WO2017159730A1

WO2017159730A1 - Impeller, rotary machine, and impeller manufacturing method

Info

Publication number: WO2017159730A1
Application number: PCT/JP2017/010391
Authority: WO
Inventors: 山下　修一; 伸一郎得山
Original assignee: 三菱重工業株式会社; 三菱重工コンプレッサ株式会社
Priority date: 2016-03-18
Filing date: 2017-03-15
Publication date: 2017-09-21
Also published as: JP2017172344A; EP3415767A1; EP3415767A4; US20190078583A1

Abstract

An impeller equipped with: a disc (3a) forming a discoid shape centered around an axial line; multiple blades (3b) formed in the circumferential direction of the axial line, with intervals therebetween, on the surface of the disk (3a) facing a first side in the axial line direction; and a cover (3c) enclosing these multiple blades (3b) from the first side in the axial line direction. The cover (3c) has an inner circumferential surface (11), the diameter of which decreases from the second side toward the first side in the axial line direction, and which is connected to the blades (3b), and a tip-end surface (13), which extends from the end part of the inner circumferential surface (11) on the first side in the axial line direction toward the outside in the radial direction, and faces the first side in the axial line direction. A front-edge part (20), which is the edge part of the blades (3b) on the first side in the axial line direction, extends from the boundary (K) of the inner circumferential surface (11) and the tip-end surface (13) toward the inside in the radial direction.

Description

Impeller, rotating machine, and method of manufacturing impeller

The present invention relates to an impeller, a rotating machine, and a method for manufacturing an impeller.
This application claims priority on March 18, 2016 based on Japanese Patent Application No. 2016-56045 filed in Japan, the contents of which are incorporated herein by reference.

2. Description of the Related Art As rotary machines used for industrial compressors, turbo refrigerators, small gas turbines, etc., those equipped with an impeller in which a plurality of blades are attached to a disk fixed to a rotary shaft are known. The rotating machine gives pressure energy and velocity energy to the gas by rotating the impeller.

Patent Document 1 describes a centrifugal compressor having a so-called closed impeller in which a cover is integrally attached to a blade.
In such an impeller, the cover, the blade, and the disk are not formed by the three-piece manufacturing method that is assembled after individually forming the cover, the blade, and the disk, or the cover, the blade, and the disk are formed from the beginning. There is a case where a one-piece manufacturing method of forming in an integrated state is adopted. In particular, when the working fluid is a corrosive gas, it is difficult to weld a corrosion-resistant material, and thus a one-piece manufacturing method is often employed.
This one-piece manufacturing method is also adopted when the flow path width becomes extremely narrow, such as a small-diameter impeller, and the leg length of the welded portion becomes too large with respect to the flow path width, and there is a concern about performance reliability.

JP 2015-175250 A

The closed impeller described in Patent Document 1 may require complicated machining, electrical discharge machining, or the like, particularly when trying to make a single piece. At the time of this machining, since the root of the tool is thick, the depth at which the tool can be inserted in the axial direction of the impeller between the blades is limited. For this reason, when machining the disk side or the cover side of the impeller, the tool is inserted obliquely with respect to the impeller axis line. However, a cutting tool may interfere with the impeller cover or the like, and there is a problem that skill is required for processing the impeller. In the case of electric discharge machining, it is necessary to form the discharge electrode in a complicated shape, which may increase the cost.
An object of this invention is to provide the impeller which can be processed easily, a rotary machine, and the manufacturing method of an impeller.

According to the first aspect of the present invention, the impeller includes a disk, a plurality of blades, and a cover. The disc has a disk shape centered on the axis. The plurality of blades are formed on the surface facing the first side in the axial direction of the disk at intervals in the circumferential direction of the axial line. The cover surrounds the plurality of blades from the first side in the axial direction. The cover has an inner peripheral surface and a tip surface. The inner peripheral surface is reduced in diameter toward the first side from the other side in the axial direction and connected to the blade. The distal end surface extends radially outward from the first axial end of the inner peripheral surface and faces the first axial direction. A front edge portion, which is a first edge portion in the axial direction of the blade, extends radially inward from a boundary between the inner peripheral surface and the tip surface.
By comprising in this way, it can suppress that a cover protrudes in the 1st side of an axial direction rather than the front edge of a braid | blade. Therefore, it is possible to easily perform the processing while suppressing the tool from interfering with the cover and the discharge electrode from becoming complicated in shape. In addition, the swirling flow generated immediately before the front edge of the blade can be reduced by the rotation of the cover. Therefore, the boundary layer in the vicinity of the blade leading edge can be thinned on the cover side, and the impeller performance can be improved.

According to the second aspect of the present invention, the distal end surface according to the first aspect may include a convex curved surface that is disposed on the other side in the axial direction toward the radially inner side.
By comprising in this way, interference, such as a tool, can further be suppressed.

According to the third aspect of the present invention, the edge according to the first or second aspect may be provided with a protruding edge that protrudes radially outward on the first side in the axial direction.
As described above, since the protruding edge portion is provided on the radially outer portion of the front edge portion of the blade, the inner peripheral surface and the tip end are provided via the protruding edge portion regardless of the shape of the front edge portion of the blade. The leading edge of the blade can be formed continuously from the boundary with the surface.

According to the fourth aspect of the present invention, the rotating machine includes the impeller according to any one of the first to third aspects.
With this configuration, it is possible to easily create a rotating machine, and it is possible to improve the efficiency of the rotating machine as the impeller performance is improved.

According to a fifth aspect of the present invention, an impeller manufacturing method is an impeller manufacturing method including a disk, a plurality of blades, and a cover. The disc has a disk shape centered on the axis. The plurality of blades are formed on the surface facing the first side in the axial direction of the disk at intervals in the circumferential direction of the axial line. The cover surrounds the plurality of blades from the first side in the axial direction. The cover includes an inner peripheral surface and a tip surface. The inner peripheral surface is reduced in diameter toward the first side from the other side in the axial direction and connected to the blade. The distal end surface extends radially outward from the first axial end of the inner peripheral surface and faces the first axial direction. In this impeller manufacturing method, a front edge, which is a first edge in the axial direction of the blade, is formed so as to extend radially inward from the boundary between the inner peripheral surface and the tip surface. Process.
By doing in this way, the interference of the tool to a disk can be suppressed. Therefore, the impeller can be easily manufactured.

According to the impeller, it is possible to easily process.

It is a block diagram which shows schematic structure of the centrifugal compressor in 1st embodiment of this invention. It is an enlarged view of the impeller in 1st embodiment of this invention. It is a graph in which the horizontal axis represents the position in the span direction relative to the blade (Span Normalized), and the vertical axis represents the absolute value (Vt_abs) of the circumferential speed of the gas. It is a flowchart which shows the manufacturing method of the impeller in 1st embodiment of this invention. It is an enlarged view equivalent to FIG. 2 in 2nd embodiment of this invention. It is an enlarged view equivalent to FIG. 2 in the modification of 1st embodiment of this invention.

(First embodiment)
Next, an impeller and a rotary machine according to a first embodiment of the present invention will be described with reference to the drawings.
FIG. 1 is a configuration diagram showing a schematic configuration of a centrifugal compressor in the first embodiment of the present invention.
As shown in FIG. 1, the centrifugal compressor 1 includes a rotating shaft 2, an impeller 3 A, a journal bearing 5 A, a thrust bearing 5 B, and a casing 6.

The rotary shaft 2 is formed in a column shape extending in the axis O direction. The rotary shaft 2 is rotatably supported by the journal bearing 5A on the first end portion 2a side (first side in the axial direction) and the second end portion 2b side (second side in the axial direction) in the axis O direction. ing. Further, the rotary shaft 2 has a first end 2a supported by a thrust bearing 5B.

A plurality of impellers 3A are provided in the direction of the axis O with respect to the rotating shaft 2. These impellers 3A are attached to the rotary shaft 2 by fitting or the like. The impeller 3A includes a disk 3a, a blade 3b, and a cover 3c.

The disk 3a is formed in a disk shape centered on the axis O. More specifically, the disk 3a has a diameter centered on the axis O as it goes from the side near the first end 2a of the rotary shaft 2 toward the side close to the second end 2b of the rotary shaft 2 in the direction of the axis O. It is formed so as to gradually expand in the direction. The central axis of the disk 3 a overlaps the axis O of the rotary shaft 2. Therefore, in the following description, the axis of the disk 3a is also referred to as “axis O”.

The blade 3b is formed on the surface of the disk 3a facing the first end 2a side in the axis O direction, and a plurality of blades 3b are formed at intervals in the circumferential direction of the axis O. These blades 3b extend away from the disk 3a and are arranged radially about the axis O.

The cover 3c covers the plurality of blades 3b from the first end 2a side in the axis O direction. In other words, the cover 3c is provided so as to face the disk 3a with the blade 3b interposed therebetween. The inner peripheral surface 11 of the cover 3c is formed so as to decrease in diameter from the side close to the second end 2b in the axis O direction toward the side close to the first end 2a. From the inner peripheral surface 11, the blade 3b described above extends toward the disk 3a.

The impeller 3A has a surface of the disk 3a facing the first end portion 2a in the axis O direction, the blade 3b, and the inner peripheral surface 11 of the cover 3c from the side close to the first end portion 2a in the axis O direction. A flow path extending toward the radially outer side is formed toward the side closer to the second end 2b.
The rotary shaft 2 of the centrifugal compressor 1 in this embodiment is provided with a plurality of impellers 3A in the direction of the axis O, thereby constituting a multistage impeller group 3G.

The casing 6 is formed in a cylindrical shape and accommodates the rotating shaft 2, the impeller 3A, the journal bearing 5A, and the like. The casing 6 rotatably supports the rotary shaft 2 via the journal bearing 5A. As a result, the impeller 3A attached to the rotary shaft 2 can rotate relative to the casing 6.

The casing 6 includes a suction port 6a, a connection channel 6b, a casing channel 6c, a connection channel 6d, and a discharge port 6e.

The suction port 6a is formed on the side close to the first end 2a of the casing 6 in the axis O direction. The suction port 6a receives gas supplied from the outside. The suction port 6a is disposed on the outermost surface 6f of the casing 6 on the side closest to the first end portion 2a in the axis O direction.

The connection flow path 6b connects the suction port 6a and the first stage impeller 3A arranged closest to the first end 2a among the plurality of impellers 3A. That is, the connection flow path 6b supplies the gas received by the suction port 6a to the first stage impeller 3A.

The casing flow path 6c connects the flow paths of the impellers 3A adjacent in the axis O direction. More specifically, the impeller accommodating space near the outer peripheral end of the impeller 3A arranged on the upstream side and the impeller accommodating space near the front end of the impeller 3A arranged on the downstream side are communicated. The casing flow path 6c guides the gas boosted by the impeller 3A disposed on the upstream side to the radially outer side, then guides the gas to the radially inner side and supplies it to the front end portion of the impeller 3A disposed on the downstream side. Thereby, the gas flowing through the casing flow path 6c is stepped up stepwise by the plurality of impellers 3A. A diffuser, a return vane, or the like may be provided in the casing flow path 6c.

The connection flow path 6d connects the final stage impeller 3A disposed on the side closest to the second end 2b and the discharge port 6e. That is, the connection flow path 6d guides the gas boosted by the multistage impeller group 3G to the discharge port 6e.
The discharge port 6 e discharges the gas guided by the connection flow path 6 d to the outside of the casing 6. The outlet 6e is disposed on the outermost surface 6f of the casing 6 on the side closest to the second end 2b in the axis O direction.

FIG. 2 is an enlarged view of the impeller in the first embodiment of the present invention.
As shown in FIG. 2, the cover 3 c of the impeller 3 A includes an inner peripheral surface 11, an outer peripheral surface 12, a front end surface 13, and a rear end surface 14. The inner peripheral surface 11 has a shape as described above, and is a convex curved surface in a cross section including the axis O shown in FIG.

As with the inner peripheral surface 11, the outer peripheral surface 12 is directed from the side close to the second end 2 b in the axis O direction (the right side in FIG. 2) to the side close to the first end 2 a (the left side in FIG. 2). The diameter is reduced according to the above. The outer peripheral surface 12 is a concave curved surface in a cross section including the axis O shown in FIG. The outer peripheral surface 12 is disposed so as to face the inner peripheral surface 6g of the casing 6 in which the impeller 3A is accommodated via a slight gap. The outer peripheral surface 12 is gradually separated from the inner peripheral surface 11 as it goes from the side close to the second end 2b in the direction of the axis O toward the side close to the first end 2a. In other words, the cover 3c is formed so that its thickness gradually increases from the side close to the second end 2b toward the side close to the first end 2a. A seal mechanism such as a labyrinth seal may be provided between the inner peripheral surface 6g of the casing 6 and the outer peripheral surface 12 of the cover 3c.

The front end surface 13 extends from the end portion 11a of the inner peripheral surface 11 on the side close to the first end portion 2a in the axis O direction toward the outer side in the radial direction with the axis O as the center. That is, the tip surface 13 faces the first end 2a side in the axis O direction. The tip surface 13 is formed across the end 11a of the inner peripheral surface 11 and the end 12a of the outer peripheral surface 12 on the side close to the first end 2a in the axis O direction. In this embodiment, the tip surface 13 is formed by a plane perpendicular to the axis O. A flat surface facing the tip surface 13 is formed on the inner peripheral surface 6g of the casing 6 in this embodiment.

The rear end surface 14 is formed so as to extend over the end portion 11b of the inner peripheral surface 11 and the end portion 12b of the outer peripheral surface 12 on the side close to the second end portion 2b in the axis O direction.

The blade 3b includes a front edge portion 20 that is an edge portion on the side close to the first end portion 2a in the axis O direction. The front edge portion 20 extends from the boundary portion K where the inner peripheral surface 11 and the tip surface 13 intersect toward the radially inner side with the axis O as the center. The front edge portion 20 can be formed so as to be continuous with the distal end surface 13. The front edge portion 20 in this embodiment is formed in a slightly curved shape close to a straight line. The front edge 20 is further inclined with respect to a plane perpendicular to the axis O so as to be arranged on the second end 2b side from the radially outer side centering on the axis O toward the inner side. As the front edge 20 is inclined in this manner, in the blade 3b, the distance from the front edge 20 on the side close to the cover 3c to the rear edge 21 of the blade 3b and the front edge on the side close to the disk 3a. The difference from the distance from the portion 20 to the rear edge portion 21 of the blade 3b is small. The boundary K described above includes not only the position where the inner peripheral surface 11 and the tip surface 13 intersect, but also includes, for example, a position shifted by about 1 mm to 2 mm from the position where the inner peripheral surface 11 and the tip surface 13 intersect. (The same applies to the second embodiment below).

FIG. 3 is a graph in which the horizontal axis represents the position in the span direction with respect to the blade (Span Normalized), and the vertical axis represents the absolute value (Vt_abs) of the circumferential velocity of the gas.
In the graph of FIG. 3, a broken line is a comparative example. In this comparative example, the front edge 20 of the blade 3b extends radially inward from a position closer to the second end 2b than the boundary K. In the graph of FIG. 3, the solid line is the case of the above-described embodiment (example). As shown in FIG. 3, in the comparative example, in the span direction, there is a region in which the absolute value of the circumferential velocity of the gas increases immediately before the position of the front edge 20 of the blade 3b (indicated by an arrow in FIG. 3). is there. This is because the absolute value of the circumferential speed of the gas increases because the cover 3c disposed closer to the first end 2a than the front edge 20 of the blade 3b comes into contact with the gas. It is thought that.

On the other hand, in the impeller 3A in this embodiment, the front edge 20 extends from the boundary K toward the inside in the radial direction. Therefore, immediately before the front edge portion 20 of the blade 3b, an increase in the absolute value of the circumferential speed of the gas due to contact between the gas and the cover 3c does not occur.

Next, a method for manufacturing the impeller 3A described above will be described.
FIG. 4 is a flowchart showing a method for manufacturing an impeller in the first embodiment of the present invention.
First, a base material on which the outer peripheral surface 12 and the front end surface 13 of the cover 3c are formed is formed of a metal such as stainless steel (step S01).
Next, the blade 3b, the inner peripheral surface 11 of the cover 3c, and the disk 3a are formed by cutting using a cutting tool T as shown in FIG. 2 (step S02). At this time, the blade 3b is formed by cutting so that the front edge portion 20 of the blade 3b extends radially inward from the boundary K between the inner peripheral surface 11 and the tip surface 13 of the cover 3c. Then, finishing processing such as surface polishing is performed as necessary.

According to the first embodiment described above, it is possible to prevent the cover 3c from projecting toward the first end 2a side in the axis O direction from the front edge 20 of the blade 3b. Therefore, it can suppress that a tool interferes with cover 3c, and can process it easily. Further, the rotation of the cover 3c can reduce the swirling flow generated immediately before the front edge 20 of the blade 3b due to the rotation of the cover 3c. Therefore, the boundary layer in the vicinity of the front edge portion 20 of the blade 3b can be thinned on the cover 3c side, and the performance of the impeller 3A can be improved.

(Second embodiment)
Next, an impeller according to a second embodiment of the present invention will be described with reference to the drawings. This second embodiment differs from the first embodiment described above only in the shape of the tip surface 13. For this reason, the same parts as those in the first embodiment are denoted by the same reference numerals, and redundant description is omitted.

FIG. 5 is an enlarged view corresponding to FIG. 2 in the second embodiment of the present invention.
As shown in FIG. 5, the impeller 3B in the second embodiment includes a disk 3a, a blade 3b, and a cover 103c.
The cover 103c covers the plurality of blades 3b from the side close to the first end 2a in the axis O direction (the left side in FIG. 3). The cover 103c is provided so as to face the disk 3a, and includes an inner peripheral surface 11, an outer peripheral surface 12, a front end surface 113, and a rear end surface 14.

The inner peripheral surface 11, the outer peripheral surface 12, and the rear end surface 14 have the same configuration as in the first embodiment described above. That is, the inner peripheral surface 11 is formed so as to decrease in diameter from the side close to the second end 2b in the axis O direction (the right side in FIG. 3) toward the side close to the first end 2a. From this inner peripheral surface 11, the blade 3b extends toward the disk 3a.

As with the inner peripheral surface 11, the outer peripheral surface 12 is formed so that its diameter decreases from the side close to the second end 2b in the axis O direction toward the side close to the first end 2a. The outer peripheral surface 12 is disposed so as to face the inner peripheral surface 6g of the casing 6 to be accommodated with a slight gap. Since the outer peripheral surface 12 is close to the second end 2b side in the axis O direction, the outer peripheral surface 12 is gradually separated from the inner peripheral surface 11 toward the side close to the first end 2a.

The rear end surface 14 is formed across the end 11b of the inner peripheral surface 11 and the end 12b of the outer peripheral surface 12 on the side close to the second end 2b in the axis O direction.

The front end surface 113 extends from the end portion 11a of the inner peripheral surface 11 near the first end portion 2a in the axis O direction toward the outside in the radial direction with the axis O as the center. This front end surface 113 faces the first end portion 2a side in the axis O direction. The tip surface 113 is formed across the end portion 11a of the inner peripheral surface 11 and the end portion 12a on the side of the outer peripheral surface 12 near the first end portion 2a. At least a part of the front end surface 113 in the second embodiment is a convex curved surface in a cross section including the axis O.

More specifically, the cover 103c in the second embodiment has an outer periphery on the side where the end 11a of the inner peripheral surface 11 on the side close to the first end 2a is close to the first end 2a in the axis O direction. It arrange | positions rather than the edge part 12a of the surface 12 at the side near the 2nd edge part 2b. The tip surface 113 formed across the end portion 11a and the end portion 12a is a first inner portion in the axis O direction from the outer side in the radial direction toward the inner side in the radial direction around the axis O. A convex curved surface is formed so as to curve from the side close to the end 2a toward the side close to the second end 2b.

The blade 3b extends from the boundary K between the tip surface 113 and the inner peripheral surface 11 inward in the radial direction with the axis O as the center, as in the first embodiment. Also in the second embodiment, the front edge portion 20 of the blade 3b is formed in a curved shape that is slightly curved close to a straight line. The blade 3b is further inclined with respect to a plane perpendicular to the axis O so as to be arranged closer to the second end portion 2b as it goes inward from the outer side in the radial direction around the axis O.

According to the second embodiment described above, the tip surface 113 has a convex curved surface, so that interference of the cutting tool T can be suppressed when performing cutting or the like. Even if the cutting tool T interferes, since it is a convex curved surface, it is possible to suppress the movement of the cutting tool T or the like, or the cutting tool T from being damaged.

(Other variations)
In each embodiment mentioned above, the case where the whole front edge part 20 of the braid | blade 3b extended in the radial direction inner side from the boundary part K was demonstrated to an example. However, it is not limited to this configuration.
FIG. 6 is an enlarged view corresponding to FIG. 2 in a modification of the first embodiment of the present invention.
As shown in FIG. 6, the front edge portion 20 C of the impeller 3 C may include a protruding edge portion 22 that protrudes on the outer side in the radial direction on the side close to the first end portion 2 a in the axis O direction.
The protruding edge portion 22 is formed so as to protrude toward the first end portion 2a side in the axis O direction from the extension line E of the remaining portion 20Cb of the front edge portion 20C, and the axis O is centered from the boundary K. It is formed so as to extend obliquely toward the inner side in the radial direction and closer to the second end 2b and to be continuous with the remaining portion 20Cb. By providing the protruding edge 22, the front edge 20C cannot be arranged on the side close to the first end 2a in the direction of the axis O as in the first embodiment, for example. However, the front edge portion 20C can be formed so as to extend from the boundary portion K inward in the radial direction with the axis O as the center.

The present invention is not limited to the above-described embodiments, and includes various modifications made to the above-described embodiments without departing from the spirit of the present invention. That is, the specific shapes, configurations, and the like given in the embodiment are merely examples, and can be changed as appropriate.
In the embodiment described above, the centrifugal compressor has been described as an example. However, the present invention is not limited to the compressor, and can be applied to a rotating machine such as a turbine.

In the above-described embodiment, an example in which six impellers 3A are provided in series on the rotary shaft 2 of the centrifugal compressor 1 is shown. However, the centrifugal compressor 1 only needs to have at least one impeller 3 A with respect to the rotating shaft 2. Similarly, in the case where the

impellers

3B and 3C are provided, only one

impeller

3B and 3C may be provided.

Further, in each of the above-described embodiments, the case where the

impellers

3A, 3B, and 3C are cut has been described. However, they may be formed by electric discharge machining. Also in the case of this electric discharge machining, by applying the present invention, it is not necessary to form the discharge electrode in a complicated shape, and an increase in cost can be suppressed.

This invention can be applied to an impeller. According to this invention, it becomes possible to process easily.

DESCRIPTION OF SYMBOLS 1 Centrifugal compressor 2 Rotating shaft 2a 1st end part 2b

2nd end part

3A, 3B, 3C Impeller 3a Disk 3b Blade 3c, 103c Cover 3G Multistage impeller group 5A Journal bearing 5B Thrust bearing 6 Casing 6a Inlet 6b Connection flow Road 6c Casing flow path 6d Connection flow path 6e Discharge port 6f Outer peripheral surface 6g Inner peripheral surface 7 Sealing device 11 Inner peripheral surface

11a End portion

11b End portion 12 Outer peripheral surface

12a End portion

12b End portion 13 Front end surface 14 Rear end surface 20 Front edge Part 21 Rear edge 22 Projecting edge E Extension line T Cutting tool

Claims

A disc having a disc shape centered on the axis,
A plurality of blades formed on the surface facing the first side in the axial direction of the disk, spaced apart in the circumferential direction of the axial line;
A cover surrounding the plurality of blades from the first side in the axial direction,
The cover is
An inner peripheral surface connected to the blade with a reduced diameter toward the first side from the second side in the axial direction;
A distal end surface extending radially outward from the end on the first axial side of the inner peripheral surface and facing the first side in the axial direction,
A front edge portion that is an edge portion on a first side in the axial direction of the blade is an impeller that extends radially inward from a boundary between the inner peripheral surface and the tip surface.
The tip surface is
2. The impeller according to claim 1, further comprising a convex curved surface arranged on the second side in the axial direction toward a radially inner side.
The front edge is
3. The impeller according to claim 1, further comprising a protruding edge that protrudes to a first side in the axial direction on a radially outer side thereof.
A rotary machine comprising the impeller according to any one of claims 1 to 3.
A disc having a disc shape centered on the axis,
A plurality of blades formed on the surface facing the first side in the axial direction of the disk, spaced apart in the circumferential direction of the axial line;
A cover surrounding the plurality of blades from the first side in the axial direction,
The cover is reduced in diameter toward the first side from the second side in the axial direction and is connected to the blade, and from the end on the first side in the axial direction of the inner peripheral surface A front end surface extending radially outward and facing a first side in the axial direction, the front edge being an edge on the first side in the axial direction of the blade. A method for manufacturing an impeller, including a step of forming a radially inner side from a boundary between the inner peripheral surface and the tip surface.