Abstract:
An article of manufacture having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth TABLE I. The X and Y values are distances in inches, which when connected by smooth continuing arcs define airfoil profile sections at each distance Z in inches The X, Y and Z distances may be scalable as a function of the same constant or number to provide a scaled up or scaled down airfoil section for the nozzle. The nominal airfoil given by the X, Y and Z distances lies within an envelope of +0.130 inch to −0.030 inch.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    The present invention relates to airfoils for a gas turbine and particularly relates to an airfoil profile for a first stage stator nozzle of a gas turbine. 
         [0002]    In a gas turbine, many system requirements should be met at each stage of a gas turbine&#39;s flow path section to meet design goals. These design goals include, but are not limited to, overall improved efficiency and airfoil loading capability. For example, and in no way limiting of the invention, a nozzle of a turbine should achieve thermal and mechanical operating requirements for that particular stage. 
         [0003]    Airfoil points have been patented as demonstrated by Barry et al. in U.S. Pat. No. 5,980,209. Barry et al. identified from 100-150 points per section with each section at a spacing of 0.52″. The number of points defined may be dependent upon the rate of change of curvature of the section. In other words, for areas with higher curvature more points may be used to define that region. 
       BRIEF DESCRIPTION OF THE INVENTION 
       [0004]    In accordance with one exemplary aspect of the present invention, there is provided an article of manufacture having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in TABLE I. X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z in inches. The profile sections at the Z distances may be joined smoothly with one another to form a complete airfoil shape. 
         [0005]    In accordance with another exemplary aspect of the present invention, a turbine comprises a turbine stage. The turbine stage includes a plurality of articles of manufacture. Each of the articles of manufacture includes an airfoil having an airfoil shape. The airfoil comprises a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z in inches. The profiles at the Z distances are joined smoothly with one another to form a complete airfoil shape. 
         [0006]    In accordance with a third aspect of the present invention, a turbine including a turbine stage having a plurality of articles of manufacture is provided. Each of the articles of manufacture may include an airfoil having a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z set forth in TABLE I. X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z in inches, the profile sections at the Z distances being joined smoothly with one another to form a complete airfoil shape. The X, Y and Z distances may be scalable as a function of the same constant or number to provide a scaled-up or scaled-down nozzle airfoil. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]      FIG. 1  schematically illustrates a turbine having a first stage turbine nozzle arrangement employing the nozzles and nozzle airfoils; 
           [0008]      FIG. 2  illustrates a coordinate system for the article of manufacture employed as an embodiment of the inventive airfoil for a stage in a gas turbine. 
           [0009]      FIG. 3  illustrates a forward view including the trailing edge of an embodiment of the inventive airfoil; 
           [0010]      FIG. 4  illustrates an aft view of an embodiment of the present invention; 
           [0011]      FIG. 5  illustrates a typical section through an embodiment of the inventive airfoil including exemplary section points at section Z 1 . 
           [0012]      FIG. 6  illustrates factors affecting variation of the nominal profile for an embodiment of the inventive airfoil. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    The embodiments of the present invention have many advantages, including defining airfoils for nozzles satisfying the restrictive thermal and mechanical operating requirements for that particular stage that a nozzle of a turbine should achieve. 
         [0014]    According to one aspect of the present invention, a unique profile for an article of manufacture is provided. The article of manufacture may find application as an airfoil for the nozzles of a turbine stage, preferably the first stage nozzle of a gas turbine. The airfoil shape for the exemplary first stage nozzle of a gas turbine is established to optimize design factors including: 1) turbine work split; 2) airflow through the gas turbine; 3) aerodynamic efficiency; 4) mechanical life requirements and 5) impingement tube insertability. The airfoil shape can be defined by a loci of points in space that meet stage design requirements and can also be manufactured. These points are unique and specific to the system and are not obvious to those skilled in the art. 
         [0015]    A gas turbine includes a plurality of turbine stages employing nozzles and nozzle airfoils. The turbine includes multiple stages. Each stage includes a nozzle arrangement in conjunction with a respective bucket associated with a stage of a rotor wheel. Each of the rotor wheels is connected to a shaft of the turbine, which in turn may be coupled with an electrical generator and a compressor. It will be appreciated, however, that turbines come in many different configurations of and numbers of stages, nozzle arrangements, and buckets. 
         [0016]      FIG. 1  schematically illustrates an axial turbine flow path  12  of a gas turbine  10  that includes a plurality of turbine stages employing nozzles and buckets. The direction of hot gas through the turbine flow path  12 , as embodied by the invention, is indicated by arrow. For example, the gas turbine  10  may include a first stage with nozzle  20  and bucket  16 , a second stage with nozzle  26  and bucket  22  and third stage with nozzle  32  and bucket  28 . Each bucket is supported by a respective stage of the turbine wheel (not shown). 
         [0017]    The first stage nozzle  20  includes an airfoil  25  and sidewalls  38 ,  40 . The nozzle  20  includes an outer sidewall  40  and an inner sidewall wall  38 . The sidewalls provide suitable mounting and support with surrounding turbine hardware. 
         [0018]    The airfoil  25  has a profile including a 3-dimensional shape with suction and pressure sides, respectively, as well as a leading edge and trailing edge. 
         [0019]    It will be appreciated that a plurality of the nozzles are spaced circumferentially, one from the other, about the first stage nozzle arrangement. 
         [0020]    An important aspect of the present invention is the cold airfoil shape, which was optimized for performance and includes a three-dimensional (3D) trailing edge profile. The list of X, Y, and Z coordinates for the airfoil, which are included as TABLE I, meets gas turbine requirements for stage efficiency, nozzle throat area, turbine work split, impingement tube insertability and mechanical life. The points are arrived at by iteration between aerodynamic and mechanical design optimization. These are the only loci of points that allow the gas turbine to run in an efficient, safe and smooth manner. 
         [0021]    The airfoil shape may be represented as a set of 1342 points from 11 sections of 122 points each. The X, Y &amp; Z coordinates, which represent the airfoil shape, are created in a coordinate system which is defined relative to the cold engine centerline axis of the gas turbine (Y coordinate axis). The origin of the coordinate system on the cold engine centerline axis is defined as X=0.0, Y=0.0 and Z=0.0. The Z coordinate axis is defined as a radial line from the Y coordinate axis. The X coordinate axis is defined as being normal to the Y-Z plane. Airfoil sections are cut normal to the radial line (Z coordinate axis). The X &amp; Y points, which make up the airfoil shape at each section, are in inches. The radial Z values in inches for the section planes extend from Z=46.628 farthest from the cold engine centerline axis to Z=40.628 inches, closest to the cold engine centerline axis. 
         [0022]    The radial distance between each of the adjacent sections is 0.6 inches apart, such that the total radial distance from Z 0  to Z 10  is 6 inches. The bottom and top sections Z 0  and Z 10 , respectively, may be obscured by fillets in the actual part casting where the airfoil meets the sidewalls. The remaining 9 sections are taken in a region of the airfoil where no fillets exist. 
         [0023]    All of the 1342 points are taken from a nominal cold or room temperature for each cross-section of the nozzle. Each section is joined smoothly with the other sections to form the complete airfoil shape. 
         [0024]    It will be appreciated that as each nozzle airfoil heats up in use, the profile will change as a result of stress and temperature. Thus, the cold or room temperature profile is given by the X, Y and Z coordinates for manufacturing purposes. Since the manufactured nozzle airfoil profile may be different from the nominal airfoil profile given by the following table, a distance of plus 0.130 inch to minus 0.030 inch from the nominal profile in a direction normal to any airfoil surface location defines the profile envelope for this nozzle airfoil. The design is robust to this variation without impairment of the mechanical and aerodynamic functions. 
         [0025]    It will also be appreciated that the airfoil can be scaled up or scaled down geometrically for introduction into similar turbine designs, with smaller or larger frame size. Consequently, the X, Y, and Z coordinates in inches of the nominal airfoil profile given below are a function of the same constant or number. That is, the X, Y and the Z coordinate values in inches may be multiplied or divided by the same constant or number to provide a scaled up or scaled down version of the nozzle airfoil profile while retaining the airfoil section shape and unique properties. 
         [0026]    Referring now to  FIGS. 2 ,  3 , and  4 , there is illustrated an airfoil for a first stage turbine nozzle constructed in accordance with one aspect of the present invention including an airfoil mounted between an inner and an outer sidewall. Fillets are not included in point definition. 
         [0027]      FIG. 2  illustrates a coordinate system for the article of manufacture employed as an embodiment of the inventive airfoil for a stage in a gas turbine. The coordinate system  200  is defined relative to the cold engine centerline axis  205  of the gas turbine. The coordinate system includes X, Y and Z coordinates X C    215 , Y C    220  and Z C    225 . The origin of the XYZ coordinate system (X=0, Y=0, Z=0) is centered on the cold engine centerline axis  205 . Y C    220  is coincident with the cold engine centerline axis  205 . Z C    225  is directed along a radial line normal to the cold engine centerline axis  205 . The positive radial direction of Z C    225  is defined as toward the airfoil  230 . X C    215  is defined as being normal to the plane formed by Y C    220  and Z C    225 . The positive direction of X C    215  and Y C    220  are identified by the label placement. 
         [0028]    The nozzle includes an airfoil with a trailing edge, a leading edge, an outer sidewall and an inner sidewall.  FIG. 3  illustrates a forward view including the trailing edge of an embodiment of the inventive airfoil. The nozzle  300  may include the airfoil  310 , an outer sidewall  320  and an inner sidewall  330  and a leading edge  340 . The trailing edge  350  of the airfoil  310  is optimized, according to the coordinate points provided in TABLE I, to provide improved aerodynamic stage efficiency. Sections Z 0  to Z 10  are designated  470 ,  460 ,  450 ,  440 ,  430 ,  420 ,  410 ,  400 ,  390 ,  380 , and  370 , respectively. 
         [0029]    Referring again to  FIG. 2 , airfoil sections  250  are cut normal to Z C    225  (radial line passing from the cold engine centerline axis  205 ). The X and Y points, which make up the airfoil shape at each section, are in inches. The radial Z values for the section planes extend from Z 0  ( 470 )=40.628 closest to the cold engine centerline axis to Z 10  ( 370 )=46.628 inch, farthest from the cold engine centerline axis. The engine centerline is located at Z=0 inch. The points defining the nozzle sections are specified in TABLE I. 
         [0030]      FIG. 4  illustrates an aft view of an embodiment of the inventive airfoil, including the outer sidewall  320 , the inner sidewall  330 , the leading edge  340 , and the trailing edge  350  for the airfoil  310  are identified. 
         [0031]      FIG. 5  illustrates a typical section through an embodiment of the inventive airfoil including exemplary section points at Z 1 =41.228. A typical section  500  through the airfoil may include a pressure side  510 , a suction side  520 , a leading edge  540 , and a trailing edge  550 .  FIG. 5  illustrates the typical distribution of coordinate points  560  taken from TABLE I for a section. The coordinate points  560  are defined such that the trailing edge  550  includes a greater concentration of points representing the greater curvature for this surface, thereby capturing the true intent and criticality of the airfoil shape for operation. 
         [0032]      FIG. 6  illustrates factors affecting variation of the nominal profile for an embodiment of the inventive airfoil. A profile is the range of the variation between measured points on an airfoil surface and their ideal position. The TABLE I values are generated and shown for determining the profile of the airfoil. There are typical manufacturing tolerances, as well as coatings, which must be accounted for in the actual profile of the airfoil. Accordingly, the values for the profile given in TABLE I are for a nominal airfoil. It will therefore be appreciated that ±typical manufacturing tolerances, i.e. ±values, are additive to the X and Y values given in TABLE I below. Two components characterize the changes, namely, the mean and the standard deviation for each process. In consideration of process capability data a tolerance  610  of 0.160 inches overlapping the nominal profile  630  is appropriate for this design. This means that there is a +0.130 inch tolerance  615  and a −0.030 inch tolerance  620  around the nominal profile  630 . The design is robust to this variation meaning that mechanical and aerodynamic function is not impaired. Accordingly, a distance of +0.130 inch/−0.030 inch in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular nozzle airfoil design and turbine. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 The coordinate values given below provide the preferred 
               
               
                 nominal profile envelope information regarding the XYZ 
               
               
                 coordinates that are optimized for the 1st stage nozzle. 
               
             
          
           
               
                 X 
                 Y 
                 Z 
               
               
                   
               
             
          
           
               
                 1.252 
                 −0.100 
                 40.628 
               
               
                 1.300 
                 0.081 
                 40.628 
               
               
                 1.357 
                 −0.158 
                 40.628 
               
               
                 1.407 
                 0.040 
                 40.628 
               
               
                 1.567 
                 −0.277 
                 40.628 
               
               
                 1.462 
                 −0.217 
                 40.628 
               
               
                 1.131 
                 0.118 
                 40.628 
               
               
                 1.513 
                 0.000 
                 40.628 
               
               
                 1.097 
                 0.067 
                 40.628 
               
               
                 1.193 
                 0.121 
                 40.628 
               
               
                 1.146 
                 −0.041 
                 40.628 
               
               
                 1.104 
                 0.005 
                 40.628 
               
               
                 1.620 
                 −0.041 
                 40.628 
               
               
                 2.047 
                 −0.205 
                 40.628 
               
               
                 3.003 
                 −0.583 
                 40.628 
               
               
                 2.781 
                 −1.061 
                 40.628 
               
               
                 2.472 
                 −0.372 
                 40.628 
               
               
                 2.684 
                 −0.989 
                 40.628 
               
               
                 1.940 
                 −0.164 
                 40.628 
               
               
                 2.897 
                 −0.541 
                 40.628 
               
               
                 2.366 
                 −0.330 
                 40.628 
               
               
                 2.487 
                 −0.850 
                 40.628 
               
               
                 1.834 
                 −0.123 
                 40.628 
               
               
                 2.388 
                 −0.782 
                 40.628 
               
               
                 2.791 
                 −0.498 
                 40.628 
               
               
                 2.287 
                 −0.715 
                 40.628 
               
               
                 2.260 
                 −0.288 
                 40.628 
               
               
                 1.775 
                 −0.398 
                 40.628 
               
               
                 3.247 
                 −1.442 
                 40.628 
               
               
                 3.156 
                 −1.363 
                 40.628 
               
               
                 1.727 
                 −0.082 
                 40.628 
               
               
                 1.671 
                 −0.337 
                 40.628 
               
               
                 3.064 
                 −1.285 
                 40.628 
               
               
                 2.186 
                 −0.650 
                 40.628 
               
               
                 2.685 
                 −0.456 
                 40.628 
               
               
                 2.971 
                 −1.209 
                 40.628 
               
               
                 3.215 
                 −0.669 
                 40.628 
               
               
                 2.877 
                 −1.134 
                 40.628 
               
               
                 2.084 
                 −0.585 
                 40.628 
               
               
                 2.153 
                 −0.246 
                 40.628 
               
               
                 3.109 
                 −0.626 
                 40.628 
               
               
                 2.586 
                 −0.919 
                 40.628 
               
               
                 1.982 
                 −0.522 
                 40.628 
               
               
                 2.578 
                 −0.414 
                 40.628 
               
               
                 1.879 
                 −0.459 
                 40.628 
               
               
                 3.532 
                 −0.799 
                 40.628 
               
               
                 4.321 
                 −2.736 
                 40.628 
               
               
                 3.981 
                 −2.238 
                 40.628 
               
               
                 4.190 
                 −2.533 
                 40.628 
               
               
                 3.848 
                 −0.931 
                 40.628 
               
               
                 4.476 
                 −1.206 
                 40.628 
               
               
                 4.814 
                 −3.564 
                 40.628 
               
               
                 3.832 
                 −2.049 
                 40.628 
               
               
                 4.507 
                 −3.046 
                 40.628 
               
               
                 3.674 
                 −1.867 
                 40.628 
               
               
                 4.580 
                 −1.254 
                 40.628 
               
               
                 3.593 
                 −1.778 
                 40.628 
               
               
                 4.372 
                 −1.159 
                 40.628 
               
               
                 3.908 
                 −2.143 
                 40.628 
               
               
                 4.384 
                 −2.839 
                 40.628 
               
               
                 3.754 
                 −1.957 
                 40.628 
               
               
                 3.953 
                 −0.976 
                 40.628 
               
               
                 4.446 
                 −2.942 
                 40.628 
               
               
                 3.336 
                 −1.524 
                 40.628 
               
               
                 4.256 
                 −2.634 
                 40.628 
               
               
                 4.268 
                 −1.112 
                 40.628 
               
               
                 4.123 
                 −2.433 
                 40.628 
               
               
                 4.785 
                 −1.353 
                 40.628 
               
               
                 3.743 
                 −0.887 
                 40.628 
               
               
                 4.683 
                 −1.303 
                 40.628 
               
               
                 4.751 
                 −3.461 
                 40.628 
               
               
                 3.321 
                 −0.712 
                 40.628 
               
               
                 4.163 
                 −1.066 
                 40.628 
               
               
                 4.629 
                 −3.254 
                 40.628 
               
               
                 4.690 
                 −3.358 
                 40.628 
               
               
                 3.637 
                 −0.842 
                 40.628 
               
               
                 4.053 
                 −2.335 
                 40.628 
               
               
                 4.058 
                 −1.021 
                 40.628 
               
               
                 4.568 
                 −3.150 
                 40.628 
               
               
                 3.509 
                 −1.692 
                 40.628 
               
               
                 3.423 
                 −1.607 
                 40.628 
               
               
                 3.426 
                 −0.755 
                 40.628 
               
               
                 4.989 
                 −1.457 
                 40.628 
               
               
                 5.707 
                 −4.006 
                 40.628 
               
               
                 6.128 
                 −3.469 
                 40.628 
               
               
                 5.838 
                 −2.032 
                 40.628 
               
               
                 5.790 
                 −3.927 
                 40.628 
               
               
                 6.283 
                 −2.925 
                 40.628 
               
               
                 5.866 
                 −3.842 
                 40.628 
               
               
                 4.887 
                 −1.405 
                 40.628 
               
               
                 6.183 
                 −2.483 
                 40.628 
               
               
                 4.878 
                 −3.666 
                 40.628 
               
               
                 6.278 
                 −2.811 
                 40.628 
               
               
                 5.288 
                 −1.625 
                 40.628 
               
               
                 5.171 
                 −4.046 
                 40.628 
               
               
                 6.065 
                 −2.287 
                 40.628 
               
               
                 5.083 
                 −3.964 
                 40.628 
               
               
                 5.995 
                 −2.197 
                 40.628 
               
               
                 6.129 
                 −2.383 
                 40.628 
               
               
                 5.664 
                 −1.883 
                 40.628 
               
               
                 5.013 
                 −3.866 
                 40.628 
               
               
                 5.919 
                 −2.112 
                 40.628 
               
               
                 6.259 
                 −2.698 
                 40.628 
               
               
                 5.189 
                 −1.567 
                 40.628 
               
               
                 6.179 
                 −3.366 
                 40.628 
               
               
                 4.945 
                 −3.767 
                 40.628 
               
               
                 6.007 
                 −3.662 
                 40.628 
               
               
                 5.391 
                 −4.119 
                 40.628 
               
               
                 6.254 
                 −3.151 
                 40.628 
               
               
                 5.753 
                 −1.956 
                 40.628 
               
               
                 6.275 
                 −3.039 
                 40.628 
               
               
                 5.938 
                 −3.754 
                 40.628 
               
               
                 5.504 
                 −4.107 
                 40.628 
               
               
                 6.227 
                 −2.588 
                 40.628 
               
               
                 5.090 
                 −1.511 
                 40.628 
               
               
                 6.221 
                 −3.260 
                 40.628 
               
               
                 5.385 
                 −1.685 
                 40.628 
               
               
                 5.573 
                 −1.814 
                 40.628 
               
               
                 6.070 
                 −3.567 
                 40.628 
               
               
                 5.279 
                 −4.100 
                 40.628 
               
               
                 5.611 
                 −4.067 
                 40.628 
               
               
                 5.480 
                 −1.748 
                 40.628 
               
               
                 1.540 
                 −0.382 
                 41.228 
               
               
                 1.049 
                 −0.437 
                 41.228 
               
               
                 0.710 
                 −0.282 
                 41.228 
               
               
                 0.970 
                 −0.215 
                 41.228 
               
               
                 0.673 
                 −0.247 
                 41.228 
               
               
                 1.426 
                 −0.348 
                 41.228 
               
               
                 1.603 
                 −0.717 
                 41.228 
               
               
                 1.383 
                 −0.602 
                 41.228 
               
               
                 1.272 
                 −0.546 
                 41.228 
               
               
                 0.856 
                 −0.183 
                 41.228 
               
               
                 0.692 
                 −0.156 
                 41.228 
               
               
                 0.665 
                 −0.197 
                 41.228 
               
               
                 1.312 
                 −0.315 
                 41.228 
               
               
                 1.084 
                 −0.248 
                 41.228 
               
               
                 0.742 
                 −0.150 
                 41.228 
               
               
                 0.936 
                 −0.384 
                 41.228 
               
               
                 1.198 
                 −0.282 
                 41.228 
               
               
                 0.823 
                 −0.333 
                 41.228 
               
               
                 1.493 
                 −0.659 
                 41.228 
               
               
                 1.160 
                 −0.491 
                 41.228 
               
               
                 3.239 
                 −0.911 
                 41.228 
               
               
                 2.874 
                 −1.492 
                 41.228 
               
               
                 2.108 
                 −0.554 
                 41.228 
               
               
                 2.772 
                 −1.422 
                 41.228 
               
               
                 2.669 
                 −1.352 
                 41.228 
               
               
                 2.565 
                 −1.284 
                 41.228 
               
               
                 2.448 
                 −0.659 
                 41.228 
               
               
                 2.461 
                 −1.216 
                 41.228 
               
               
                 3.075 
                 −1.638 
                 41.228 
               
               
                 1.994 
                 −0.519 
                 41.228 
               
               
                 2.561 
                 −0.694 
                 41.228 
               
               
                 2.356 
                 −1.150 
                 41.228 
               
               
                 2.221 
                 −0.588 
                 41.228 
               
               
                 3.013 
                 −0.838 
                 41.228 
               
               
                 2.250 
                 −1.086 
                 41.228 
               
               
                 2.975 
                 −1.564 
                 41.228 
               
               
                 2.144 
                 −1.022 
                 41.228 
               
               
                 2.036 
                 −0.959 
                 41.228 
               
               
                 2.900 
                 −0.802 
                 41.228 
               
               
                 2.674 
                 −0.730 
                 41.228 
               
               
                 1.929 
                 −0.897 
                 41.228 
               
               
                 2.334 
                 −0.623 
                 41.228 
               
               
                 2.787 
                 −0.765 
                 41.228 
               
               
                 1.821 
                 −0.836 
                 41.228 
               
               
                 1.767 
                 −0.450 
                 41.228 
               
               
                 1.881 
                 −0.484 
                 41.228 
               
               
                 3.174 
                 −1.713 
                 41.228 
               
               
                 1.712 
                 −0.776 
                 41.228 
               
               
                 3.126 
                 −0.874 
                 41.228 
               
               
                 1.653 
                 −0.416 
                 41.228 
               
               
                 4.137 
                 −1.218 
                 41.228 
               
               
                 4.249 
                 −1.258 
                 41.228 
               
               
                 3.834 
                 −2.278 
                 41.228 
               
               
                 3.914 
                 −1.138 
                 41.228 
               
               
                 4.762 
                 −3.438 
                 41.228 
               
               
                 4.471 
                 −1.342 
                 41.228 
               
               
                 3.559 
                 −2.026 
                 41.228 
               
               
                 4.627 
                 −3.230 
                 41.228 
               
               
                 3.744 
                 −2.192 
                 41.228 
               
               
                 4.557 
                 −3.127 
                 41.228 
               
               
                 3.652 
                 −2.108 
                 41.228 
               
               
                 4.695 
                 −3.334 
                 41.228 
               
               
                 3.689 
                 −1.061 
                 41.228 
               
               
                 4.026 
                 −1.178 
                 41.228 
               
               
                 3.577 
                 −1.023 
                 41.228 
               
               
                 3.465 
                 −1.945 
                 41.228 
               
               
                 4.411 
                 −2.926 
                 41.228 
               
               
                 3.464 
                 −0.985 
                 41.228 
               
               
                 4.800 
                 −1.477 
                 41.228 
               
               
                 4.335 
                 −2.828 
                 41.228 
               
               
                 4.828 
                 −3.543 
                 41.228 
               
               
                 4.691 
                 −1.431 
                 41.228 
               
               
                 4.257 
                 −2.732 
                 41.228 
               
               
                 3.272 
                 −1.789 
                 41.228 
               
               
                 4.360 
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                 46.628 
               
               
                 1.452 
                 −0.772 
                 46.628 
               
               
                 0.606 
                 −0.128 
                 46.628 
               
               
                 0.700 
                 −0.396 
                 46.628 
               
               
                 0.186 
                 −0.171 
                 46.628 
               
               
                 1.120 
                 −0.254 
                 46.628 
               
               
                 0.315 
                 −0.225 
                 46.628 
               
               
                 2.274 
                 −0.542 
                 46.628 
               
               
                 1.633 
                 −0.381 
                 46.628 
               
               
                 2.894 
                 −1.639 
                 46.628 
               
               
                 2.786 
                 −0.674 
                 46.628 
               
               
                 3.011 
                 −1.717 
                 46.628 
               
               
                 1.698 
                 −0.907 
                 46.628 
               
               
                 2.146 
                 −0.509 
                 46.628 
               
               
                 2.776 
                 −1.562 
                 46.628 
               
               
                 2.658 
                 −0.640 
                 46.628 
               
               
                 2.018 
                 −0.477 
                 46.628 
               
               
                 3.170 
                 −0.775 
                 46.628 
               
               
                 3.243 
                 −1.874 
                 46.628 
               
               
                 2.659 
                 −1.486 
                 46.628 
               
               
                 2.530 
                 −0.607 
                 46.628 
               
               
                 2.540 
                 −1.411 
                 46.628 
               
               
                 1.890 
                 −0.445 
                 46.628 
               
               
                 3.042 
                 −0.741 
                 46.628 
               
               
                 2.422 
                 −1.336 
                 46.628 
               
               
                 2.302 
                 −1.263 
                 46.628 
               
               
                 2.402 
                 −0.574 
                 46.628 
               
               
                 2.183 
                 −1.190 
                 46.628 
               
               
                 2.062 
                 −1.118 
                 46.628 
               
               
                 1.761 
                 −0.413 
                 46.628 
               
               
                 1.941 
                 −1.047 
                 46.628 
               
               
                 2.914 
                 −0.707 
                 46.628 
               
               
                 3.127 
                 −1.795 
                 46.628 
               
               
                 1.820 
                 −0.976 
                 46.628 
               
               
                 4.059 
                 −1.029 
                 46.628 
               
               
                 3.425 
                 −0.845 
                 46.628 
               
               
                 4.466 
                 −2.812 
                 46.628 
               
               
                 4.562 
                 −1.194 
                 46.628 
               
               
                 4.361 
                 −2.719 
                 46.628 
               
               
                 3.933 
                 −0.991 
                 46.628 
               
               
                 4.255 
                 −2.627 
                 46.628 
               
               
                 3.297 
                 −0.810 
                 46.628 
               
               
                 4.146 
                 −2.538 
                 46.628 
               
               
                 4.437 
                 −1.151 
                 46.628 
               
               
                 3.806 
                 −0.953 
                 46.628 
               
               
                 4.568 
                 −2.909 
                 46.628 
               
               
                 4.037 
                 −2.451 
                 46.628 
               
               
                 3.701 
                 −2.198 
                 46.628 
               
               
                 3.473 
                 −2.034 
                 46.628 
               
               
                 4.311 
                 −1.109 
                 46.628 
               
               
                 3.814 
                 −2.281 
                 46.628 
               
               
                 4.666 
                 −3.008 
                 46.628 
               
               
                 3.679 
                 −0.917 
                 46.628 
               
               
                 3.587 
                 −2.115 
                 46.628 
               
               
                 4.810 
                 −1.286 
                 46.628 
               
               
                 4.186 
                 −1.069 
                 46.628 
               
               
                 4.851 
                 −3.219 
                 46.628 
               
               
                 3.358 
                 −1.954 
                 46.628 
               
               
                 3.552 
                 −0.881 
                 46.628 
               
               
                 4.761 
                 −3.112 
                 46.628 
               
               
                 3.926 
                 −2.365 
                 46.628 
               
               
                 4.686 
                 −1.239 
                 46.628 
               
               
                 6.247 
                 −2.196 
                 46.628 
               
               
                 5.294 
                 −1.497 
                 46.628 
               
               
                 6.512 
                 −2.791 
                 46.628 
               
               
                 5.613 
                 −4.119 
                 46.628 
               
               
                 6.363 
                 −3.424 
                 46.628 
               
               
                 5.482 
                 −4.107 
                 46.628 
               
               
                 6.326 
                 −2.302 
                 46.628 
               
               
                 5.859 
                 −4.030 
                 46.628 
               
               
                 6.516 
                 −2.923 
                 46.628 
               
               
                 5.964 
                 −1.918 
                 46.628 
               
               
                 6.296 
                 −3.538 
                 46.628 
               
               
                 5.355 
                 −4.048 
                 46.628 
               
               
                 5.253 
                 −3.953 
                 46.628 
               
               
                 5.528 
                 −1.622 
                 46.625 
               
               
                 6.395 
                 −2.415 
                 46.628 
               
               
                 5.860 
                 −1.837 
                 46.628 
               
               
                 5.960 
                 −3.946 
                 46.628 
               
               
                 6.500 
                 −3.054 
                 46.628 
               
               
                 5.054 
                 −1.386 
                 46.628 
               
               
                 6.064 
                 −2.005 
                 46.628 
               
               
                 6.220 
                 −3.646 
                 46.628 
               
               
                 4.935 
                 −3.332 
                 46.628 
               
               
                 5.131 
                 −3.702 
                 46.628 
               
               
                 6.450 
                 −2.535 
                 46.628 
               
               
                 5.011 
                 −3.449 
                 46.628 
               
               
                 6.052 
                 −3.850 
                 46.628 
               
               
                 6.468 
                 −3.182 
                 46.628 
               
               
                 5.077 
                 −3.573 
                 46.628 
               
               
                 5.412 
                 −1.558 
                 46.628 
               
               
                 5.742 
                 −4.091 
                 46.628 
               
               
                 6.159 
                 −2.097 
                 46.628 
               
               
                 6.138 
                 −3.750 
                 46.628 
               
               
                 6.490 
                 −2.661 
                 46.628 
               
               
                 5.641 
                 −1.689 
                 46.628 
               
               
                 4.933 
                 −1.335 
                 46.628 
               
               
                 5.175 
                 −1.440 
                 46.628 
               
               
                 6.421 
                 −3.306 
                 46.628 
               
               
                 5.752 
                 −1.761 
                 46.628 
               
               
                 5.181 
                 −3.833 
                 46.628 
               
               
                   
               
             
          
         
       
     
         [0033]    While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment. On the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.