Patent Publication Number: US-11384640-B2

Title: Airfoil shape and platform contour for turbine rotor blades

Description:
BACKGROUND OF THE INVENTION 
     The subject matter disclosed herein relates to turbine engine airfoils and, more specifically, to airfoils of turbine rotor blades. 
     Some aircraft and/or power plant systems, for example certain jet aircraft, gas turbines, and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. These turbines employ airfoils (e.g., turbine rotor blades, blades, airfoils, etc.) which during operation are exposed to fluid flows. These airfoils—and the endwalls or platforms from which the airfoils extend—are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of the airfoils and platforms have an impact on system and turbine operation, performance, thrust, efficiency, and power. 
     BRIEF DESCRIPTION OF THE INVENTION 
     Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice. 
     According to an exemplary embodiment, a turbine rotor blade includes an airfoil that extends from a platform, with the platform including a first portion of a nominal platform contour substantially in accordance with Cartesian coordinate values of X′, Y′, and Z′ as set forth in Table II. The Cartesian coordinate values of X′, Y′, and Z′ are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X′, Y′, and Z′ by a height of the airfoil defined along a Z′ axis. The X′ and Y′ values of the first portion are coordinate values that, when connected by smooth continuing arcs, define contour lines of the first portion of the nominal airfoil profile at each Z′ coordinate value. The contour lines may be joined smoothly with one another to form the first portion. 
     According to an exemplary embodiment, the present disclosure includes a turbine rotor blade having an airfoil that includes a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. 
     According to another exemplary embodiment, the present disclosure includes a turbine rotor blade having an airfoil that includes a suction side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion. 
     According to another exemplary embodiment, the present disclosure includes a turbine engine that has an airfoil that includes: a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I; and a suction side portion of the nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of the suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion. 
     Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which: 
         FIG. 1  is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and stationary vanes or nozzles according to at least one embodiment of the present disclosure; 
         FIG. 2  is a perspective view of a turbine rotor blade according to at least one embodiment of the present disclosure; and 
         FIG. 3  is a top view showing surface contour of a platform at the base of a turbine rotor blade according to at least one embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     One or more specific embodiments of the present subject matter will be described below. In an effort to provide a concise description of these embodiments, all features of an actual implementation may not be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
     When introducing elements of various embodiments of the present subject matter, the articles “a,” “an,” “the,” and “said” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. 
     Referring now to the drawings, particularly to  FIG. 1 , there is illustrated an exemplary turbine  10  of a combustion or gas turbine  12  including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine vanes or nozzles disposed axially adjacent to a corresponding row of turbine rotor blades. While four stages are illustrated in  FIG. 1 , the exact number of stages of the turbine  10  is a choice of engineering design, and may be more or less than the four stages shown. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner. 
     As depicted in  FIG. 1 , the first stage includes a plurality of turbine nozzles  100  and a plurality of turbine rotor blades  150 . The turbine nozzles  100  are annularly arranged about an axis of a turbine rotor  14 , while the turbine rotor blades  150  are annularly arranged about and coupled to the turbine rotor  14 . A second stage of the turbine  12  includes a plurality of turbine nozzles  200  annularly arranged about the axis of the turbine rotor  14  and a plurality of turbine rotor blades  250  annularly arranged about and coupled to the turbine rotor  14 . A third stage of the turbine  12  includes a plurality of turbine nozzles  300  annularly arranged about the axis of the turbine rotor  14  and a plurality of turbine rotor blades  350  annularly arranged about and coupled to the turbine rotor  14 . In particular embodiments, the turbine  12  may further include a fourth stage, which also includes plurality of turbine nozzles  400  annularly arranged about the axis of the turbine rotor  14  and a plurality of turbine rotor blades  450  annularly arranged about and coupled to the turbine rotor  14 . As will be appreciated, the turbine nozzles  100 ,  200 ,  300 ,  400  may be coupled to the casing  16  of the turbine  12  and remain stationary during operation. 
     It will be appreciated that the turbine nozzles  100 ,  200 ,  300  and  400  and turbine rotor blades  150 ,  250 ,  350  and  450  are disposed or at least partially disposed within a working fluid or hot gas path  18  of the turbine  12 . The various stages of the turbine  10  at least partially define the hot gas path  18  through which combustion gases, as indicated by arrows  20 , flow during operation of the gas turbine  12 . 
       FIG. 2  provides a perspective view of an exemplary turbine rotor blade  50 , as may be incorporated in one of the stages of the turbine  12 . In a preferred embodiment of the present disclosure, the turbine rotor blade  50  (as described by the points included within Table I below) is a turbine rotor blade  250  of the plurality of the turbine rotor blades  250  used in the second stage of the turbine  12 , as shown in  FIG. 1 . As illustrated in  FIG. 2 , the turbine rotor blade  50  includes an airfoil  52  having a pressure side  54  and an opposing suction side  56 . The pressure side  54  and the suction side  56  meet or intersect at a leading edge  58  and a trailing edge  60  of the airfoil  52 . A chord line  62  extends between the leading edge  58  and the trailing edge  60  such that pressure and suction sides  54 ,  56  can be said to extend in chord or chordwise between the leading edge  58  and the trailing edge  60 . 
     The airfoil  52  further includes a first end or root  64  which intersects with and extends radially outwardly from an endwall or platform  66  of the turbine rotor blade  50 . The airfoil  52  terminates radially at a second end or radial tip  68  of the airfoil  52 . The pressure and suction sides  54 ,  56  can be said to extend in span or in a span-wise direction  70  (along the height of the airfoil  52 ) between the platform  66  and the radial tip  68  of the airfoil  52 . In other words, each turbine rotor blade  50  includes an airfoil  52  having opposing pressure and suction sides  54 ,  56  that extend in chord or chordwise  62  between opposing leading and trailing edges  58 ,  60  and that extend in span or span-wise  70  between the platform  66  and the radial tip  68  of the airfoil  52 . 
     In particular configurations, the airfoil  52  may include a fillet  72  formed between the platform  66  and the airfoil  52  proximate to the root  64 . The fillet  72  can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that reduces fluid dynamic losses. In particular embodiments, the platform  66 , airfoil  52  and the fillet  72  can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered. In particular configurations, the turbine rotor blade  50  includes a mounting portion  74  which is formed to connect and/or to secure the turbine rotor blade  50  to the rotor shaft  14 . 
     The airfoil  52  of the turbine rotor blade  50  has a profile at any cross-section taken between the platform  66  or the root  64  and the radial tip  68 . In accordance with the present disclosure, the X, Y, and Z values of the profile are given in Table I as percentage values of the airfoil span or height. As one example only, the height of the airfoil  52  of the rotor blade  50  may be from about 2 inches to about 50 inches. As another example, the height of the airfoil  52  of the rotor blade  50  may be from about 3 inches to about 10 inches. However, it is to be understood that heights below or above this range may also be employed as desired in the specific application. 
     A hot gas path of a gas turbine requires airfoils that meet system requirements of aerodynamic and mechanical blade loading and efficiency. Additionally, the platforms and fillets at the base of the airfoils impact aerodynamic characteristics and losses of the rotor blade and must also satisfy system requirements of aerodynamic and mechanical blade loading and efficiency. That is, the aerodynamic characteristics and losses associated with each of the airfoil, the fillet, and/or the platform, separately, as well as the manner in which these components function together, significantly impact performance, thrust, efficiency, and power. As will be seen, to define the shape of these components, there is a unique set or locus of points in space that meet the stage requirements and that can be manufactured. This unique locus 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. These points are unique and specific to the system. 
     In accordance with the embodiments of the present disclosure, the locus of points that defines an airfoil profile of a turbine rotor blade includes a set of points with X, Y, and Z dimensions relative to a reference origin coordinate system, as provided in Table I and shown in  FIG. 2 , respectively. As provided below, in accordance with an alternative embodiment of the present disclosure, the locus of points that defines a contour of a platform of a turbine rotor blade includes a set of points with X′, Y′, and Z′ dimensions relative to a reference origin coordinate system, as provided in Table II and shown in  FIG. 3 , respectively. 
     The Cartesian coordinate system of X, Y, and Z values given in Table I below defines the airfoil profile of the turbine rotor blade  50  at various locations along its length or span or, as used herein, height. As shown in  FIG. 2 , the point data origin  76  is defined at or proximate to the root  64  at the leading edge  58  of the airfoil  52 . In one embodiment, as presented in Table I below, the point data origin  76  is defined at or proximate to (above or below) a transition or intersection line  78  defined between the fillet  72  and the airfoil  52 . The point data origin  76  corresponds to the non-dimensional Z value of Table I (presented below) at Z equals 0. 
     The coordinate values for the X, Y, and Z coordinates are set forth in non-dimensionalized units in Table I, although units of dimensions may be used when the values are appropriately converted. That is, the X, Y, and Z values set forth in Table I are expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the radial span  70  or height of the airfoil. As one example, the Cartesian coordinate values of X, Y, and Z may be convertible to dimensional distances by multiplying the X, Y, and Z values by a height of the airfoil at the leading edge  58  and/or multiplying by a constant number. As another example, the Cartesian coordinate values of X, Y, and Z may be convertible to dimensional distances by multiplying the X, Y, and Z values by a height of the airfoil at the trailing edge  60  and/or multiplying by a constant number. Thus, to convert the Z value to a Z coordinate value, for example, in inches, the non-dimensional Z value given in Table I is multiplied by the height of the airfoil in inches. 
     As described above, the Cartesian coordinate system has orthogonally-related X, Y, and Z axes, where the X axis lies generally parallel to a centerline of the rotor shaft  14 , i.e., the rotary axis and a positive X coordinate value is axial toward an aft, i.e., exhaust end, of the turbine  10 . The positive Y coordinate value extends tangentially in the direction of rotation of the rotor, and the positive Z coordinate value extends radially outwardly toward the radial tip  68  of the airfoil  52 . All the values in Table I are given at room temperature and do not include the fillet  72 . 
     By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the airfoil shape or profile sections (which may be referred to as “pressure side sections” on the pressure side and “suction side sections” on the suction side) of the airfoil  52  of the turbine rotor blade  50  along the span or height of the airfoil  52  can be ascertained. Thus, by connecting the X and Y values with smooth continuing arcs, each profile section (which may include a pressure side section and suction side section) at each distance Z is determined. The airfoil profiles of the various surface locations between the distances Z can then be determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile. 
     The Table I values are generated and shown to four decimal places for determining the profile of the airfoil. As the turbine rotor blade heats up during operation of the gas turbine, mechanical stresses and elevated temperatures will cause a change in the X, Y, and Z values. Accordingly, it should be understood that the values for the nominal airfoil profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil. 
     There are typical manufacturing tolerances as well as coatings which may be accounted for in the actual profile of the airfoil  52 . It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, may be additive to the X and Y values given in Table I below. Accordingly, a distance of +/−5% in a direction normal to any airfoil surface location or about +/−5% of the chord  62  in a direction nominal to any airfoil surface location may define an airfoil profile envelope for this particular airfoil design, 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 Table I below at the same temperature. According to another example, a tolerance of about 10-20% of a thickness of the airfoil&#39;s trailing edge  60  in a direction normal to any airfoil surface location may define a range of variation between measured points on an actual airfoil surface and ideal positions as embodied by the invention in Table I. As should further be understood, the data provided in Table I is scalable and the geometry pertains to all aerodynamic scales and/or RPM ranges. The design of the airfoil  52  for the turbine rotor blade  50  is robust to this range of variation without impairment of mechanical and aerodynamic functions. 
     
       
         
           
               
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 SUCTION SIDE 
                 PRESSURE SIDE 
               
            
           
           
               
               
               
               
               
               
            
               
                 X (%) 
                 Y (%) 
                 Z (%) 
                 X (%) 
                 Y (%) 
                 Z (%) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0.0000 
                 0.0000 
                 0.0000 
                 0.0000 
                 0.0000 
                 0.0000 
               
               
                 −0.2028 
                 0.5110 
                 0.0000 
                 0.2343 
                 −0.3110 
                 0.0000 
               
               
                 −0.2630 
                 1.0617 
                 0.0000 
                 0.5384 
                 −0.5589 
                 0.0000 
               
               
                 −0.2082 
                 1.6138 
                 0.0000 
                 0.8891 
                 −0.7261 
                 0.0000 
               
               
                 −0.0699 
                 2.1454 
                 0.0000 
                 1.2590 
                 −0.8480 
                 0.0000 
               
               
                 0.1260 
                 2.6591 
                 0.0000 
                 1.6357 
                 −0.9549 
                 0.0000 
               
               
                 0.3576 
                 3.1550 
                 0.0000 
                 2.0166 
                 −1.0480 
                 0.0000 
               
               
                 0.6151 
                 3.6372 
                 0.0000 
                 2.4002 
                 −1.1247 
                 0.0000 
               
               
                 0.8960 
                 4.1058 
                 0.0000 
                 2.7851 
                 −1.1932 
                 0.0000 
               
               
                 1.1973 
                 4.5647 
                 0.0000 
                 3.1701 
                 −1.2576 
                 0.0000 
               
               
                 1.5138 
                 5.0113 
                 0.0000 
                 3.5564 
                 −1.3234 
                 0.0000 
               
               
                 1.8426 
                 5.4469 
                 0.0000 
                 3.9427 
                 −1.3891 
                 0.0000 
               
               
                 2.1823 
                 5.8703 
                 0.0000 
                 4.3291 
                 −1.4535 
                 0.0000 
               
               
                 2.5317 
                 6.2854 
                 0.0000 
                 4.7168 
                 −1.5111 
                 0.0000 
               
               
                 2.8879 
                 6.6922 
                 0.0000 
                 5.1031 
                 −1.5590 
                 0.0000 
               
               
                 3.2523 
                 7.0936 
                 0.0000 
                 5.4908 
                 −1.6015 
                 0.0000 
               
               
                 3.6208 
                 7.4909 
                 0.0000 
                 5.8812 
                 −1.6439 
                 0.0000 
               
               
                 3.9934 
                 7.8827 
                 0.0000 
                 6.2730 
                 −1.6905 
                 0.0000 
               
               
                 4.3743 
                 8.2677 
                 0.0000 
                 6.6662 
                 −1.7426 
                 0.0000 
               
               
                 4.7606 
                 8.6472 
                 0.0000 
                 7.0580 
                 −1.7933 
                 0.0000 
               
               
                 5.1551 
                 9.0198 
                 0.0000 
                 7.4471 
                 −1.8179 
                 0.0000 
               
               
                 5.5552 
                 9.3869 
                 0.0000 
                 7.8307 
                 −1.7837 
                 0.0000 
               
               
                 5.9607 
                 9.7486 
                 0.0000 
                 8.2088 
                 −1.6850 
                 0.0000 
               
               
                 6.3758 
                 10.1021 
                 0.0000 
                 8.5814 
                 −1.5508 
                 0.0000 
               
               
                 6.7964 
                 10.4487 
                 0.0000 
                 8.9554 
                 −1.4138 
                 0.0000 
               
               
                 7.2183 
                 10.7980 
                 0.0000 
                 9.3294 
                 −1.2891 
                 0.0000 
               
               
                 7.6663 
                 11.0980 
                 0.0000 
                 9.7061 
                 −1.1754 
                 0.0000 
               
               
                 8.1595 
                 11.3131 
                 0.0000 
                 10.0829 
                 −1.0672 
                 0.0000 
               
               
                 8.6732 
                 11.4912 
                 0.0000 
                 10.4610 
                 −0.9644 
                 0.0000 
               
               
                 9.1979 
                 11.6309 
                 0.0000 
                 10.8418 
                 −0.8672 
                 0.0000 
               
               
                 9.7322 
                 11.7296 
                 0.0000 
                 11.2227 
                 −0.7822 
                 0.0000 
               
               
                 10.2733 
                 11.7857 
                 0.0000 
                 11.6076 
                 −0.7083 
                 0.0000 
               
               
                 10.8158 
                 11.8035 
                 0.0000 
                 11.9940 
                 −0.6480 
                 0.0000 
               
               
                 11.3583 
                 11.7844 
                 0.0000 
                 12.3830 
                 −0.6028 
                 0.0000 
               
               
                 11.8981 
                 11.7268 
                 0.0000 
                 12.7748 
                 −0.5713 
                 0.0000 
               
               
                 12.4324 
                 11.6364 
                 0.0000 
                 13.1708 
                 −0.5535 
                 0.0000 
               
               
                 12.9612 
                 11.5131 
                 0.0000 
                 13.5653 
                 −0.5521 
                 0.0000 
               
               
                 13.4790 
                 11.3583 
                 0.0000 
                 13.9599 
                 −0.5685 
                 0.0000 
               
               
                 13.9873 
                 11.1747 
                 0.0000 
                 14.3517 
                 −0.6000 
                 0.0000 
               
               
                 14.4859 
                 10.9651 
                 0.0000 
                 14.7407 
                 −0.6466 
                 0.0000 
               
               
                 14.9750 
                 10.7309 
                 0.0000 
                 15.1257 
                 −0.7069 
                 0.0000 
               
               
                 15.4531 
                 10.4761 
                 0.0000 
                 15.5079 
                 −0.7795 
                 0.0000 
               
               
                 15.9203 
                 10.2034 
                 0.0000 
                 15.8888 
                 −0.8644 
                 0.0000 
               
               
                 16.3765 
                 9.9116 
                 0.0000 
                 16.2682 
                 −0.9617 
                 0.0000 
               
               
                 16.8231 
                 9.6061 
                 0.0000 
                 16.6450 
                 −1.0699 
                 0.0000 
               
               
                 17.2587 
                 9.2856 
                 0.0000 
                 17.0203 
                 −1.1905 
                 0.0000 
               
               
                 17.6848 
                 8.9527 
                 0.0000 
                 17.3930 
                 −1.3206 
                 0.0000 
               
               
                 18.0999 
                 8.6088 
                 0.0000 
                 17.7601 
                 −1.4604 
                 0.0000 
               
               
                 18.5040 
                 8.2554 
                 0.0000 
                 18.1259 
                 −1.6070 
                 0.0000 
               
               
                 18.8999 
                 7.8910 
                 0.0000 
                 18.4862 
                 −1.7604 
                 0.0000 
               
               
                 19.2876 
                 7.5197 
                 0.0000 
                 18.8424 
                 −1.9207 
                 0.0000 
               
               
                 19.6657 
                 7.1388 
                 0.0000 
                 19.1958 
                 −2.0878 
                 0.0000 
               
               
                 20.0384 
                 6.7511 
                 0.0000 
                 19.5465 
                 −2.2591 
                 0.0000 
               
               
                 20.4014 
                 6.3552 
                 0.0000 
                 19.8931 
                 −2.4358 
                 0.0000 
               
               
                 20.7590 
                 5.9525 
                 0.0000 
                 20.2370 
                 −2.6166 
                 0.0000 
               
               
                 21.1097 
                 5.5442 
                 0.0000 
                 20.5781 
                 −2.8029 
                 0.0000 
               
               
                 21.4549 
                 5.1291 
                 0.0000 
                 20.9165 
                 −2.9947 
                 0.0000 
               
               
                 21.7919 
                 4.7085 
                 0.0000 
                 21.2521 
                 −3.1906 
                 0.0000 
               
               
                 22.1248 
                 4.2839 
                 0.0000 
                 21.5850 
                 −3.3920 
                 0.0000 
               
               
                 22.4522 
                 3.8537 
                 0.0000 
                 21.9138 
                 −3.6002 
                 0.0000 
               
               
                 22.7742 
                 3.4194 
                 0.0000 
                 22.2399 
                 −3.8126 
                 0.0000 
               
               
                 23.0906 
                 2.9824 
                 0.0000 
                 22.5632 
                 −4.0318 
                 0.0000 
               
               
                 23.4030 
                 2.5413 
                 0.0000 
                 22.8810 
                 −4.2565 
                 0.0000 
               
               
                 23.7112 
                 2.0974 
                 0.0000 
                 23.1975 
                 −4.4880 
                 0.0000 
               
               
                 24.0153 
                 1.6508 
                 0.0000 
                 23.5085 
                 −4.7264 
                 0.0000 
               
               
                 24.3167 
                 1.2028 
                 0.0000 
                 23.8167 
                 −4.9688 
                 0.0000 
               
               
                 24.6154 
                 0.7521 
                 0.0000 
                 24.1208 
                 −5.2182 
                 0.0000 
               
               
                 24.9099 
                 0.3000 
                 0.0000 
                 24.4222 
                 −5.4730 
                 0.0000 
               
               
                 25.2045 
                 −0.1534 
                 0.0000 
                 24.7195 
                 −5.7305 
                 0.0000 
               
               
                 25.4963 
                 −0.6083 
                 0.0000 
                 25.0140 
                 −5.9922 
                 0.0000 
               
               
                 25.7867 
                 −1.0631 
                 0.0000 
                 25.3058 
                 −6.2580 
                 0.0000 
               
               
                 26.0771 
                 −1.5179 
                 0.0000 
                 25.5963 
                 −6.5251 
                 0.0000 
               
               
                 26.3676 
                 −1.9741 
                 0.0000 
                 25.8826 
                 −6.7950 
                 0.0000 
               
               
                 26.6580 
                 −2.4303 
                 0.0000 
                 26.1675 
                 −7.0662 
                 0.0000 
               
               
                 26.9484 
                 −2.8851 
                 0.0000 
                 26.4498 
                 −7.3402 
                 0.0000 
               
               
                 27.2389 
                 −3.3413 
                 0.0000 
                 26.7292 
                 −7.6156 
                 0.0000 
               
               
                 27.5293 
                 −3.7975 
                 0.0000 
                 27.0073 
                 −7.8951 
                 0.0000 
               
               
                 27.8183 
                 −4.2537 
                 0.0000 
                 27.2827 
                 −8.1759 
                 0.0000 
               
               
                 28.1074 
                 −4.7099 
                 0.0000 
                 27.5553 
                 −8.4595 
                 0.0000 
               
               
                 28.3965 
                 −5.1661 
                 0.0000 
                 27.8252 
                 −8.7458 
                 0.0000 
               
               
                 28.6842 
                 −5.6237 
                 0.0000 
                 28.0923 
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     It will also be appreciated that the airfoil  52  disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y, and Z coordinate values of Table I, with the X, Y, and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number. 
     An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I. The actual profile on a manufactured turbine rotor blade will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a +or −5% profile tolerance is used herein. The X, Y, and Z values are all non-dimensionalized relative to the airfoil height. 
     The disclosed airfoil shape optimizes and is specific to machine conditions and specifications. It provides a unique profile to achieve: 1) interaction between other stages in the turbine  10 ; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical rotor blade or airfoil loadings. The disclosed locus of points defined in Table I allows the gas turbine  12  or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil  52  may be adopted as long as 1) interaction between other stages in the pressure turbine  10 ; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings are maintained in the scaled turbine. The airfoil  52  described herein thus improves overall gas turbine  12  efficiency. Specifically, the airfoil  52  provides a desired turbine efficiency lapse rate (ISO, hot, cold, part load, etc.). The airfoil  52  also meets all aeromechanics and stress requirements. The turbine rotor blade  50  described herein has very specific aerodynamic design requirements. Significant cross-functional design effort was required to meet these design goals. The airfoil  52  of the rotor blade  50 , thus, is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost-effective manner. 
     Further, it should be understood that exemplary embodiments of the present disclosure may include the entirety of the nominal airfoil profile set forth in Table I or portions thereof. Such portions may include a portion of the pressure side (or “pressure side portion”), a portion of the suction side (or “suction side portion”), and portions of both the pressure side and suction side. 
     Thus, exemplary embodiments of the present disclosure include a turbine rotor blade having an airfoil that includes a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. 
     Exemplary embodiments of the present disclosure also include a turbine rotor blade having an airfoil that includes a suction side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion. 
     Exemplary embodiment of the present disclosure also include a turbine engine that has an airfoil, wherein the airfoil includes: a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I; and a suction side portion of the nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of the suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y, and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion. 
     A height of each of the suction side portion and pressure side portion may be defined along the Z axis. The height of each of the suction side portion and pressure side portion may be less than or equal to the height of the airfoil. According to exemplary embodiments, the height of the suction side portion and the height of the pressure side portion may be substantially the same. According to exemplary embodiments, the suction side portion and the pressure side portion may each start at a common distance relative to a base of the airfoil and extend toward a tip of the airfoil. According to exemplary embodiments, the height of the suction side portion and/or the height of the pressure side portion may each be equal to or greater than 50% of the height of the airfoil. According to other embodiments, the height of the suction side portion and/or the height of the pressure side portion may each be equal to or greater than 75% of the height of the airfoil. According to other embodiments, the height of the suction side portion and the height of the pressure side portion may each be equal to 100% of the height of the airfoil. 
     As already described, the turbine engine may include a combustion or gas turbine that has a compressor, combustor, and turbine. The airfoil of the present disclosure may be configured as the airfoil of a turbine rotor blade, i.e., a rotor blade within the turbine of the gas turbine. More specifically, the airfoil of the present disclosure may be the airfoil of a turbine rotor blade that is configured to function as a second stage rotor blade in the turbine. In such cases, the turbine includes a second stage that has a row of circumferentially spaced nozzles and a row of circumferentially spaced rotor blades, with the airfoil of the present disclosure being configured as the airfoil of each of the rotor blades within the second stage row of rotor blades. 
     Along with the airfoil, the shape or contour of the platform impacts aerodynamic characteristics and losses of the rotor blade and must satisfy system requirements of aerodynamic and mechanical blade loading and efficiency. The aerodynamic characteristics and losses associated with the platform, separately, as well as the manner in which the platform functions in tandem with the airfoil and/or fillets, may significantly affect performance, thrust, efficiency, and power. Embodiments of the present disclosure include one or more platform contours, which may be coupled with the airfoil shape disclosed above or used separately. As provided below, to define the contour of the platform there is a unique set or locus of points in space that meets the stage requirements and that can be manufactured. This unique locus of points satisfies the requirements for stage efficiency and, as will be appreciated, are arrived at by iteration between aerodynamic and mechanical loadings and enable the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system. 
     Turning now to  FIG. 3 , a top view is provided of an endwall or platform  66 , which is the surface from which the airfoil  52  of the rotor blade  50  extends. In accordance with an alternative embodiment of the present disclosure, a locus of points—each of which are represented as points on the illustrated contour lines of  FIG. 3 —defines a specific contour of a platform of a turbine rotor blade. These points are set forth in a set of points with X′, Y′, and Z′ dimensions relative to a reference origin coordinate system, as provided in Table II and shown in  FIG. 3 , respectively. Embodiments of the present disclosure, thus, include a rotor blade having a non-axisymmetric platform contour that enhances performance, efficiency and/or durability of the rotor blade when compared with conventional designs. 
     More specifically, the Cartesian coordinate system of X′, Y′, and Z′ values given in Table II below defines a surface shape or contour of the platform  66  of the turbine rotor blade  50  at various locations or points defined along contour lines  87 . The contour lines  87  extend along the surface or surface area (also “total surface area”) of the platform  66 , which is defined between an outer periphery or edges of the platform  66 —which, as indicated, may be referred to as a leading edge  88 , trailing edge  90 , pressure edge  94 , and suction edge  96 —and a base of the airfoil  52  (i.e., the location at which the platform  66  terminates into the airfoil  52  and/or the fillet  72  of airfoil  52 ). As shown in  FIG. 3 , the point data origin  96  is defined at a position within the footprint of the airfoil  52 , for example, a point near a central portion of the footprint of the airfoil  52 . The point data origin  96  corresponds to the non-dimensional Z′ value of Table II (presented below) at Z′ equals 0. 
     The coordinate values for the X′, Y′, and Z′ coordinates are set forth in non-dimensionalized units in Table II, although units of dimensions may be used when the values are appropriately converted. That is, the X′, Y′, and Z′ values set forth in Table II are expressed in non-dimensional form (X′, Y′, and Z′) from 0% to 100%, which may be converted by multiplying each by the radial span  70  (as shown in  FIG. 2 ) or height of the airfoil. As one example, the Cartesian coordinate values of X′, Y′, and Z′ may be convertible to dimensional distances by multiplying the X′, Y′, and Z′ values by a height of the airfoil at the leading edge  58  and/or multiplying by a constant number. As another example, the Cartesian coordinate values of X′, Y′, and Z′ may be convertible to dimensional distances by multiplying the X′, Y′, and Z′ values by a height of the airfoil at the trailing edge  60  and/or multiplying by a constant number. Thus, to convert the Z′ value to a Z′ coordinate value of the platform contour, for example, in inches, the non-dimensional Z′ value given in Table II is multiplied by the height of the airfoil in inches. 
     As described above, the Cartesian coordinate system has orthogonally-related X′, Y′, and Z′ axes, where the X′ axis lies generally parallel to a centerline of the rotor shaft  14 , i.e., the rotary axis and a positive X′ coordinate value is axial toward an aft, i.e., exhaust end, of the turbine  10 . The positive Y′ coordinate value extends tangentially in the direction of rotation of the rotor, and the positive Z′ coordinate value extends radially outwardly toward the radial tip  68  of the airfoil  52 . All the values in Table II are given at room temperature. 
     By defining X′ and Y′ coordinate values at selected locations in a Z′ direction normal to the X′, Y′ plane, the contour of the platform is shown via several contour lines, each of which having multiple points spaced at regular intervals. By connecting the X′ and Y′ values of each of the contour lines with smooth continuing arcs, each contour line at each distance Z′ can be determined. The overall contour of the platform of the various surface locations can then be determined by smoothly connecting the adjacent contour lines to one another to form the platform. 
     The Table II values are generated and shown to four decimal places for determining the contour of the platform. As the turbine rotor blade heats up during operation of the gas turbine, mechanical stresses and elevated temperatures will cause a change in the X′, Y′, and Z′ values. Accordingly, it should be understood that the values for the nominal platform contour given in Table II represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated platform. Further, there are typical manufacturing tolerances as well as coatings which may be accounted for in the actual shape or contour of the platform  66 . It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/−values, including any coating thicknesses, may be additive to the X′, Y′ and/or Z′ values given in Table II below. Accordingly, a distance of +/−5% in a direction normal to any platform surface location or about +/−5% of the length of the chord  62  in a direction nominal to any platform surface location may define a platform contour envelope for this particular platform design, i.e., a range of variation between measured points on the actual platform surface at nominal cold or room temperature and the ideal position of those points as given in Table II below at the same temperature. According to another example, a tolerance of about 10-20% of a thickness of the airfoil&#39;s trailing edge  60  in a direction normal to any platform surface location may define a range of variation between measured points on an actual platform surface and ideal positions as embodied by the invention in Table II. As should further be understood, the data provided in Table II is scalable and the geometry pertains to all aerodynamic scales and/or RPM ranges. The design of the platform  66  for the turbine rotor blade  50  is robust to this range of variation without impairment of mechanical and aerodynamic functions. 
     
       
         
           
               
               
               
             
               
                 TABLE II 
               
               
                   
               
               
                 X′ (%) 
                 Y′ (%) 
                 Z′ (%) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 7.10323 
                 −4.29481 
                 −3.30160 
               
               
                 7.90465 
                 −4.22084 
                 −3.30160 
               
               
                 6.39633 
                 −3.90438 
                 −3.30160 
               
               
                 8.65676 
                 −3.90301 
                 −3.30160 
               
               
                 9.41434 
                 −3.60299 
                 −3.30160 
               
               
                 10.18289 
                 −3.33037 
                 −3.30160 
               
               
                 10.96239 
                 −3.08788 
                 −3.30160 
               
               
                 11.75149 
                 −2.88376 
                 −3.30160 
               
               
                 12.55017 
                 −2.71799 
                 −3.30160 
               
               
                 13.35708 
                 −2.59470 
                 −3.30160 
               
               
                 14.16809 
                 −2.50976 
                 −3.30160 
               
               
                 14.98185 
                 −2.46044 
                 −3.30160 
               
               
                 13.17077 
                 −12.99815 
                 −3.93178 
               
               
                 12.48853 
                 −12.57209 
                 −3.93178 
               
               
                 11.77067 
                 −12.20906 
                 −3.93178 
               
               
                 11.01582 
                 −11.92958 
                 −3.93178 
               
               
                 8.63210 
                 −11.81314 
                 −3.93178 
               
               
                 10.23084 
                 −11.75560 
                 −3.93178 
               
               
                 9.42941 
                 −11.71039 
                 −3.93178 
               
               
                 7.53887 
                 −6.85663 
                 −3.93178 
               
               
                 6.70457 
                 −6.57168 
                 −3.93178 
               
               
                 8.35811 
                 −6.48538 
                 −3.93178 
               
               
                 9.14446 
                 −6.03329 
                 −3.93178 
               
               
                 5.99630 
                 −6.00452 
                 −3.93178 
               
               
                 9.94452 
                 −5.60312 
                 −3.93178 
               
               
                 5.27707 
                 −5.45243 
                 −3.93178 
               
               
                 10.76238 
                 −5.20995 
                 −3.93178 
               
               
                 11.60079 
                 −4.86335 
                 −3.93178 
               
               
                 4.61264 
                 −4.83595 
                 −3.93178 
               
               
                 12.45976 
                 −4.57292 
                 −3.93178 
               
               
                 13.33790 
                 −4.34413 
                 −3.93178 
               
               
                 14.23111 
                 −4.18385 
                 −3.93178 
               
               
                 17.85328 
                 −4.13179 
                 −3.93178 
               
               
                 15.13391 
                 −4.08932 
                 −3.93178 
               
               
                 16.94774 
                 −4.07562 
                 −3.93178 
               
               
                 16.04082 
                 −4.05781 
                 −3.93178 
               
               
                 6.88266 
                 12.43236 
                 −3.93178 
               
               
                 12.96527 
                 12.62552 
                 −3.93178 
               
               
                 11.75286 
                 13.18446 
                 −3.93178 
               
               
                 7.82245 
                 13.35708 
                 −3.93178 
               
               
                 10.46373 
                 13.52832 
                 −3.93178 
               
               
                 9.13076 
                 13.60093 
                 −3.93178 
               
               
                 15.64765 
                 −13.37078 
                 −4.56196 
               
               
                 14.78046 
                 −12.35427 
                 −4.56196 
               
               
                 13.89410 
                 −11.35420 
                 −4.56196 
               
               
                 6.00589 
                 −11.00760 
                 −4.56196 
               
               
                 13.02281 
                 −10.34043 
                 −4.56196 
               
               
                 5.95246 
                 −9.78834 
                 −4.56196 
               
               
                 12.30221 
                 −9.22118 
                 −4.56196 
               
               
                 5.51682 
                 −8.64991 
                 −4.56196 
               
               
                 12.24604 
                 −7.91972 
                 −4.56196 
               
               
                 4.75786 
                 −7.69094 
                 −4.56196 
               
               
                 13.05432 
                 −6.87581 
                 −4.56196 
               
               
                 3.88657 
                 −6.82923 
                 −4.56196 
               
               
                 14.20645 
                 −6.20864 
                 −4.56196 
               
               
                 20.80142 
                 −6.06480 
                 −4.56196 
               
               
                 3.09473 
                 −5.89629 
                 −4.56196 
               
               
                 19.47668 
                 −5.88670 
                 −4.56196 
               
               
                 15.48736 
                 −5.83465 
                 −4.56196 
               
               
                 18.14919 
                 −5.74012 
                 −4.56196 
               
               
                 16.81485 
                 −5.69902 
                 −4.56196 
               
               
                 2.55360 
                 −4.80307 
                 −4.56196 
               
               
                 16.38057 
                 11.37886 
                 −4.56196 
               
               
                 5.13734 
                 11.63230 
                 −4.56196 
               
               
                 15.76409 
                 12.04329 
                 −4.56196 
               
               
                 15.12843 
                 12.68991 
                 −4.56196 
               
               
                 5.60038 
                 12.83924 
                 −4.56196 
               
               
                 14.47222 
                 13.31598 
                 −4.56196 
               
               
                 6.00452 
                 14.06809 
                 −4.56196 
               
               
                 6.25522 
                 15.33393 
                 −4.56196 
               
               
                 17.41900 
                 −13.82971 
                 −5.19214 
               
               
                 16.83129 
                 −12.85978 
                 −5.19214 
               
               
                 16.27372 
                 −11.87479 
                 −5.19214 
               
               
                 15.79012 
                 −10.85006 
                 −5.19214 
               
               
                 3.99479 
                 −10.52264 
                 −5.19214 
               
               
                 15.50106 
                 −9.75820 
                 −5.19214 
               
               
                 3.49476 
                 −9.52531 
                 −5.19214 
               
               
                 15.69012 
                 −8.65813 
                 −5.19214 
               
               
                 2.82211 
                 −8.63210 
                 −5.19214 
               
               
                 23.17556 
                 −8.08686 
                 −5.19214 
               
               
                 22.05494 
                 −7.92383 
                 −5.19214 
               
               
                 16.49291 
                 −7.87999 
                 −5.19214 
               
               
                 2.13850 
                 −7.74711 
                 −5.19214 
               
               
                 20.94116 
                 −7.71423 
                 −5.19214 
               
               
                 19.82191 
                 −7.53750 
                 −5.19214 
               
               
                 17.56285 
                 −7.52654 
                 −5.19214 
               
               
                 18.69306 
                 −7.44983 
                 −5.19214 
               
               
                 1.58915 
                 −6.77307 
                 −5.19214 
               
               
                 1.23981 
                 −5.71272 
                 −5.19214 
               
               
                 1.09323 
                 −4.60442 
                 −5.19214 
               
               
                 18.62319 
                 10.09932 
                 −5.19214 
               
               
                 3.47284 
                 10.68840 
                 −5.19214 
               
               
                 18.03000 
                 10.94596 
                 −5.19214 
               
               
                 17.43270 
                 11.78848 
                 −5.19214 
               
               
                 3.93452 
                 12.42962 
                 −5.19214 
               
               
                 16.83540 
                 12.62963 
                 −5.19214 
               
               
                 4.12905 
                 14.21741 
                 −5.19214 
               
               
                 3.75916 
                 15.96959 
                 −5.19214 
               
               
                 18.93280 
                 −14.25303 
                 −5.82232 
               
               
                 18.46565 
                 −13.19679 
                 −5.82232 
               
               
                 18.09713 
                 −12.10220 
                 −5.82232 
               
               
                 17.96150 
                 −10.95965 
                 −5.82232 
               
               
                 25.10994 
                 −10.14179 
                 −5.82232 
               
               
                 23.96329 
                 −10.01849 
                 −5.82232 
               
               
                 18.39441 
                 −9.91712 
                 −5.82232 
               
               
                 1.29872 
                 −9.83218 
                 −5.82232 
               
               
                 22.83581 
                 −9.76505 
                 −5.82232 
               
               
                 21.70834 
                 −9.51435 
                 −5.82232 
               
               
                 19.41503 
                 −9.41297 
                 −5.82232 
               
               
                 20.56579 
                 −9.35818 
                 −5.82232 
               
               
                 0.83019 
                 −8.97733 
                 −5.82232 
               
               
                 0.43017 
                 −8.08960 
                 −5.82232 
               
               
                 0.12193 
                 −7.16487 
                 −5.82232 
               
               
                 −0.08220 
                 −6.21275 
                 −5.82232 
               
               
                 −0.17398 
                 −5.24282 
                 −5.82232 
               
               
                 −0.14659 
                 −4.27016 
                 −5.82232 
               
               
                 20.64251 
                 8.60607 
                 −5.82232 
               
               
                 1.77957 
                 9.45544 
                 −5.82232 
               
               
                 20.14522 
                 9.45681 
                 −5.82232 
               
               
                 19.65340 
                 10.31030 
                 −5.82232 
               
               
                 2.02480 
                 10.66100 
                 −5.82232 
               
               
                 19.17392 
                 11.17200 
                 −5.82232 
               
               
                 2.16727 
                 11.88301 
                 −5.82232 
               
               
                 18.71772 
                 12.04466 
                 −5.82232 
               
               
                 2.13302 
                 13.11186 
                 −5.82232 
               
               
                 1.87958 
                 14.31468 
                 −5.82232 
               
               
                 1.47544 
                 15.47640 
                 −5.82232 
               
               
                 1.07131 
                 16.63950 
                 −5.82232 
               
               
                 20.41921 
                 −14.68868 
                 −6.45250 
               
               
                 20.17672 
                 −13.74889 
                 −6.45250 
               
               
                 20.10960 
                 −12.78444 
                 −6.45250 
               
               
                 26.04288 
                 −12.26522 
                 −6.45250 
               
               
                 27.00322 
                 −12.17070 
                 −6.45250 
               
               
                 25.08939 
                 −12.09672 
                 −6.45250 
               
               
                 20.45483 
                 −11.89534 
                 −6.45250 
               
               
                 24.15371 
                 −11.84054 
                 −6.45250 
               
               
                 23.21529 
                 −11.59120 
                 −6.45250 
               
               
                 21.29461 
                 −11.44736 
                 −6.45250 
               
               
                 22.26043 
                 −11.42270 
                 −6.45250 
               
               
                 −0.73978 
                 −9.21433 
                 −6.45250 
               
               
                 −1.13158 
                 −7.87999 
                 −6.45250 
               
               
                 −1.39188 
                 −6.51277 
                 −6.45250 
               
               
                 −1.48503 
                 −5.12501 
                 −6.45250 
               
               
                 −1.37544 
                 −3.73861 
                 −6.45250 
               
               
                 22.60977 
                 6.78540 
                 −6.45250 
               
               
                 −0.10001 
                 7.66354 
                 −6.45250 
               
               
                 22.06179 
                 7.91013 
                 −6.45250 
               
               
                 0.13015 
                 8.82800 
                 −6.45250 
               
               
                 21.53709 
                 9.04583 
                 −6.45250 
               
               
                 0.28358 
                 10.00479 
                 −6.45250 
               
               
                 21.05350 
                 10.20070 
                 −6.45250 
               
               
                 0.32057 
                 11.18981 
                 −6.45250 
               
               
                 20.67813 
                 11.39256 
                 −6.45250 
               
               
                 0.23015 
                 12.37345 
                 −6.45250 
               
               
                 0.03699 
                 13.54339 
                 −6.45250 
               
               
                 −0.20823 
                 14.70512 
                 −6.45250 
               
               
                 −0.44798 
                 15.86684 
                 −6.45250 
               
               
                 −0.62881 
                 17.03952 
                 −6.45250 
               
               
                 22.07685 
                 −15.20515 
                 −7.08268 
               
               
                 27.82245 
                 −14.52154 
                 −7.08268 
               
               
                 26.84567 
                 −14.48455 
                 −7.08268 
               
               
                 22.25906 
                 −14.24892 
                 −7.08268 
               
               
                 25.89903 
                 −14.22015 
                 −7.08268 
               
               
                 28.69101 
                 −14.08453 
                 −7.08268 
               
               
                 24.96061 
                 −13.92013 
                 −7.08268 
               
               
                 24.00575 
                 −13.68724 
                 −7.08268 
               
               
                 23.02760 
                 −13.68450 
                 −7.08268 
               
               
                 −2.46866 
                 −8.68827 
                 −7.08268 
               
               
                 −2.66320 
                 −7.70601 
                 −7.08268 
               
               
                 −2.79745 
                 −6.71416 
                 −7.08268 
               
               
                 −2.86595 
                 −5.71546 
                 −7.08268 
               
               
                 −2.86458 
                 −4.71539 
                 −7.08268 
               
               
                 −2.78101 
                 −3.71806 
                 −7.08268 
               
               
                 −2.62484 
                 −2.72895 
                 −7.08268 
               
               
                 −2.33304 
                 4.36057 
                 −7.08268 
               
               
                 24.53045 
                 4.70580 
                 −7.08268 
               
               
                 24.12083 
                 5.70313 
                 −7.08268 
               
               
                 −2.07137 
                 5.97438 
                 −7.08268 
               
               
                 23.73587 
                 6.71142 
                 −7.08268 
               
               
                 −1.78779 
                 7.58408 
                 −7.08268 
               
               
                 23.39475 
                 7.73478 
                 −7.08268 
               
               
                 23.15775 
                 8.78690 
                 −7.08268 
               
               
                 −1.58641 
                 9.20611 
                 −7.08268 
               
               
                 22.89883 
                 9.83218 
                 −7.08268 
               
               
                 −1.53161 
                 10.83910 
                 −7.08268 
               
               
                 22.62210 
                 10.87472 
                 −7.08268 
               
               
                 −1.62888 
                 12.47072 
                 −7.08268 
               
               
                 −1.82204 
                 14.09412 
                 −7.08268 
               
               
                 −2.01932 
                 15.71614 
                 −7.08268 
               
               
                 −2.12480 
                 17.34776 
                 −7.08268 
               
               
                 28.73484 
                 −17.00801 
                 −7.71286 
               
               
                 27.69779 
                 −16.84362 
                 −7.71286 
               
               
                 29.68833 
                 −16.60525 
                 −7.71286 
               
               
                 26.70320 
                 −16.49154 
                 −7.71286 
               
               
                 25.69628 
                 −16.17782 
                 −7.71286 
               
               
                 24.65374 
                 −16.15727 
                 −7.71286 
               
               
                 30.33221 
                 −15.77368 
                 −7.71286 
               
               
                 −4.44688 
                 −7.55942 
                 −7.71286 
               
               
                 −4.59758 
                 −5.77848 
                 −7.71286 
               
               
                 −4.63319 
                 −3.99068 
                 −7.71286 
               
               
                 −4.57292 
                 −2.20426 
                 −7.71286 
               
               
                 −4.55237 
                 −0.41784 
                 −7.71286 
               
               
                 −4.56470 
                 1.36996 
                 −7.71286 
               
               
                 26.54154 
                 2.17138 
                 −7.71286 
               
               
                 −4.53730 
                 3.15775 
                 −7.71286 
               
               
                 26.18947 
                 3.20570 
                 −7.71286 
               
               
                 25.88396 
                 4.25372 
                 −7.71286 
               
               
                 −4.36879 
                 4.93595 
                 −7.71286 
               
               
                 25.68121 
                 5.32502 
                 −7.71286 
               
               
                 29.89246 
                 5.37434 
                 −7.71286 
               
               
                 28.83622 
                 5.60723 
                 −7.71286 
               
               
                 30.68292 
                 6.01822 
                 −7.71286 
               
               
                 27.85670 
                 6.08946 
                 −7.71286 
               
               
                 25.84972 
                 6.37989 
                 −7.71286 
               
               
                 26.85937 
                 6.53058 
                 −7.71286 
               
               
                 −4.12083 
                 6.70731 
                 −7.71286 
               
               
                 30.57333 
                 7.08953 
                 −7.71286 
               
               
                 30.07877 
                 8.06083 
                 −7.71286 
               
               
                 −3.88931 
                 8.47866 
                 −7.71286 
               
               
                 29.43215 
                 8.94034 
                 −7.71286 
               
               
                 −3.76327 
                 10.26235 
                 −7.71286 
               
               
                 −3.73724 
                 12.04877 
                 −7.71286 
               
               
                 −3.76190 
                 13.83656 
                 −7.71286 
               
               
                 −3.77423 
                 15.62436 
                 −7.71286 
               
               
                 −3.71258 
                 17.41078 
                 −7.71286 
               
               
                 31.60627 
                 −18.15467 
                 −8.34304 
               
               
                 31.86520 
                 −17.54778 
                 −8.34304 
               
               
                 32.11864 
                 −16.93952 
                 −8.34304 
               
               
                 32.39948 
                 −16.34359 
                 −8.34304 
               
               
                 32.70224 
                 −15.75724 
                 −8.34304 
               
               
                 29.34174 
                 −2.46592 
                 −8.34304 
               
               
                 29.01158 
                 −1.48366 
                 −8.34304 
               
               
                 28.72526 
                 −0.48770 
                 −8.34304 
               
               
                 28.52387 
                 0.52743 
                 −8.34304 
               
               
                 32.46935 
                 0.60963 
                 −8.34304 
               
               
                 31.46380 
                 0.61237 
                 −8.34304 
               
               
                 30.52538 
                 1.04802 
                 −8.34304 
               
               
                 33.02418 
                 1.45763 
                 −8.34304 
               
               
                 29.60477 
                 1.52476 
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                 28.64306 
                 1.53435 
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                 33.21734 
                 2.47414 
                 −8.34304 
               
               
                 33.29954 
                 3.50709 
                 −8.34304 
               
               
                 33.33516 
                 4.54278 
                 −8.34304 
               
               
                 33.25570 
                 5.57298 
                 −8.34304 
               
               
                 32.92280 
                 6.55250 
                 −8.34304 
               
               
                 32.42962 
                 7.46352 
                 −8.34304 
               
               
                 31.84328 
                 8.31838 
                 −8.34304 
               
               
                 33.69546 
                 −18.73005 
                 −8.97322 
               
               
                 33.81601 
                 −18.25467 
                 −8.97322 
               
               
                 33.96397 
                 −17.78615 
                 −8.97322 
               
               
                 34.18042 
                 −17.34502 
                 −8.97322 
               
               
                 34.45167 
                 −16.93678 
                 −8.97322 
               
               
                   
               
            
           
         
       
     
     It will also be appreciated that the platform contour disclosed in the above Table II may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table II may be scaled upwardly or downwardly such that the relative surface shape of the platform remains unchanged. A scaled version of the coordinates in Table II would be represented by X′, Y′, and Z′ coordinate values of Table II, with the X′, Y′, and Z′ non-dimensional coordinate values converted to inches, multiplied or divided by a constant number. 
     An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an actual platform surface and the ideal position listed in Table II. The actual profile on a manufactured turbine rotor blade will be different than those in Table II and the design is robust to the variation or tolerances described above, meaning that mechanical and aerodynamic function are not impaired. 
     The disclosed platform contour optimizes and is specific to machine conditions and specifications. In use, it provides a unique surface contour that achieves: 1) aerodynamic efficiency; and 2) normalized aerodynamic and mechanical rotor blade or platform loading. The disclosed locus of points defined in Table II allows the gas turbine  12  or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed platform contour may be adopted as long as 1) aerodynamic efficiency; and 2) normalized aerodynamic and mechanical rotor blade loadings are maintained in the scaled turbine. The platform  66  described herein thus improves overall gas turbine  12  efficiency. The disclosed platform  66  also meets all aeromechanics and stress requirements. The turbine rotor blade  50  described herein has very specific aerodynamic design requirements. Significant cross-functional design effort was required to meet these design goals. The platform  66  of the rotor blade  50 , thus, is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost-effective manner, and, in accordance with alternative embodiments, may be use to advantage in conjunction with the airfoil disclosed above. 
     Further, it should be understood that exemplary embodiments of the present disclosure may include the entirety of the nominal surface shape or contour set forth in Table II or portions thereof. Such portions may include a portion of the platform  66  adjacent to or near the pressure edge  94 , a portion of the platform  66  adjacent to or near the suction edge  96 , a portion of the platform  66  adjacent to or near the leading edge  88 , and/or a portion of the platform  66  adjacent or near the trailing edge  90 . Such portions further may include a portion of the platform  66  defined between the pressure edge  94  and the airfoil  52 , a portion of the platform  66  defined between the suction edge  96  and the airfoil. 
     Thus, embodiments of the present disclosure may include a turbine rotor blade including an airfoil that extends from a platform. The platform may include a first portion of a nominal platform contour substantially in accordance with Cartesian coordinate values of X′, Y′, and Z′ as set forth in Table II. The Cartesian coordinate values of X′, Y′, and Z′ are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X′, Y′, and Z′ by a height of the airfoil defined along a Z′ axis. The X′ and Y′ values of the first portion are coordinate values that, when connected by smooth continuing arcs, define contour lines of the first portion of the nominal airfoil profile at each Z′ coordinate value. The contour lines may be joined smoothly with one another to form the first portion. As defined above, a total surface area of the platform is defined between an outer periphery of the platform, which is defined by a leading edge, trailing edge, pressure edge, and suction edge of the platform, and a base of the airfoil. In accordance with exemplary embodiments, the surface area of the first portion may be equal to or greater than 50% of the total surface area of the platform. In accordance with other embodiments, the surface area of the first portion may be equal to or greater than 75% of the total surface area of the platform. In accordance with still other embodiments, the surface area of the first portion may be equal to 100% of the total surface area of the platform. 
     This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.