing the existing “serrated” trailing edge additional devices with a suitable number of the additional devices of the present invention. Furthermore, while the rotor blade can be equipped with several identical additional devices, each having the same serrated teeth, it is equally possible to equip the rotor blade with additional devices having different serrated tooth geometries while maintaining a favorable high aspect ratio; for example, the length or width of the serrated teeth of the additional devices can vary, while each has the same high aspect ratio. It should be noted that the wind turbine rotor blade can also be equipped with one or more additional devices that include one or more serrated teeth with shapes different from those described above.

[0045] For clarity, it should be understood that the use of “a” or “an” throughout this application does not exclude multiple, and “includes” does not exclude other steps or elements.

Claims

1. A wind turbine rotor blade attachment (1), comprising a plurality of attachments arranged from the trailing edge (20) of the rotor blade (20). TE ) outwardly extending serrated teeth (10), wherein, The serrated tooth (10) has an elongated shape defined by a base (10B) and two at least partially curved side edges (10S) converging at a tip (10T), and wherein, - The length of the serrated tooth (10) from its base (10B) to its tip (10T) is at least 6 times the width of the serrated tooth (10) at its base (10B); - The side edges (10S) of the serrated teeth (10) include raised curved portions; and - The closest distance (1G) between adjacent sawtooth teeth (10) is at most 1 mm.

2. The additional device according to the preceding claim, wherein, The length (L) of the serrated tooth (10) is at most 12 times the width (W) of the serrated tooth (10) at its base (10B).

3. The additional device according to any one of the preceding claims, wherein, The closest distance (1G) between adjacent sawtooth teeth (10) is at most 1.0 mm, more preferably at most 0.5 mm.

4. The additional device according to any one of the preceding claims, wherein, The side edge (10S) of the serrated tooth (10) intersects its base (10B) at an angle of up to 4.76° relative to the normal, more preferably at an angle of up to 2° relative to the normal.

5. The additional device according to any one of the preceding claims, wherein, At least a portion of the side edge (10S) of the serrated tooth (10) has the form of a conical arc.

6. The additional device according to claim 5, wherein, The conical arc is a secant arch line.

7. The additional device according to claim 5, wherein, The conical arc is a tangential arch line.

8. The additional device according to any one of the preceding claims, wherein, The thickness (h) of the serrated teeth (10) x ) and width (w x The ratio is at most 0.

8.

9. The additional device according to any one of the preceding claims, wherein, The serrated teeth (10) are symmetrical about the longitudinal axis (10L).

10. The additional device according to any of the preceding claims, comprising an elongated belt (11) to facilitate mounting to a wind turbine rotor blade (20), and wherein, The serrated teeth (10) extend outward from the elongated strip (11).

11. The additional device according to any of the preceding claims, wherein it is manufactured from a thermoplastic material as an injection molded part.

12. A wind turbine rotor blade (20) comprising a root portion and an airfoil portion, and further comprising a plurality of trailing edges (20' ... TE The additional device (1) according to any one of claims 1 to 11.

13. The wind turbine rotor blade according to the preceding claim, wherein, The auxiliary device (1) is installed within the outermost 30% of the rotor blade span onto the trailing edge (20). TE ).

14. The wind turbine rotor blade according to claim 12 or claim 13, wherein, The additional device (1) is tilted toward the pressure side of the airfoil (20A) at an angle (α) of at least 0° relative to the zero lift position (20Z) of the airfoil.

15. A wind turbine (2) equipped with a plurality of rotor blades (20) according to any one of claims 12 to 14.

Images