Abstract:
The present invention relates to the improved aerodynamic design of a pair of blade profiles valid over a wide range of flow regime. The so formed blades, pertain to high pressure, intermediate pressure and first few stages of low pressure cylinders of axial steam turbines. 
     The invented blades cover a wide range of stagger angles; pitch/chord ratios; inlet flow angles and outlet Mach numbers.

Description:
FIELD OF INVENTION 
   This invention relates to aerodynamically wide range applicable cylindrical blade profiles for axial steam turbines. 
   BACKGROUND OF THE INVENTION AND PRIOR ART 
   The designers of steam turbines seek for quick selection of useful blades with a minimum number of inventory. One would prefer a few efficient blades to cover a wide flow range prevailing in turbine stages. There are publications such as Deich et al. (Atlas of Blades Profiles for Axial Turbines 1965) for a set of profiles. Further, two patents U.S. Pat. No. 5,211,703 (1993) and U.S. Pat. No. 5,192,190 (1993) on stationary blade have been filed by the authors, viz. Ferteger, Jurek and Evans, David H. Such patents were for a twisted stationary blade with varying stagger angle from hub to tip (from 42 deg at hub to 52 deg at shroud). The blade is non-cylindrical and twisted over the span. A recent patent by the present author (U.S. Pat. No. 6,709,239) is for design of three dimensional twisted blade for use in entry stages of HP/IP cylinders of axial steam turbines. A related patent by Purcaru et al. (U.S. Pat. No. 4.695.228) deals with the construction of profiles through ellipse, parabola and circle segments. The present author has also filed an application (Pub. No. U.S. 2003/0231961A1, U.S. Pat. No. 6,979,178B2) for two cylindrical profiles for subsonic flow application and for a specified range of stagger angles. One of the profiles, P2822 is the reference profile for the present invention which concerns with a new blade profile; that can be used for forming a cylindrical blade i.e. with constant stagger from hub to tip. The blades formed by this profile are untwisted or cylindrical in shape. In addition, the present invention deals with both stationary (guide) and rotating (moving) type of blades for axial steam turbines. 
   While converting heat energy into kinetic energy, turbines blades suffer two kinds of aerodynamic losses; one—the profile loss due to stream wise boundary layer growth (along blade surfaces), and, mixing in blade wakes, the second—the profile loss due to secondary flow resulting from boundary layer growth along the hub and casing and flows resulting from turning of inlet boundary layer (passage vortex; pressure face to suction face in a cascade passage). The reduction in losses is achieved by various means such as smooth surface and aft-loaded pressure distribution along the blade surfaces (instead of fore-loaded or flat-topped design). Smooth contour variation usually ensures lower profile losses for incompressible and subsonic flows. The lower velocity and cross-channel pressure gradient in the first part of cascade passage where the secondary flow originates; and higher diffusion in the rear part of suction face are the desired features in aft-loaded profiles which in turn reduces secondary flow losses. 
   The cylindrical blade is defined herein as one of constant cross-section over the blade height.  FIG. 1  shows a schematic base profile. At any cross-section, the shape of the profile remains same as shown typically in  FIG. 2 . The profile or section is made of two surfaces; suction face and pressure face, each joining leading edge to trailing edge. X-axis and U-axis coincide with the turbine axis and circumferential directions, respectively. Usually the center of gravity lies at the origin of co-ordinate axes. The blade or profile is set at angle betabi or y, tg, also known as stagger or setting angle with respect of U-axis. Chord is defined as axial distance of base profile measured between two farthest tangents to the profile; one at leading edge side and other at trailing edge side. The tangents are normal to the chord. Axial chord is the projected length of the profile on X-axis; hence varies with profile stagger. Inlet and exit flow angles β 1 , tg and β 2 , tg are fluid flow angles with respect to tangent (U-axis); also referred as beta  1   x  and beta  2   x  with reference to turbine axis, respectively. The profile faces can be specified by various ways; e.g. through discrete points (x, y co-ordinates), through a set of arcs and through Bezier points. The basic difference between any two cylindrical blades is the profile shape and what is being claimed here is the unique quantitative shape of the proposed blade (e.g. geometrical ratios as shown in  FIG. 3 ). 
   OBJECTS OF THE INVENTION 
   An object of the present invention is to propose an aerodynamic efficient blade profile and relate and complement with another profile from application point of view. 
   Another object of the present invention is to propose an aerodynamic efficient blade profile which is applicable for a wide stagger variation. 
   Still another object of the present invention is to propose an aerodynamic efficient blade profile and wherein tooling is minimum. 
   DESCRIPTION OF THE INVENTION 
   According to this invention there is provided two cylindrical blades for axial steam turbines comprising a leading edge and a trailing edge with specified circles and a pressure face and suction face and joining at said trailing and leading edges and an inlet flow angle characterized in that the trailing edge is below the base line. 

   
     BRIEF DESCRIPTION OF DRAWINGS 
     The nature of invention, its objective and further, advantages residing in the same will be apparent from the following description made with reference to the non-limiting exemplary embodiments of the invention represented in the accompanying drawings. 
       FIG. 1  Profile Geometry Description (Base Profile) 
       FIG. 2  Profile Geometry Description (Stacked Profile) 
       FIG. 3  e 3  Profile: Geometrical Ratios 
       FIG. 4  e 3  Profile: (Stacked View) 
       FIG. 5  e 3  Profile: Loss characteristics as function of M 2 , S/c &amp; gamatg 
       FIG. 6  e 3  Profile: Outlet flow angles as function of M 2 , S/c &amp; gamatg 
       FIG. 7  e 3  Profile: Loss characteristics as function of inlet angle &amp; gamatg 
       FIG. 8  e 3  Profile: Outlet flow angle as function of inlet angle &amp; gamatg 
       FIG. 9  e 3  Profile: Loss characteristics as function of Gamatg &amp; M 2   
       FIG. 10  e 3  Profile: Outlet flow angle as function of Gamatg &amp; M 2   
       FIG. 11  e 9  Profile: Geometrical Ratios 
       FIG. 12  e 9  Profile: (Stacked View) 
       FIG. 13  e 9  Profile: Loss characteristics as function of inlet angle &amp; gamatg 
       FIG. 14  e 9  Profile: Outlet flow angle as function of inlet angle &amp; gamatg 
       FIG. 15  e 9  Profile: Loss characteristics as function of M 2 , s/c &amp; gamatg 
       FIG. 16  e 9  Profile: Outlet Flow angles as function of M 2 , s/c &amp; gamatg 
       FIG. 17  e 9  Profile: Loss characteristics as function of Gamatg &amp; M 2   
       FIG. 18  e 9  Profile: Outlet Flow angle as function of Gamatg &amp; M 2   
   

   The Profile Geometry:  FIG. 1  indicates a typical profile geometry L (or C) denotes the length of base chord, Diameters of leading edge circle, nearly the largest in-circle and trailing edge circles, are denoted by d 1 , D and d 2 . The peak locations (maximum height) of suction and pressure faces are denoted by ( 11 ,b 1 ) and ( 12 ,b 2 ); respectively. The coordinates of center of largest in-circle is ( 13 ,b 3 ). B 4  is the difference (b 1 −b 2 ). The vertical shift of lowest point at trailing edge (pressure face) from base line is denoted by b 5 . Pitch s is the circumferential distance between two adjacent blades in a turbine blade row. It is defined mathematically as S=2 nr/z; r being section radius of the blade where profile section is taken and z is no of blades in the blade-row. Blade turning angle (from inlet edge to outlet edge) is called as camber angle. 
   Performance Analysis: The proposed blade profiles are analyzed by a CFD (Computational Fluid Dynamics) software for various flow conditions to simulate incompressible as well as subsonic flow regime. The profiles are numerically experimented for a set of stagger angle y,tg (gamatg); pressure ratios (hence exit Mach no.), inlet flow angles and pitch-by-chord ratios to result outlet flow angles β 2 ,tg (or beta 2   x ) and energy loss coefficient. In total; result from 148 successful CFD runs are included herein to establish the nomograms. 
   Energy loss coefficient is defined as 
           ϛ   =     1   -       {     1   -       (     p2   /   po2     )         k   -   1     k         }     /     {     {     1   -       (     p2   /   po2     )         k   -   1     k         }                 
Where p 2  is mass-averaged static pressure at the outlet; po 1  and po 2  are mass averaged stagnation pressure at the inlet and exit of the cascade. K is the ratio of specific heats of working fluid (1.4 for air). Also note that beta 2   x =β 2 ,tg−90; beta 1   x =90−β 1 ,tg. It may be noted that the results quoted herein for energy loss coefficient ζ, is more indicative in nature than the absolute value, since it may vary quantitatively with the use of other CFD software. However the graphical patterns may not change significantly.
 
   The reference blade profile e 3 :
         1. Geometry:  FIG. 3  indicates a typical profile geometry e 3  having profile thickness value as 38% of chord located at 25% of chord distance from the leading edge. Other geometrical ratios are also shown in the same figure. The unique geometrical feature of the base profile is that the trailing edge (depth b 5 ) is below the base line. The stacked views of profiles for 2 extreme stagger angles (gamatg=43 and 63 degrees) are shown in  FIG. 4 .   2. Performance Analysis: The first proposed blade profile is analyzed and results are shown in graphical forms for quick use during design. ( FIGS. 5–10 ).       

     FIGS. 5 and 6  show the effect of exit Mach number M 2 ; pitch-chord ratio s/c and two useful extreme range of stagger angles; gamatg (47 and 57 deg) on energy loss coefficient ζ and outlet flow angles (beta 2   x ). The range of s/c and M 2  chosen is very wide: 0.65–1.05 and 0.3 to 1.2; respectively. The following observations may be noted:
         1. Higher the stagger angle, the lower is the loss at every exit Match on M 2     2. Loss increases with M 2  except at s/c=0.65 and gamatg=57   3. The suggested profile is useful for a range for a range of M 2 (M&lt;0.9)   4. Loss is minimum for s/c=0.85 and any M 2  (N 2 &lt;0.9)   5. Loss is maximum for s/c=0.65 for any M 2  (M 2 &lt;0.7) and also for s/c=1.05 for a M 2 ; M 2 &gt;0.7   6. Exit flow angle beta 2   x  decreases with increase in M 2  for M 2 =0.9 and below. The trend is opposite for M 2 &gt;0.9   7. Higher the stagger, the higher the exit flow angle beta 2   x      8. Beta  2   x  increases with increase in pitch-chord ratio s/c.   9.  FIGS. 5 and 6  indicate that s/c=0.85 is optimum ratio, from the point of view of loss.       
     FIGS. 7 and 8  show the behavior of profile for various inflow angle (incidence effects). The loss is independent of large variation of beta 1   x  (−10 to 30 degree) for both extreme stagger (gamatg=47 and 57) at s/c=0.85 and M 2 =0.6. Similarly there is very negligible change in outlet angle for a large variation in beta 1   x . The trend is valid for other M 2  and intermediate stagger angles.  FIGS. 9 and 10  are summary nomograms of performance for optimum pitch chord ratio=0.85. They indicate that the profile is useful for stagger angle range 47–63 resulting beta 2   x=− 76 to −60 for exit Mach no. range M 2 =0.3–0.9. 
   The invented blade profile e 9 :
         1. Geometry:  FIG. 11  indicates a typical profile geometry e 9  having profile thickness value as 33% of chord located at 27.8% of chord distance from the leading edge. Other geometrical ratios are also shown in the same FIG. It is more cambered profile then e 3  hence useful for low reaction blade. The unique geometrical feature of the base profile is that the trailing edge (depth b 5 ) is below the base line. The stacked views of profiles for 2 extreme stagger angles (gamatg=50 and 70 degrees) are shown in  FIG. 12 .       

   II. Performance Analysis: The first proposed blade profile is analyzed and results are shown in graphical forms for quick use during design ( FIGS. 13–18 ). 
   This profile shows the outlet angle variation independent of inlet flow angle (10–50 degree) for two extreme stagger angles 57 and 67 degrees for s/c=0.85 and M 2 =0.6. However, there is noticeable variation in loss coefficient and outlet angles as function of M 2 , s/c and stagger angles is shown in  FIGS. 15 and 16 . There is little variation in beta 2   x  for M 2 =0.9 and below. Beta 2   x  increases with M 2  for M 2 &gt;0.9. Energy loss coefficient is minimum for s/c=0.85 for M 2 &lt;0.9 and below. Two summary performance graphs are shown for optimum s/c=0.85 in  FIGS. 17 and 18 . Profile behavior is reasonably good for stagger angle range 57–67 covering beta 2   x=− 75 to −65 with relatively low loss. Thus with the help a pair profiles e 3  and e 9 , a range of inlet flow angles (10 to 50 degrees), exit Mach numbers (0.3 to 0.9) and stagger angles (47 to 67 degrees), the requirement of cylindrical blades with low energy loss can be accomplished.