Patent Publication Number: US-6910868-B2

Title: Airfoil shape for a turbine bucket

Description:
This invention was made with Government support under Contract No. DE-FC21-95MC31176 awarded by the Department of Energy. The Government has certain rights in this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to an airfoil for a bucket of a stage of a gas turbine and particularly relates to a third stage turbine bucket airfoil profile. 
     Many system requirements must be met for each stage of the hot gas path section of a gas turbine in order to meet design goals including overall improved efficiency and airfoil loading. Particularly, the buckets of the third stage of the turbine section must meet the operating requirements for that particular stage and also be capable of efficient manufacture. 
     BRIEF DESCRIPTION OF THE INVENTION 
     In accordance with a preferred embodiment of the present invention there is provided a unique airfoil shape for a bucket of a gas turbine, preferably the third stage bucket, that enhances the performance of the gas turbine. The airfoil shape hereof improves aerodynamic efficiency and fourth stage airfoil aerodynamic and mechanical loading. The bucket airfoil profile is defined by a unique loci of points to achieve the necessary efficiency and loading requirements whereby improved turbine performance is obtained. These unique loci of points define the nominal airfoil profile and are identified by the X, Y and Z Cartesian coordinates of Table I which follows. The 2040 points for the coordinate values shown in Table I are relative to the turbine centerline and for a cold, i.e., room temperature bucket airfoil at various cross-sections along its length. The positive X, Y and Z directions are axially parallel to the turbine rotor centerline looking aft toward the turbine exhaust, tangentially in the direction of engine rotation looking aft and radially outwardly toward the bucket tip, respectively. The X and Y coordinates are given in distance dimensions, e.g., units of inches, and are joined smoothly at each Z location to form a smooth continuous airfoil cross-section. The Z coordinates are given in non-dimensionalized form from 0 to 0.938. By multiplying the airfoil height dimension, e.g., in inches, by the non-dimensional Z value of Table I, the airfoil shape, i.e., the profile, of the bucket is obtained. Each defined airfoil section in the X and Y plane is joined smoothly with adjacent airfoil sections in the Z direction to form the complete airfoil shape. 
     It will be appreciated that as each bucket 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. Because a manufactured bucket airfoil profile may be different from the nominal airfoil profile given by the following table, a distance of plus or minus 0.150 inches from the nominal profile in a direction normal to any surface location along the nominal profile and which includes any coating process, defines a profile envelope for this bucket airfoil. The airfoil shape is robust to this variation without impairment of the mechanical and aerodynamic functions of the bucket. 
     It will also be appreciated that the airfoil can be scaled up or scaled down geometrically for introduction into similar turbine designs. Consequently, the X and Y coordinates in inches of the nominal airfoil profile given below may be a function of the same constant or number. That is, the X, Y 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 bucket airfoil profile while retaining the airfoil section shape. Similarly, the Z coordinate value, when converted to inches, may remain the same or be multiplied by the same or a different number as the X and Y coordinate values for scalability. 
     In a preferred embodiment according to the present invention, there is provided a turbine bucket including a bucket airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Z values are non-dimensional values from 0 to 0.938 convertible to Z distances in inches by multiplying the Z values by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the profile sections at the Z distances being joined smoothly with one another to form a complete airfoil shape. 
     In a further preferred embodiment according to the present invention, there is provided a turbine bucket including a bucket airfoil having an uncoated nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Z values are non-dimensional values from 0 to 0.938 convertible to Z distances in inches by multiplying the Z values by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the profile sections at the Z distances being joined smoothly with one another to form a complete airfoil shape, the X and Y values being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil. 
     In a further preferred embodiment according to the present invention, there is provided a turbine comprising a turbine wheel having a plurality of buckets, each of the buckets including an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Z values are non-dimensional values from 0 to 0.938 convertible to Z distances in inches by multiplying the Z values by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define the airfoil profile sections at each distance Z, the profile sections at the Z distances being joined smoothly with one another to form a complete airfoil shape. 
     In a further preferred embodiment according to the present invention, there is provided a turbine comprising a turbine wheel having a plurality of buckets, each of the buckets including an airfoil having an uncoated nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Z values are non-dimensional values from 0 to 0.938 convertible to Z distances in inches by multiplying the Z values by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the profile sections at the Z distances being joined smoothly with one another to form a complete airfoil shape, the X and Y distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down bucket airfoil. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic representation of a hot gas path through multiple stages of a gas turbine and illustrates a third stage bucket airfoil according to a preferred embodiment of the present invention; 
         FIG. 2  is a view of the bucket looking radially inwardly from the tip shroud; 
         FIGS. 3 and 4  are perspective views of the bucket hereof as viewed from 180° opposite angles; 
         FIGS. 5 and 6  are respective side elevational views of the bucket hereof; 
         FIG. 7  is an end elevational view of the bucket as viewed from its trailing edge; and 
         FIG. 8  is a representative illustration of a profile section of the bucket airfoil. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to the drawings, particularly to  FIG. 1 , there is illustrated a hot gas path, generally designated  10 , of a gas turbine  12  including a plurality of turbine stages. Four stages are illustrated. For example, the first stage comprises a plurality of circumferentially spaced nozzles  14  and buckets  16 . The nozzles are circumferentially spaced one from the other and fixed about the axis of the rotor. The first stage buckets  16 , of course, are mounted on a turbine rotor wheel  19 . A second stage of the turbine  12  is also illustrated, including a plurality of circumferentially spaced nozzles  18  and a plurality of circumferentially spaced buckets  20  mounted on a turbine rotor wheel  21 . The third stage includes a plurality of circumferentially spaced nozzles  22  and buckets  24  mounted on a turbine rotor wheel  26 . The fourth stage includes a plurality of circumferentially spaced nozzles  28  and buckets  30  mounted on a turbine rotor wheel  32 . It will be appreciated that the nozzles and buckets lie in the hot gas path  10  of the turbine, the direction of flow of the hot gas through the hot gas path  10  being indicated by the arrow  10 . The buckets and turbine wheels, as well as ancillary parts, form a turbine rotor. 
     It will be appreciated that the buckets, for example, the buckets  24  of the third stage, are mounted on the associated rotor wheel, e.g., wheel  26 , forming part of the rotor. 
     Referring now to  FIGS. 3 and 4 , illustrating the bucket  24 , there is illustrated an airfoil  34  constructed in accordance with the present invention. The airfoil  34  is mounted on a platform  36  carried by a shank  38  of bucket  24 . The shank  38  carries a dovetail  40  for coupling the bucket  24  to the turbine wheel  26 . The airfoil  34  also carries a tip shroud  42  adjacent its tip. It will be appreciated that the tip shroud  42 , airfoil  34 , platform  36 , shank  38  and dovetail  40  are collectively referred to as a bucket  24 . The airfoil  34  has a compound curvature with suction and pressure sides  44  and  46 , respectively (FIG.  2 ). The buckets  24  are, of course, spaced circumferentially one from the other about the wheel  26  and in the third stage of the turbine section of this preferred embodiment, there are ninety-two buckets. Also illustrated are wheelspace seals  48  in the form of angel wings on opposite ends of the shank  38  of each bucket. Preferably, the bucket is integrally cast with generally radially extending cooling passages internal to the bucket and exiting through the tip shroud  42  and into the hot gas path  10 , the exit openings being illustrated at  50 . 
     To define the airfoil shape of each third stage bucket airfoil  24 , there is a unique set or loci of points in space that meet the stage requirements and enable the airfoil to be manufactured. This unique loci of points meets the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. The loci which defines the bucket airfoil profile comprises a set of 2040 points relative to the axis of rotation of the turbine. A Cartesian coordinate system of X, Y and Z values given in Table 1 below defines the profile of the bucket airfoil at various locations along its length. The coordinate values for the X and Y coordinates are set forth in inches in Table I although other units of dimensions may be used when the values are appropriately converted. The Z values are set forth in Table I in non-dimensional form from 0 to 0.938. To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in the table is multiplied by the height of airfoil in inches. The Cartesian coordinate system has orthogonally-related X, Y and Z axes and the X axis lies parallel to the turbine rotor centerline, i.e., the rotary axis and a positive x coordinate value is axial toward the aft, i.e., exhaust end of the turbine. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor looking aft and the positive Z coordinate value is radially outwardly toward the bucket tip. 
     By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section of the bucket airfoil, e.g., the representative profile section  50  illustrated in  FIG. 8 , at each Z distance along the length of the airfoil can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section  50  at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections  50  to one another to form the airfoil profile. These values represent the airfoil profiles at ambient, non-operating or non-hot conditions and are for an uncoated airfoil. 
     The Table I values are generated and shown to three decimal places 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, including any coating thicknesses, are additive to the X and Y values given in Table I below. Accordingly, a distance of ±0.150 inches in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular bucket airfoil design and turbine, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in the Table below at the same temperature. The bucket airfoil design is robust to this range of variation without impairment of mechanical and aerodynamic functions. 
     The coordinate values given in Table I below provide the preferred nominal profile envelope. 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                 TABLE I 
               
               
                   
               
               
                 X 
                 Y 
                 Z 
                 X 
                 Y 
                 Z 
                 X 
                 Y 
                 Z 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 −1.274 
                 0.911 
                 0.000 
                 −1.291 
                 0.964 
                 0.063 
                 −0.845 
                 1.160 
                 0.125 
               
               
                 −0.798 
                 1.056 
                 0.000 
                 −0.823 
                 1.107 
                 0.063 
                 −1.303 
                 1.015 
                 0.125 
               
               
                 −1.090 
                 0.988 
                 0.000 
                 −1.016 
                 1.070 
                 0.063 
                 −1.035 
                 1.124 
                 0.125 
               
               
                 −1.363 
                 0.863 
                 0.000 
                 −0.725 
                 1.115 
                 0.063 
                 −1.465 
                 0.910 
                 0.125 
               
               
                 −0.896 
                 1.040 
                 0.000 
                 −1.459 
                 0.862 
                 0.063 
                 −0.749 
                 1.166 
                 0.125 
               
               
                 −1.183 
                 0.952 
                 0.000 
                 −1.202 
                 1.006 
                 0.063 
                 −1.217 
                 1.059 
                 0.125 
               
               
                 −0.698 
                 1.066 
                 0.000 
                 −0.920 
                 1.092 
                 0.063 
                 −0.941 
                 1.145 
                 0.125 
               
               
                 −1.448 
                 0.811 
                 0.000 
                 −1.110 
                 1.041 
                 0.063 
                 −1.386 
                 0.966 
                 0.125 
               
               
                 −0.994 
                 1.017 
                 0.000 
                 −1.376 
                 0.915 
                 0.063 
                 −1.127 
                 1.095 
                 9.125 
               
               
                 −0.012 
                 0.951 
                 0.000 
                 −0.431 
                 1.098 
                 0.063 
                 −1.541 
                 0.850 
                 0.125 
               
               
                 −0.299 
                 1.038 
                 0.000 
                 −0.054 
                 0.986 
                 0.063 
                 −0.365 
                 1.123 
                 0.125 
               
               
                 0.170 
                 0.866 
                 0.000 
                 −0.239 
                 1.054 
                 0.063 
                 −0.093 
                 1.025 
                 0.125 
               
               
                 −0.598 
                 1.069 
                 0.000 
                 −0.528 
                 1.110 
                 0.063 
                 −0.555 
                 1.158 
                 0.125 
               
               
                 −0.106 
                 0.986 
                 0.000 
                 0.035 
                 0.945 
                 0.063 
                 0.160 
                 0.883 
                 0.125 
               
               
                 −0.398 
                 1.054 
                 0.000 
                 0.206 
                 0.848 
                 0.063 
                 −0.273 
                 1.096 
                 0.125 
               
               
                 0.080 
                 0.911 
                 0.000 
                 −0.146 
                 1.023 
                 0.063 
                 −0.006 
                 0.982 
                 0.125 
               
               
                 −0.202 
                 1.015 
                 0.000 
                 −0.626 
                 1.116 
                 0.063 
                 −0.460 
                 1.144 
                 0.125 
               
               
                 0.257 
                 0.816 
                 0.000 
                 0.122 
                 0.899 
                 0.063 
                 0.239 
                 0.827 
                 0.125 
               
               
                 −0.498 
                 1.065 
                 0.000 
                 −0.334 
                 1.079 
                 0.063 
                 −0.182 
                 1.063 
                 0.125 
               
               
                 −1.685 
                 0.626 
                 0.000 
                 −1.538 
                 0.803 
                 0.063 
                 −0.652 
                 1.166 
                 0.125 
               
               
                 −1.891 
                 0.408 
                 0.000 
                 −1.878 
                 0.449 
                 0.063 
                 0.078 
                 0.935 
                 0.125 
               
               
                 −1.530 
                 0.753 
                 0.000 
                 −1.817 
                 0.527 
                 0.063 
                 −1.680 
                 0.716 
                 0.125 
               
               
                 −1.757 
                 0.557 
                 0.000 
                 −1.685 
                 0.672 
                 0.063 
                 −1.958 
                 0.322 
                 0.125 
               
               
                 −1.953 
                 0.329 
                 0.000 
                 −1.613 
                 0.740 
                 0.063 
                 −1.804 
                 0.567 
                 0.125 
               
               
                 −1.609 
                 0.692 
                 0.000 
                 −1.753 
                 0.601 
                 0.063 
                 −1.612 
                 0.785 
                 0.125 
               
               
                 −1.826 
                 0.484 
                 0.000 
                 −1.935 
                 0.369 
                 0.063 
                 −1.912 
                 0.407 
                 0.125 
               
               
                 0.423 
                 0.705 
                 0.000 
                 0.367 
                 0.735 
                 0.063 
                 −1.744 
                 0.643 
                 0.125 
               
               
                 0.341 
                 0.763 
                 0.000 
                 0.444 
                 0.674 
                 0.063 
                 −1.860 
                 0.489 
                 0.125 
               
               
                 −2.190 
                 −0.212 
                 0.000 
                 0.288 
                 0.793 
                 0.063 
                 0.315 
                 0.768 
                 0.125 
               
               
                 −2.011 
                 0.248 
                 0.000 
                 −1.006 
                 −0.076 
                 0.063 
                 0.390 
                 0.706 
                 0.125 
               
               
                 −1.173 
                 −0.155 
                 0.000 
                 −1.495 
                 −0.126 
                 0.063 
                 0.462 
                 0.642 
                 0.125 
               
               
                 −2.154 
                 −0.016 
                 0.000 
                 −1.981 
                 −0.201 
                 0.063 
                 −0.945 
                 −0.009 
                 0.125 
               
               
                 −0.873 
                 −0.134 
                 0.000 
                 −2.111 
                 −0.082 
                 0.063 
                 −1.715 
                 −0.063 
                 0.125 
               
               
                 −1.469 
                 −0.205 
                 0.000 
                 −1.203 
                 −0.087 
                 0.063 
                 −1.235 
                 −0.012 
                 0.125 
               
               
                 −2.065 
                 0.163 
                 0.000 
                 −1.689 
                 −0.161 
                 0.063 
                 −1.523 
                 −0.038 
                 0.125 
               
               
                 −2.183 
                 −0.112 
                 0.000 
                 −0.908 
                 −0.075 
                 0.063 
                 −2.026 
                 0.141 
                 0.125 
               
               
                 −1.371 
                 −0.186 
                 0.000 
                 −1.398 
                 −0.110 
                 0.063 
                 −1.041 
                 −0.007 
                 0.125 
               
               
                 −1.567 
                 −0.227 
                 0.000 
                 −2.035 
                 0.200 
                 0.063 
                 −1.811 
                 −0.073 
                 0.125 
               
               
                 −1.073 
                 −0.145 
                 0.000 
                 −1.883 
                 −0.196 
                 0.063 
                 −1.331 
                 −0.018 
                 0.125 
               
               
                 −2.113 
                 0.075 
                 0.000 
                 −2.075 
                 0.110 
                 0.063 
                 −1.997 
                 −0.041 
                 0.125 
               
               
                 −1.272 
                 −0.169 
                 0.000 
                 −1.105 
                 −0.080 
                 0.063 
                 −2.035 
                 0.046 
                 0.125 
               
               
                 −0.973 
                 −0.138 
                 0.000 
                 −2.103 
                 0.016 
                 0.063 
                 −1.619 
                 −0.050 
                 0.125 
               
               
                 −0.375 
                 −0.178 
                 0.000 
                 −1.592 
                 −0.143 
                 0.063 
                 −1.997 
                 0.234 
                 0.125 
               
               
                 −0.573 
                 −0.148 
                 0.000 
                 −2.072 
                 −0.170 
                 0.063 
                 −1.138 
                 −0.008 
                 0.125 
               
               
                 −0.277 
                 −0.198 
                 0.000 
                 −1.988 
                 0.286 
                 0.063 
                 −1.908 
                 −0.074 
                 0.125 
               
               
                 −0.474 
                 −0.161 
                 0.000 
                 −1.301 
                 −0.097 
                 0.063 
                 −1.427 
                 −0.027 
                 0.125 
               
               
                 −0.180 
                 −0.223 
                 0.000 
                 −1.786 
                 −0.180 
                 0.063 
                 −0.466 
                 −0.076 
                 0.125 
               
               
                 −0.773 
                 −0.135 
                 0.000 
                 −0.517 
                 −0.112 
                 0.063 
                 −0.009 
                 −0.229 
                 0.125 
               
               
                 −0.673 
                 −0.139 
                 0.000 
                 −0.229 
                 −0.179 
                 0.063 
                 −0.752 
                 −0.025 
                 0.125 
               
               
                 0.796 
                 0.372 
                 0.000 
                 −0.712 
                 −0.086 
                 0.063 
                 −0.280 
                 −0.128 
                 0.125 
               
               
                 0.579 
                 0.580 
                 0.000 
                 −0.420 
                 −0.130 
                 0.063 
                 −0.561 
                 −0.055 
                 0.125 
               
               
                 0.930 
                 0.223 
                 0.000 
                 −0.135 
                 −0.208 
                 0.063 
                 −0.098 
                 −0.192 
                 0.125 
               
               
                 1.117 
                 −0.013 
                 0.000 
                 −0.614 
                 −0.097 
                 0.063 
                 −0.848 
                 −0.015 
                 0.125 
               
               
                 0.726 
                 0.443 
                 0.000 
                 −0.324 
                 −0.153 
                 0.063 
                 −0.373 
                 −0.100 
                 0.125 
               
               
                 1.056 
                 0.067 
                 0.000 
                 −0.810 
                 −0.079 
                 0.063 
                 −0.656 
                 −0.038 
                 0.125 
               
               
                 1.176 
                 −0.094 
                 0.000 
                 0.519 
                 0.609 
                 0.063 
                 −0.189 
                 −0.158 
                 0.125 
               
               
                 0.502 
                 0.644 
                 0.000 
                 0.859 
                 0.254 
                 0.063 
                 0.967 
                 0.058 
                 0.125 
               
               
                 0.864 
                 0.296 
                 0.000 
                 1.155 
                 −0.138 
                 0.063 
                 0.664 
                 0.435 
                 0.125 
               
               
                 1.233 
                 −0.176 
                 0.000 
                 0.661 
                 0.473 
                 0.063 
                 1.132 
                 −0.181 
                 0.125 
               
               
                 0.654 
                 0.513 
                 0.000 
                 0.982 
                 0.101 
                 0.063 
                 0.851 
                 0.212 
                 0.125 
               
               
                 0.994 
                 0.146 
                 0.000 
                 0.795 
                 0.329 
                 0.063 
                 0.531 
                 0.575 
                 0.125 
               
               
                 −2.149 
                 −0.300 
                 0.000 
                 1.098 
                 −0.058 
                 0.063 
                 1.024 
                 −0.021 
                 0.125 
               
               
                 0.012 
                 −0.282 
                 0.000 
                 0.591 
                 0.543 
                 0.063 
                 0.728 
                 0.362 
                 0.125 
               
               
                 −2.056 
                 −0.331 
                 0.000 
                 0.921 
                 0.178 
                 0.063 
                 0.910 
                 0.136 
                 0.125 
               
               
                 0.380 
                 −0.439 
                 0.000 
                 1.210 
                 −0.220 
                 0.063 
                 0.599 
                 0.506 
                 0.125 
               
               
                 0.106 
                 −0.317 
                 0.000 
                 0.729 
                 0.402 
                 0.063 
                 1.079 
                 −0.100 
                 0.125 
               
               
                 −1.761 
                 −0.275 
                 0.000 
                 1.041 
                 0.022 
                 0.063 
                 0.790 
                 0.288 
                 0.125 
               
               
                 0.198 
                 −0.354 
                 0.000 
                 0.229 
                 −0.357 
                 0.063 
                 0.419 
                 −0.454 
                 0.125 
               
               
                 −1.859 
                 −0.300 
                 0.000 
                 0.049 
                 −0.276 
                 0.063 
                 0.166 
                 −0.312 
                 0.125 
               
               
                 −0.084 
                 −0.251 
                 0.000 
                 0.317 
                 −0.401 
                 0.063 
                 0.336 
                 −0.404 
                 0.125 
               
               
                 −1.956 
                 −0.322 
                 0.000 
                 0.140 
                 −0.315 
                 0.063 
                 0.080 
                 −0.269 
                 0.125 
               
               
                 −1.664 
                 −0.251 
                 0.000 
                 0.403 
                 −0.447 
                 0.063 
                 0.252 
                 −0.357 
                 0.125 
               
               
                 0.290 
                 −0.395 
                 0.000 
                 −0.043 
                 −0.241 
                 0.063 
                 0.815 
                 −0.731 
                 0.125 
               
               
                 0.643 
                 −0.584 
                 0.000 
                 0.655 
                 −0.601 
                 0.063 
                 0.581 
                 −0.559 
                 0.125 
               
               
                 0.728 
                 −0.637 
                 0.000 
                 0.489 
                 −0.497 
                 0.063 
                 0.738 
                 −0.672 
                 0.125 
               
               
                 0.469 
                 −0.485 
                 0.000 
                 0.737 
                 −0.656 
                 0.063 
                 0.501 
                 −0.505 
                 0.125 
               
               
                 0.812 
                 −0.692 
                 0.000 
                 0.573 
                 −0.548 
                 0.063 
                 0.660 
                 −0.615 
                 0.125 
               
               
                 0.557 
                 −0.533 
                 0.000 
                 0.817 
                 −0.713 
                 0.063 
                 1.237 
                 −0.343 
                 0.125 
               
               
                 1.290 
                 −0.259 
                 0.000 
                 1.317 
                 −0.385 
                 0.063 
                 1.387 
                 −0.592 
                 0.125 
               
               
                 1.554 
                 −0.684 
                 0.000 
                 1.419 
                 −0.553 
                 0.063 
                 1.288 
                 −0.426 
                 0.125 
               
               
                 1.503 
                 −0.598 
                 0.000 
                 1.264 
                 −0.302 
                 0.063 
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                 −0.578 
                 0.894 
                 0.938 
               
               
                 0.110 
                 0.871 
                 0.938 
               
               
                 −0.320 
                 0.591 
                 0.938 
               
               
                 0.379 
                 0.293 
                 0.938 
               
               
                 0.215 
                 0.656 
                 0.938 
               
               
                 −0.470 
                 0.777 
                 0.938 
               
               
                 −0.226 
                 0.462 
                 0.938 
               
               
                 0.315 
                 0.439 
                 0.938 
               
               
                 0.146 
                 0.800 
                 0.938 
               
               
                 −0.369 
                 0.654 
                 0.938 
               
               
                 −0.136 
                 0.331 
                 0.938 
               
               
                 0.249 
                 0.584 
                 0.938 
               
               
                 −0.273 
                 0.527 
                 0.938 
               
               
                 0.347 
                 0.366 
                 0.938 
               
               
                 0.181 
                 0.728 
                 0.938 
               
               
                 −0.419 
                 0.716 
                 0.938 
               
               
                 −0.181 
                 0.397 
                 0.938 
               
               
                 0.282 
                 0.512 
                 0.938 
               
               
                 −0.050 
                 0.197 
                 0.938 
               
               
                 0.156 
                 −0.144 
                 0.938 
               
               
                 0.034 
                 0.062 
                 0.938 
               
               
                 −0.092 
                 0.264 
                 0.938 
               
               
                 0.116 
                 −0.075 
                 0.938 
               
               
                 0.410 
                 0.220 
                 0.938 
               
               
                 −0.007 
                 0.129 
                 0.938 
               
               
                 0.195 
                 −0.213 
                 0.938 
               
               
                 0.075 
                 −0.007 
                 0.938 
               
               
                 0.441 
                 0.147 
                 0.938 
               
               
                 0.531 
                 −0.075 
                 0.938 
               
               
                 0.471 
                 0.073 
                 0.938 
               
               
                 0.561 
                 −0.149 
                 0.938 
               
               
                 0.502 
                 −0.001 
                 0.938 
               
               
                 0.590 
                 −0.223 
                 0.938 
               
               
                 0.350 
                 −0.492 
                 0.938 
               
               
                 0.234 
                 −0.283 
                 0.938 
               
               
                 0.426 
                 −0.632 
                 0.938 
               
               
                 0.312 
                 −0.422 
                 0.938 
               
               
                 0.388 
                 −0.562 
                 0.938 
               
               
                 0.273 
                 −0.352 
                 0.938 
               
               
                 0.676 
                 −0.446 
                 0.938 
               
               
                 0.761 
                 −0.669 
                 0.938 
               
               
                 0.619 
                 −0.297 
                 0.938 
               
               
                 0.704 
                 −0.520 
                 0.938 
               
               
                 0.789 
                 −0.744 
                 0.938 
               
               
                 0.648 
                 −0.371 
                 0.938 
               
               
                 0.733 
                 −0.595 
                 0.938 
               
               
                 0.463 
                 −0.702 
                 0.938 
               
               
                 0.816 
                 −0.818 
                 0.938 
               
               
                 0.538 
                 −0.843 
                 0.938 
               
               
                 0.722 
                 −1.196 
                 0.938 
               
               
                 0.899 
                 −1.043 
                 0.938 
               
               
                 0.612 
                 −0.984 
                 0.938 
               
               
                 0.844 
                 −0.893 
                 0.938 
               
               
                 0.500 
                 −0.773 
                 0.938 
               
               
                 0.685 
                 −1.125 
                 0.938 
               
               
                 0.926 
                 −1.117 
                 0.938 
               
               
                 0.575 
                 −0.914 
                 0.938 
               
               
                 0.872 
                 −0.968 
                 0.938 
               
               
                 0.648 
                 −1.055 
                 0.938 
               
               
                 0.954 
                 −1.192 
                 0.938 
               
               
                 0.832 
                 −1.408 
                 0.938 
               
               
                 1.035 
                 −1.417 
                 0.938 
               
               
                 0.904 
                 −1.550 
                 0.938 
               
               
                 1.116 
                 −1.642 
                 0.938 
               
               
                 0.981 
                 −1.267 
                 0.938 
               
               
                 0.795 
                 −1.338 
                 0.938 
               
               
                 0.977 
                 −1.692 
                 0.938 
               
               
                 1.062 
                 −1.492 
                 0.938 
               
               
                 0.868 
                 −1.479 
                 0.938 
               
               
                 1.142 
                 −1.717 
                 0.938 
               
               
                 1.008 
                 −1.342 
                 0.938 
               
               
                 0.759 
                 −1.267 
                 0.938 
               
               
                 0.941 
                 −1.621 
                 0.938 
               
               
                 1.089 
                 −1.567 
                 0.938 
               
               
                 1.013 
                 −1.763 
                 0.938 
               
               
                 1.197 
                 −2.115 
                 0.938 
               
               
                 1.169 
                 −1.792 
                 0.938 
               
               
                 1.085 
                 −1.905 
                 0.938 
               
               
                 1.249 
                 −2.017 
                 0.938 
               
               
                 1.156 
                 −2.047 
                 0.938 
               
               
                 1.196 
                 −1.867 
                 0.938 
               
               
                 1.049 
                 −1.834 
                 0.938 
               
               
                 1.121 
                 −1.976 
                 0.938 
               
               
                 1.223 
                 −1.942 
                 0.938 
               
               
                 1.264 
                 −2.093 
                 0.938 
               
               
                   
               
            
           
         
       
     
     In a preferred embodiment of this third stage turbine bucket airfoil, the hub radius at the leading edge of the airfoil is 40.716 inches. However, the Z=0 coordinate value is measured from a hub radius of 41.891 inches, i.e., at the leading edge of the airfoil. That is, the Z=0 coordinate value is 1.175 inches along a radius outwardly from the hub. The height of the airfoil bucket from Z=0 to Z=0.938 is 15.146 inches. 
     It will also be appreciated that the airfoil disclosed in the above Table may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table 1 may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table 1 would be represented by X and Y coordinate values of Table 1, multiplied or divided by a constant number. Similarly, the Z coordinate value, when converted to inches, may remain the same or be multiplied by the same or a different number as the X and Y coordinate values for scalability. 
     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, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.