CROSS-REFERENCE TO RELATED APPLICATIONS
Not applicable.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not applicable.
BACKGROUND
This invention relates to a transition duct for a gas turbine engine, specifically to a novel and improved profile for a transition duct that results in lower operating stresses and extended component life.
In a typical can annular gas turbine engine, a plurality of combustors are arranged in a generally annular array about the engine. The combustors receive pressurized air from the engine's compressor, adds fuel to create a fuel/air mixture, and combusts that mixture to produce hot gases. The hot gases exiting the combustors are utilized to turn a turbine, which is coupled to a shaft that drives a generator for generating electricity.
The hot gases are transferred from the combustor to the turbine by a transition duct. Due to the position of the combustors relative to the turbine inlet, the transition duct must change cross-sectional shape from a generally cylindrical shape at the combustor exit to a generally rectangular arc-like shape at the turbine inlet. In addition, the transition duct undergoes a change in radial position, since the combustors are typically mounted outboard of the turbine. Extreme care must be taken with respect to the design of these ducts in order to avoid sharp geometric changes, otherwise regions of high stress and stress concentrations can occur. The combination of complex geometry changes as well as extreme mechanical and thermal loading seen by the transition duct can create a harsh operating environment that can lead to premature deterioration, requiring repair and replacement of the transition ducts. To withstand the hot temperatures from the combustor gases, transition ducts are typically air-cooled. A variety of methods are available to provide cooling such as through internal channels, impingement cooling, or effusion cooling.
Severe cracking, resulting in component failure and forcing engine shutdown, has been known to occur in transition ducts having extremely sharp geometry changes and internal air-cooled channels. In such an incident, the engine requires transition ducts replacement or repair prior to returning to operational status. The present invention seeks to overcome the shortfalls of these prior art designs.
SUMMARY
The present invention is defined by the claims below. Embodiments of the present invention solve at least the above problems by providing an apparatus for a transition duct having a geometric profile that results in lower operating stresses and improved component life.
In an aspect of the present invention, a transition duct is provided having an inlet ring, an aft frame, and a panel assembly having an internal profile defined by a series of X, Y, and Z Cartesian coordinates taken along a sweep angle θ.
A novel and improved transition duct having an enhanced profile for improved performance and durability is provided. The internal flowpath geometry of the transition duct has been configured to remove areas of sharp geometric change. The sharp geometric changes, in combination with high thermal and mechanical loading, caused regions of high steady and vibratory stresses and local stress concentrations in prior art ducts that often lead to cracking and premature failure. Furthermore, due to a rounder profile, certain natural frequencies of the transition duct are raised to avoid potential vibratory issues.
A variety of cooling methods can be used in combination with the enhanced profile of the present invention transition duct. In an embodiment, the cooling system continues to use air, but the air is directed through a plurality of effusion holes in the panel assembly of the transition duct. Effusion cooling provides more uniform cooling of the transition duct than the plurality of internal cooling channels used in the prior art, which were also a source of stress concentrations.
In an embodiment of the present invention, there is provided a transition duct with a panel assembly having an inlet end of generally circular cross section and an outlet end having a generally rectangular arc-like cross section with an uncoated internal profile substantially in accordance with the coordinate values θ, X, Y, and Z as set forth in Table 1. The origin of the coordinate system is positioned at the center of the panel assembly inlet end along a centerline axis. It will be appreciated that the coordinate values given are for manufacturing purposes, in a room temperature condition. The coordinate values X, Y, and Z in Table 1 are standard Cartesian coordinates, and correspond to a specific sweep angle θ, which together, define a cross section of the panel assembly. Each cross section is joined smoothly with adjacent cross sections to define a panel assembly for the transition duct. It will also be appreciated that as the transition duct transfers hot combustion gases from a combustor to the turbine inlet, the transition duct absorbs heat, and therefore the coordinates provided in Table 1 do not necessarily correspond to the panel assembly position when in operation at an elevated temperature.
In an alternate embodiment, there is provided a transition duct with a panel assembly having an inlet end of generally circular cross section and outlet end having a generally rectangular arc-like cross section with an uncoated internal profile within an envelope of +/−0.250 inches in a direction normal to any surface of the panel assembly substantially in accordance with the coordinate values θ, X, Y, and Z as set forth in Table 1. The origin of the Cartesian coordinate system is positioned at the center of the panel assembly inlet end along a centerline axis. A distance of +/−0.250 inches in a direction normal to any surface location along the panel assembly defines an envelope for this particular panel assembly and ensures that manufacturing tolerances are accommodated within the envelope of the panel assembly. As with the embodiment previously disclosed, it will be appreciated that the coordinate values given are for manufacturing purposes, in a room temperature condition. Each set of coordinate values X, Y, and Z in Table 1 is in standard Cartesian coordinates and corresponds to a specific sweep angle θ, which, when taken together defines a cross section of the panel assembly. Each cross section is joined smoothly with adjacent cross sections to define a panel assembly for the transition duct. It will also be appreciated that as the transition duct transfers hot combustion gases from a combustor to the turbine inlet, the transition duct heats up and therefore the Cartesian coordinates for a given θ value provided in Table 1 may not necessarily correspond to the panel assembly position when in operation at an elevated temperature.
The instant invention will now be described with particular reference to the accompanying drawings.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING
Illustrative embodiments of the present invention are described in detail below with reference to the attached drawing figures, which are incorporated by reference herein and wherein:
FIG. 1 is a perspective view of a transition duct of the prior art.
FIG. 2 is a cross section view of the transition duct in FIG. 1.
FIG. 3 is a perspective view of an embodiment of the present invention.
FIG. 4 is a cross section view of the embodiment of the present invention of FIG. 3.
FIG. 5 is a cross section view of the preferred embodiment of the panel assembly of present invention.
FIGS. 6
a, 6b, 6c, and 6d are section views taken through the panel assembly of the present invention at various sweep angles.
FIG. 7 is a perspective view showing each of the cross sections that define the panel assembly of the present invention.
DETAILED DESCRIPTION
Embodiments of the present invention provide apparatus for a gas turbine transition duct that are configured geometrically to have lower operating stresses. Lower stresses, both mechanical and thermal, result in improved component life.
Referring to FIGS. 1 and 2, a transition duct 10 of the prior art is shown. The transition duct 10 contains an inlet ring 11, a panel assembly 12, and an aft frame 13. The inlet ring 11 is of generally circular cross section while the aft frame 13 is of generally rectangular arc-like cross section where the generally rectangular are-like shape is defined by a pair of concentric arcs of different diameters connected by a pair of radial lines. The transition duct 10, which is used to transfer hot combustion gases from a combustor to a turbine, has geometric profile that must transition from a generally circular cross section to that of a generally arc-like cross section at the turbine inlet as well as to change radial positions. The geometric profile of the transition duct 10 contains a sharp transition from circular to rectangular arc-like over a short axial and radial distance thereby resulting in high stress regions throughout the aft end of the transition duct 10.
The present invention is shown in FIGS. 3-7. Referring to FIGS. 3 and 4, a transition duct 20 includes a panel assembly 23 having an inlet end 21 of generally circular cross section and an outlet end 22 having a generally rectangular arc-like cross section. The panel assembly 23 comprises a first panel 24 and a second panel 25 joined together along a plurality of axial seams 26 by a means such as welding. In an embodiment of the present invention, the panel assembly 23 also contains a plurality of cooling holes 27 extending throughout the first panel 24 and the second panel 25 to provide cooling air to the panels. The transition duct 20 further comprises an inlet ring 28 fixed to the inlet end 21 and an aft frame 29 fixed to the outlet end 22. The panel assembly 23 of the transition duct 20 is preferably manufactured from a high temperature nickel base alloy such as Haynes 230.
The panel assembly 23, formed from the first panel 24 and the second panel 25, has an uncoated internal profile substantially in accordance with coordinate values X, Y, and Z as set forth in Table 1, carried only to three decimal places. Although the preferred unit of measure for the values given in Table 1 is inches, those skilled in the art will appreciate that the values of Table 1 for X, Y, and Z can be scaled up or down depending on the diameter of the particular combustion liner with which the present invention is to be used. This uncoated internal profile provides an optimized transition from a generally circular inlet end to a generally rectangular arc-like outlet end over the allowable axial and radial distance for a gas turbine engine, such that high steady stresses and stress concentrations in the transition duct 20 are minimized. For the purpose of describing the present invention, the coordinate values X, Y, and Z of Table 1 are taken at various sweep angles θ wherein θ is an angle measured from the inlet end 21 and increases to its maximum value at the outlet end 22. Sweep angle θ originates at an intersection line 90 formed from a first plane 100, that is defined by the inlet end 21 of the panel assembly 23, and a second plane 102, that is defined by the outlet end 22 of the panel assembly 23, as shown in FIGS. 4 and 5. An origin 92 of the Cartesian coordinate system, from which data in Table 1 is generated, is positioned at center of the inlet end 21 along an axis A-A that runs through the center of the inlet end 21, and is perpendicular to the first plane 100. The Cartesian coordinate system is oriented such that X and Y extend radially out from the origin 92, or center point of the inlet end 21, and Z extends axially along axis A-A towards the outlet end 22, as shown in FIG. 5. Coordinate values X, Y, and Z are listed in Table 1 for each sweep angle θ, measured in one-half degree increments, sufficient to define the optimized internal profile of the panel assembly 23. The data compiled in Table 1 is computer generated and though it represents the nominal uncoated internal profile, the data will vary depending on manufacturing tolerances. Therefore, it will be appreciated that a gas turbine component of this size having the panel assembly 23 fabricated primarily from formed and welded sheet metal can be expected to have manufacturing tolerances upwards of +/−0.125 inches.
For the data listed in Table 1 a plurality of wireframe sections can be created when applying a best-fit curve to the section data for each sweep angle θ. For example, FIGS. 6a-6d show wireframe cross sections taken at various sweep angles from the inlet end 21 to the outlet end 22 of the panel assembly 23 as well as the Cartesian coordinates (each shown as an “x” in FIGS. 6a-6d) used to define each section taken. For clarity purposes, the wireframe sections are shown progressively stacked to show the change from the previous section(s). In each of FIGS. 6a-6d, the relevant section is the one with multiple “x” markings; the other sections shown are merely for reference purposes. At the inlet end 21, a section is taken corresponding to θ=0.0 degrees and is shown in section view in FIG. 6a, while FIG. 6b shows a section taken where the sweep angle θ=5.0 degrees. In FIG. 6c, where a section is taken with θ=11.0 degrees, panel assembly 23 is shown transitioning from a generally circular cross section to a rectangular arc-like shape. A final section demonstrating this transition is shown in FIG. 6d and taken at θ=17.0 degrees, at the outlet end 22 of the panel assembly 23. It can be seen in FIGS. 6a-6d how the section geometry of the panel assembly 23 transitions from a generally circular cross section to a generally rectangular arc-like cross section. FIG. 7 shows, in perspective view, each wireframe section formed at each respective sweep angle θ, that when compiled, define the internal flowpath of the panel assembly 23.
An additional feature of the transition duct 20 is a protective two-layer coating applied along the internal profile of the panel assembly 23 to protect the transition duct 20 from deterioration associated with prolonged exposure to elevated temperatures. The two-layer air plasma sprayed coating preferably comprises a MCrAlY bond coating applied directly to the panel assembly 23 and a Yttra Stabilized Zirconia top coating applied over the bond coating, the combined coating having a thickness of at least 0.019 inches. The two-layer coating is preferably applied once the panel assembly 23 has been formed and welded in accordance with the profile as defined in Table 1.
In an alternate embodiment of the present invention there is provided a transition duct similar to that of the preferred embodiment except for the uncoated internal profile of the panel assembly 23 is within an envelope of +/−0.250 inches in a direction normal to any surface of the panel assembly substantially in accordance with the Cartesian coordinate values X, Y, and Z as set forth in Table 1. A distance of +/−0.250 inches in a direction normal to any surface of the panel assembly thereby defines a profile envelope for this specific transition duct panel assembly. This envelope ensures that all reasonable manufacturing tolerances are accommodated within the profile.
The X, Y, Z Cartesian coordinate data and corresponding sweep angles θ are summarized in the following Table 1.
TABLE 1
|
|
Theta (deg.)XYZ
|
|
0.00.0007.2260.000
0.01.0467.1500.000
0.02.0716.9260.000
0.03.0546.5490.000
0.03.9496.0520.000
0.04.7645.4330.000
0.05.4844.7050.000
0.06.0943.8830.000
0.06.5822.9820.000
0.06.9372.0220.000
0.07.1531.0210.000
0.07.2260.0000.000
0.07.164−0.9430.000
0.06.980−1.8700.000
0.06.644−2.8400.000
0.06.175−3.7520.000
0.05.581−4.5890.000
0.04.875−5.3340.000
0.04.070−5.9700.000
0.03.184−6.4870.000
0.02.233−6.8720.000
0.01.130−7.1370.000
0.00.000−7.2260.000
0.0−1.130−7.1370.000
0.0−2.233−6.8720.000
0.0−3.184−6.4870.000
0.0−4.070−5.9700.000
0.0−4.875−5.3340.000
0.0−5.581−4.5890.000
0.0−6.175−3.7520.000
0.0−6.644−2.8400.000
0.0−6.980−1.8700.000
0.0−7.164−0.9430.000
0.0−7.2260.0000.000
0.0−7.1531.0210.000
0.0−6.9372.0220.000
0.0−6.5822.9820.000
0.0−6.0943.8830.000
0.0−5.4844.7050.000
0.0−4.7645.4330.000
0.0−3.9496.0520.000
0.0−3.0546.5490.000
0.0−2.0716.9260.000
0.0−1.0467.1500.000
0.50.0007.2261.133
0.51.0467.1501.134
0.52.0716.9261.136
0.53.0546.5491.139
0.53.9496.0521.143
0.54.7645.4331.149
0.55.4844.7051.155
0.56.0943.8831.162
0.56.5822.9821.170
0.56.9372.0221.178
0.57.1531.0211.187
0.57.2260.0001.196
0.57.164−0.9431.204
0.56.980−1.8701.212
0.56.644−2.8401.221
0.56.175−3.7521.229
0.55.581−4.5891.236
0.54.875−5.3341.243
0.54.070−5.9701.248
0.53.184−6.4871.253
0.52.233−6.8721.256
0.51.130−7.1371.258
0.50.000−7.2261.259
0.5−1.130−7.1371.258
0.5−2.233−6.8721.256
0.5−3.184−6.4871.253
0.5−4.070−5.9701.248
0.5−4.875−5.3341.243
0.5−5.581−4.5891.236
0.5−6.175−3.7521.229
0.5−6.644−2.8401.221
0.5−6.980−1.8701.212
0.5−7.164−0.9431.204
0.5−7.2260.0001.196
0.5−7.1531.0211.187
0.5−6.9372.0221.178
0.5−6.5822.9821.170
0.5−6.0943.8831.162
0.5−5.4844.7051.155
0.5−4.7645.4331.149
0.5−3.9496.0521.143
0.5−3.0546.5491.139
0.5−2.0716.9261.136
0.5−1.0467.1501.134
1.00.0007.2262.266
1.01.0467.1502.267
1.02.0716.9262.271
1.03.0546.5492.278
1.03.9496.0512.287
1.04.7645.4332.297
1.05.4844.7052.310
1.06.0943.8822.325
1.06.5822.9822.340
1.06.9372.0222.357
1.07.1531.0212.374
1.07.2260.0002.392
1.07.164−0.9432.409
1.06.980−1.8702.425
1.06.644−2.8402.442
1.06.175−3.7522.458
1.05.581−4.5892.472
1.04.875−5.3332.485
1.04.071−5.9702.496
1.03.184−6.4872.506
1.02.233−6.8722.512
1.01.130−7.1372.517
1.00.000−7.2262.518
1.0−1.130−7.1372.517
1.0−2.233−6.8722.512
1.0−3.184−6.4872.506
1.0−4.071−5.9702.496
1.0−4.875−5.3332.485
1.0−5.581−4.5892.472
1.0−6.175−3.7522.458
1.0−6.644−2.8402.442
1.0−6.980−1.8702.425
1.0−7.164−0.9432.409
1.0−7.2260.0002.392
1.0−7.1531.0212.374
1.0−6.9372.0222.357
1.0−6.5822.9822.340
1.0−6.0943.8822.325
1.0−5.4844.7052.310
1.0−4.7645.4332.297
1.0−3.9496.0512.287
1.0−3.0546.5492.278
1.0−2.0716.9262.271
1.0−1.0467.1502.267
1.50.0007.2263.400
1.51.0477.1493.402
1.52.0736.9253.408
1.53.0546.5493.417
1.53.9496.0513.430
1.54.7645.4323.447
1.55.4854.7053.466
1.56.0953.8823.487
1.56.5822.9823.511
1.56.9372.0223.536
1.57.1531.0213.562
1.57.2260.0003.589
1.57.164−0.9433.614
1.56.980−1.8703.638
1.56.644−2.8403.663
1.56.175−3.7523.687
1.55.582−4.5893.709
1.54.875−5.3333.729
1.54.071−5.9703.745
1.53.184−6.4873.759
1.52.233−6.8723.769
1.51.130−7.1373.776
1.50.000−7.2263.778
1.5−1.130−7.1373.776
1.5−2.233−6.8723.769
1.5−3.184−6.4873.759
1.5−4.071−5.9703.745
1.5−4.875−5.3333.729
1.5−5.582−4.5893.709
1.5−6.175−3.7523.687
1.5−6.644−2.8403.663
1.5−6.980−1.8703.638
1.5−7.164−0.9433.614
1.5−7.2260.0003.589
1.5−7.1531.0213.562
1.5−6.9372.0223.536
1.5−6.5822.9823.511
1.5−6.0953.8823.487
1.5−5.4854.7053.466
1.5−4.7645.4323.447
1.5−3.9496.0513.430
1.5−3.0546.5493.417
1.5−2.0736.9253.408
1.5−1.0477.1493.402
2.00.0007.2244.534
2.01.0497.1484.536
2.02.0776.9224.544
2.03.0586.5454.557
2.03.9516.0474.575
2.04.7655.4284.596
2.05.4834.7004.622
2.06.0913.8784.651
2.06.5762.9784.682
2.06.9292.0194.716
2.07.1421.0194.750
2.07.212−0.0014.786
2.07.143−0.9704.820
2.06.944−1.9214.853
2.06.604−2.8794.887
2.06.134−3.7794.918
2.05.541−4.6054.947
2.04.839−5.3394.972
2.04.040−5.9664.994
2.03.160−6.4755.012
2.02.217−6.8545.025
2.01.122−7.1125.034
2.00.000−7.1995.037
2.0−1.122−7.1125.034
2.0−2.217−6.8545.025
2.0−3.160−6.4755.012
2.0−4.040−5.9664.994
2.0−4.839−5.3394.972
2.0−5.541−4.6054.947
2.0−6.134−3.7794.918
2.0−6.604−2.8794.887
2.0−6.944−1.9214.853
2.0−7.143−0.9704.820
2.0−7.212−0.0014.786
2.0−7.1421.0194.750
2.0−6.9292.0194.716
2.0−6.5762.9784.682
2.0−6.0913.8784.651
2.0−5.4834.7004.622
2.0−4.7655.4284.596
2.0−3.9516.0474.575
2.0−3.0586.5454.557
2.0−2.0776.9224.544
2.0−1.0497.1484.536
2.50.0007.1665.671
2.51.0817.0855.675
2.52.1386.8475.685
2.53.1456.4465.702
2.54.0135.9465.724
2.54.8025.3295.751
2.55.4974.6085.783
2.56.0843.7965.818
2.56.5502.9105.857
2.56.8891.9675.898
2.57.0900.9875.941
2.57.153−0.0135.984
2.57.064−1.0696.031
2.56.822−2.1026.076
2.56.472−3.0196.116
2.56.002−3.8806.153
2.55.419−4.6696.188
2.54.733−5.3716.218
2.53.956−5.9706.245
2.53.103−6.4566.266
2.52.190−6.8176.281
2.51.108−7.0656.292
2.50.000−7.1506.296
2.5−1.108−7.0656.292
2.5−2.190−6.8176.281
2.5−3.103−6.4566.266
2.5−3.956−5.9706.245
2.5−4.733−5.3716.218
2.5−5.419−4.6696.188
2.5−6.002−3.8806.153
2.5−6.472−3.0196.116
2.5−6.822−2.1026.076
2.5−7.064−1.0696.031
2.5−7.153−0.0135.984
2.5−7.0900.9875.941
2.5−6.8891.9675.898
2.5−6.5502.9105.857
2.5−6.0843.7965.818
2.5−5.4974.6085.783
2.5−4.8025.3295.751
2.5−4.0135.9465.724
2.5−3.1456.4465.702
2.5−2.1386.8475.685
2.5−1.0817.0855.675
3.00.0007.0936.811
3.01.1137.0076.815
3.02.2016.7546.829
3.03.2336.3286.851
3.04.0755.8256.877
3.04.8385.2106.910
3.05.5094.4956.947
3.06.0743.6946.989
3.06.5222.8237.035
3.06.8451.8987.083
3.07.0370.9377.134
3.07.094−0.0417.185
3.06.988−1.1807.245
3.06.701−2.2897.303
3.06.343−3.1647.348
3.05.876−3.9857.392
3.05.303−4.7387.431
3.04.635−5.4077.466
3.03.882−5.9807.496
3.03.058−6.4457.520
3.02.177−6.7907.539
3.01.101−7.0367.551
3.00.000−7.1197.556
3.0−1.101−7.0367.551
3.0−2.177−6.7907.539
3.0−3.058−6.4457.520
3.0−3.882−5.9807.496
3.0−4.635−5.4077.466
3.0−5.303−4.7387.431
3.0−5.876−3.9857.392
3.0−6.343−3.1647.348
3.0−6.701−2.2897.303
3.0−6.988−1.1807.245
3.0−7.094−0.0417.185
3.0−7.0370.9377.134
3.0−6.8451.8987.083
3.0−6.5222.8237.035
3.0−6.0743.6946.989
3.0−5.5094.4956.947
3.0−4.8385.2106.910
3.0−4.0755.8256.877
3.0−3.2336.3286.851
3.0−2.2016.7546.829
3.0−1.1137.0076.815
3.50.0007.0077.954
3.51.1476.9157.960
3.52.2666.6447.976
3.53.3236.1898.004
3.54.1375.6838.035
3.54.8745.0708.072
3.55.5194.3628.116
3.56.0613.5738.164
3.56.4902.7178.216
3.56.7991.8118.272
3.56.9810.8718.329
3.57.034−0.0858.388
3.56.914−1.3028.462
3.56.583−2.4818.534
3.56.149−3.4488.593
3.55.579−4.3428.648
3.54.882−5.1418.697
3.54.070−5.8248.739
3.53.163−6.3758.772
3.52.180−6.7758.797
3.51.102−7.0238.812
3.50.000−7.1078.817
3.5−1.102−7.0238.812
3.5−2.180−6.7758.797
3.5−3.163−6.3758.772
3.5−4.070−5.8248.739
3.5−4.882−5.1418.697
3.5−5.579−4.3428.648
3.5−6.149−3.4488.593
3.5−6.583−2.4818.534
3.5−6.914−1.3028.462
3.5−7.034−0.0858.388
3.5−6.9810.8718.329
3.5−6.7991.8118.272
3.5−6.4902.7178.216
3.5−6.0613.5738.164
3.5−5.5194.3628.116
3.5−4.8745.0708.072
3.5−4.1375.6838.035
3.5−3.3236.1898.004
3.5−2.2666.6447.976
3.5−1.1476.9157.960
4.00.0006.9069.101
4.00.8896.8519.105
4.01.7646.6859.116
4.02.6126.4129.135
4.03.4166.0299.162
4.04.3075.4389.203
4.05.0954.7179.254
4.05.7633.8849.312
4.06.2952.9589.377
4.06.6791.9629.446
4.06.9060.9209.519
4.06.974−0.1459.594
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4.04.009−5.8559.993
4.03.138−6.38310.030
4.02.197−6.77110.057
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4.0−1.112−7.02710.075
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4.56.2672.80610.565
4.56.6321.83210.642
4.56.8480.81610.722
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4.56.657−2.02310.946
4.56.353−2.88311.013
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4.50.000−7.14011.348
4.5−1.129−7.04811.341
4.5−2.228−6.77711.320
4.5−3.126−6.39911.290
4.5−3.959−5.89311.250
4.5−4.711−5.27411.201
4.5−5.367−4.55511.145
4.5−5.917−3.75211.082
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5.0−1.154−7.08612.611
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5.50.0006.51512.569
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5.51.9356.23312.597
5.52.8535.89012.630
5.53.7075.41212.676
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5.55.1754.10812.801
5.55.7513.31212.878
5.56.2072.44112.962
5.56.5361.51613.051
5.56.7300.55313.143
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5.56.488−2.38313.426
5.56.132−3.31013.515
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5.54.956−5.06313.684
5.54.179−5.78513.754
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5.52.337−6.82413.854
5.51.188−7.14113.884
5.50.000−7.24913.895
5.5−1.188−7.14113.884
5.5−2.337−6.82413.854
5.5−3.299−6.37613.811
5.5−4.179−5.78513.754
5.5−4.956−5.06313.684
5.5−5.612−4.23213.604
5.5−6.132−3.31013.515
5.5−6.488−2.38313.426
5.5−6.709−1.41513.333
5.5−6.789−0.42613.238
5.5−6.7300.55313.143
5.5−6.5361.51613.051
5.5−6.2072.44112.962
5.5−5.7513.31212.878
5.5−5.1754.10812.801
5.5−4.4894.81312.733
5.5−3.7075.41212.676
5.5−2.8535.89012.630
5.5−1.9356.23312.597
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6.00.0006.35313.737
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6.02.9395.67413.808
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6.06.6700.39514.363
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6.04.891−5.18214.950
6.04.152−5.86215.021
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6.02.414−6.86415.126
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6.0−1.230−7.21415.163
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6.0−3.320−6.42615.080
6.0−4.152−5.86215.021
6.0−4.891−5.18214.950
6.0−5.521−4.40014.867
6.0−6.025−3.53314.776
6.0−6.402−2.57714.676
6.0−6.637−1.57814.571
6.0−6.725−0.55514.463
6.0−6.6700.39514.363
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6.0−5.7463.07814.081
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6.0−1.9986.04813.769
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6.50.0006.17614.912
6.51.0456.09214.921
6.52.0645.84314.950
6.53.0305.43814.996
6.53.9154.87915.059
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6.55.4113.34115.235
6.55.9582.41515.340
6.56.3541.41715.454
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6.56.661−0.70115.695
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6.56.314−2.78115.932
6.55.920−3.76316.044
6.55.435−4.57616.137
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6.53.354−6.48716.354
6.52.507−6.91616.403
6.51.698−7.20316.436
6.50.857−7.37616.456
6.50.000−7.43616.462
6.5−0.857−7.37616.456
6.5−1.698−7.20316.436
6.5−2.507−6.91616.403
6.5−3.354−6.48716.354
6.5−4.137−5.95016.293
6.5−4.836−5.30916.220
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6.5−5.920−3.76316.044
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6.5−2.0645.84314.950
6.5−1.0456.09214.921
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8.50.9695.22019.703
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8.52.8094.62019.792
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16.06.576−6.40141.135
16.05.649−6.21641.082
16.04.716−6.06041.037
16.03.778−5.93241.000
16.02.837−5.83240.972
16.01.893−5.76040.951
16.00.947−5.71840.939
16.00.000−5.70340.935
16.0−0.947−5.71840.939
16.0−1.893−5.76040.951
16.0−2.837−5.83240.972
16.0−3.778−5.93241.000
16.0−4.716−6.06041.037
16.0−5.649−6.21641.082
16.0−6.576−6.40141.135
16.0−6.880−6.40941.137
16.0−7.165−6.30841.108
16.0−7.390−6.11241.052
16.0−7.523−5.84940.977
16.0−7.817−4.79640.675
16.0−8.110−3.74340.373
16.0−8.404−2.68940.071
16.0−8.423−2.38339.983
16.0−8.316−2.09339.900
16.0−8.100−1.86739.835
16.0−7.809−1.74039.798
16.0−6.846−1.54339.742
16.0−5.878−1.37239.693
16.0−4.905−1.22739.651
16.0−3.929−1.10839.617
16.0−2.949−1.01539.591
16.0−1.968−0.94939.572
16.0−0.984−0.90939.560
16.50.000−0.53040.754
16.50.984−0.54340.758
16.51.968−0.58340.770
16.52.949−0.64940.789
16.53.929−0.74140.817
16.54.905−0.86040.852
16.55.878−1.00540.895
16.56.846−1.17540.945
16.57.809−1.37241.003
16.58.100−1.49841.041
16.58.316−1.72441.108
16.58.423−2.01341.193
16.58.404−2.31841.284
16.58.110−3.36941.595
16.57.817−4.41941.906
16.57.523−5.47042.217
16.57.390−5.73242.295
16.57.165−5.92842.353
16.56.880−6.02842.383
16.56.576−6.02042.380
16.55.649−5.83642.326
16.54.716−5.68042.280
16.53.778−5.55242.242
16.52.837−5.45342.212
16.51.893−5.38142.191
16.50.947−5.33942.178
16.50.000−5.32442.174
16.5−0.947−5.33942.178
16.5−1.893−5.38142.191
16.5−2.837−5.45342.212
16.5−3.778−5.55242.242
16.5−4.716−5.68042.280
16.5−5.649−5.83642.326
16.5−6.576−6.02042.380
16.5−6.880−6.02842.383
16.5−7.165−5.92842.353
16.5−7.390−5.73242.295
16.5−7.523−5.47042.217
16.5−7.817−4.41941.906
16.5−8.110−3.36941.595
16.5−8.404−2.31841.284
16.5−8.423−2.01341.193
16.5−8.316−1.72441.108
16.5−8.100−1.49841.041
16.5−7.809−1.37241.003
16.5−6.846−1.17540.945
16.5−5.878−1.00540.895
16.5−4.905−0.86040.852
16.5−3.929−0.74140.817
16.5−2.949−0.64940.789
16.5−1.968−0.58340.770
16.5−0.984−0.54340.758
17.00.000−0.16441.952
17.00.984−0.17741.956
17.01.968−0.21741.968
17.02.949−0.28241.988
17.03.929−0.37542.016
17.04.905−0.49342.052
17.05.878−0.63742.096
17.06.846−0.80742.148
17.07.809−1.00442.208
17.08.100−1.13042.247
17.08.316−1.35542.316
17.08.423−1.64342.404
17.08.404−1.94842.497
17.08.110−2.99542.817
17.07.817−4.04343.137
17.07.523−5.09143.458
17.07.390−5.35243.538
17.07.165−5.54743.597
17.06.880−5.64843.628
17.06.576−5.63943.626
17.05.649−5.45643.569
17.04.716−5.30043.522
17.03.778−5.17343.483
17.02.837−5.07343.452
17.01.893−5.00243.431
17.00.947−4.96043.418
17.00.000−4.94543.413
17.0−0.947−4.96043.418
17.0−1.893−5.00243.431
17.0−2.837−5.07343.452
17.0−3.778−5.17343.483
17.0−4.716−5.30043.522
17.0−5.649−5.45643.569
17.0−6.576−5.63943.626
17.0−6.880−5.64843.628
17.0−7.165−5.54743.597
17.0−7.390−5.35243.538
17.0−7.523−5.09143.458
17.0−7.817−4.04343.137
17.0−8.110−2.99542.817
17.0−8.404−1.94842.497
17.0−8.423−1.64342.404
17.0−8.316−1.35542.316
17.0−8.100−1.13042.247
17.0−7.809−1.00442.208
17.0−6.846−0.80742.148
17.0−5.878−0.63742.096
17.0−4.905−0.49342.052
17.0−3.929−0.37542.016
17.0−2.949−0.28241.988
17.0−1.968−0.21741.968
17.0−0.984−0.17741.956
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Many different arrangements of the various components depicted, as well as components not shown, are possible without departing from the spirit and scope of the present invention. Embodiments of the present invention have been described with the intent to be illustrative rather than restrictive. Alternative embodiments will become apparent to those skilled in the art that do not depart from its scope. A skilled artisan may develop alternative means of implementing the aforementioned improvements without departing from the scope of the present invention.
It will be understood that certain features and subcombinations are of utility and may be employed without reference to other features and subcombinations and are contemplated within the scope of the claims.