Abstract:
The invention is an Exit Stay Apparatus for Francis and propeller hydraulic turbines. The purpose of the invention is to eliminate the loss of turbine efficiency and strong pulsations in draft tube caused by the axial circular vortex in all turbine operating regimes other than optimum without a noticeable decrease in maximum efficiency. It can be incorporated not only into newly fabricated hydraulic turbines, but also retrofitted into existing turbines. The Exit Stay Apparatus has a crown and exit stay vanes secured to the the crown. When installed in the turbine, the exit stay crown is located immediately after the runner crown, which is truncated at the bottom by a plane perpendicular to the central axis of the turbine. The exit stay crown together with the truncated runner crown forms water passages after the runner blade crown profile exit. The exit stay vanes are arranged in a circular array around the turbine axis, located after the runner blades, and attached at the periphery either to the draft tube cone or to an exit stay flange secured to the turbine discharge ring and to the draft tube cone.

Description:
BACKGROUND OF THE INVENTION 
   This invention relates to reaction hydraulic turbines. More specifically, the invention relates to reaction hydraulic turbines with a radial intake having a spiral casing with inlet stay vanes, a radial guide gate apparatus with wicket gates, either a mixed flow runner or an axial flow runner with runner blades secured to the runner crown, and a draft tube with a cone and an elbow. 
   At any hydroelectric plant the water level in the upper reservoir varies in time. The upper reservoir level depends on the flow of the river on which the plant is situated and on the seasonal demand of the power grid supplied by the plant. Turbine head, denoted by H t , varies along with the upper reservoir level. 
   Power output of a turbine, denoted by P t , is continually adjusted to meet the immediate demand of the power grid. Thus, P t  is also a time dependent variable. Power output of a reaction hydraulic turbine is adjusted by changing the discharge angle of the wicket gates of the guide gate apparatus. 
   Power output of a turbine P t  (kW) is given by the following formula:
 
P t =gη t Q t H t   (1)
 
where:
         η t  is the efficiency of the turbine,   H t  is the turbine head (m),   Q t  is the flow rate through the turbine (m 3 /sec), and   g is gravitational acceleration (g=9.81 m/sec 2 ).
 
Formula (1) shows that, for a fixed value of H t  power output P t  is proportional to the flow rate Q t . The flow rate of the turbine can be adjusted by varying the wicket gate discharge angle α 1 . The wicket gate discharge angle is the angle of a wicket gate exit element relative to the circumference of the turbine. The flow rate of the turbine is an increasing function of the wicket gate discharge angle.
       

   The following considerations involve the concept of an elementary turbine. The flow inside the turbine passages is partitioned into thin laminated by axisymmetric stream surfaces of averaged meridional flow. An elementary turbine is the part of a turbine located in one such thin lamina. 
   For an elementary turbine the difference between the values of whirl at the wicket gate exit and at the runner blade exit, denoted by Δ(V u R), is given by Euler&#39;s equation: 
               Δ   ⁡     (       V   u     ⁢   R     )       =       g   ⁢           ⁢     η   t     ⁢     H   t       ω             (   2   )             
 
where ω is the angular velocity of the turbine (ω=πN/30, where N is the rotation rate of the turbine in rpm). Meanwhile, for the i-th elementary turbine, the value of whirl at the wicket gate exit, denoted by [(V u R) 1 ] i , is given by
 
[(V u R) 1 ] i =[(V m R) 1 ] i  cot α 1   (3)
 
where [(V m R) 1 ] i  is the moment of velocity meridional component with respect the turbine axis at the wicket gate exit edge. Combining (2) and (3) one obtains the formula for whirl at the runner blade trailing edge for the i-th elementary turbine, denoted by [(V u R) 2 ] i . 
                 [       (       V   u     ⁢   R     )     2     ]     i     =           [       (       V   m     ⁢   R     )     1     ]     i     ⁢   cot   ⁢           ⁢     α   1       -       g   ⁢           ⁢     η   t     ⁢     H   t       ω               (   4   )             
 
   Formula (4) shows that for each elementary turbine the value of whirl at the runner blade exit varies with the values of P t  (via α 1 ) and H t . In particular, whirl does not necessarily vanish at the runner crown. If (V u R) 2 ≢0 at the runner crown, an axial circular vortex forms at the runner crown tip. Otherwise V u =(V u R) 2 /R would tend to infinity as R→0 leading to a contradiction (see L. M. Milne-Thomson,  Theoretical Hydrodynamics,  Macmillan [1960]). 
   The axial circular vortex core (0≦R≦R cv , where R cv  is the core radius) rotates as a solid body with velocity: 
               V   u     =         ω     c   ⁢           ⁢   υ       ⁢   R     2             (   5   )             
 
where ω cv  is distributed vorticity inside the core. The flow outside the axial circular vortex (R&gt;R cu ) is similar to the flow after the runner blade trailing edge and has the same values of [(V u R) 2 ] i , for the i-th elementary turbine. The axial circular vortex produces strong pulsations in draft tube. It ultimately dissipates due to the viscosity of water, causing a significant loss of head i turbine what results in a decrease of turbine efficiency given by: 
               Δη     c   ⁢           ⁢   υ       =         (       V   u     ⁢   R     )       2   ⁢   c   ⁢           ⁢   τ     2       2   ⁢     gR   dt   2     ⁢     H   t                 (   6   )             
 
where (V u R) 2cr  is whirl at the runner blade trailing edge in the elementary turbine adjacent to the runner crown and R dt  is the draft tube cone inlet radius (see G. I. Topazh,  Computation of Integral Hydraulic Indicators of Hydromachines,  Leningrad [1989]).
 
   In order to avoid strong pulsation in draft tube and a loss efficiency due to the axial circular vortex in the design regime, turbines are designed to have (V u R) 2cr =0 for the design values of power output (P t ) d  and head (H t ) d . However, with variation of H t  and especially with variation of P t , there is a significant loss of efficiency due to the axial circular vortex in prior art reaction hydraulic turbines with runner blades secured to the runner crown and having a draft tube with an elbow. For example, for a turbine with maximum efficiency η max =0.93, when H t =0.80 (H t ) d  and P 5 =0.50 (P t ) d , one may compute using (6) an efficiency loss of Δη cu =0.08 (i.e. 8%). 
   At this point Moody inventions (U.S. Pat. Nos. 1,769,887, July 1930, 1,848,738 March 1932, 1,848,739 March 1932, and 1,929,099, October 1933) should be mentioned. In all four these inventions Moody introduced draft tubes without an elbow and a horizontal diffuser. 
   Inside all Moody draft tubes there is a stationary pole mounted at the bottom. The pole is a geometrical continuation of the runner crown. The efficiency loss due to the axial circular vortex is eliminated in a reaction hydraulic turbine with runner blades secured to the runner crown and having one of Moody draft tubes. However, Moody draft tubes are inferior to the ones with an elbow and a horizontal diffuser and the turbine built with one of Moody draft tubes would have smaller efficiency at optimal operating regime. 
   For this reason turbines with Moody draft tubes with a stationary pole are not utilized at hydroelectric power plants. 
   BRIEF SUMMARY OF THE INVENTION 
   The present invention discloses an exit stay apparatus for a hydraulic turbine with runner blades secured to the runner crown. The purpose of the invention is to eliminate the loss of turbine efficiency and strong pulsations in draft tube caused by the axial circular vortex in all turbine operating regimes other than optimum without a noticeable decrease in maximum efficiency. The proposed exit stay apparatus can be incorporated not only into newly fabricated hydraulic turbines, but also retrofitted into existing Francis and propeller turbines. 
   The exit stay apparatus has an exit stay crown and exit stay vanes secured to the exit stay crown. When installed in the turbine, the exit stay crown is located immediately after the runner crown, which is truncated at the bottom by a plane perpendicular to the central axis of the turbine. The exit stay crown has the shape of a cup and together with the truncated runner crown forms water passages after the runner blade crown profile exit. The exit stay vanes are
         (a) arranged in a circular array around the turbine axis,   (b) located after the runner blades, and   (c) secured at the periphery either to the draft tube cone or to an exit stay flange secured to the turbine discharge ring and to the draft tube cone.       

   Inlet edges of the exit vanes are located near the runner blade exit edges. For each elementary turbine the distance between the runner blade exit edge and the exit stay vane inlet edge is preferably not smaller than the distance between two adjacent runner blade exit edges along the circumference, denoted by T. This is in order to avoid strong pulsations at the exit stay vane inlet edges. The solidity of the cascade formed by profiles of said exit stay vans is (L/T) ev , where L is the length of the cascade profile. The solidity of the cascade varies from values greater than 1.1 at the exit stay crown to relatively small values at the periphery. The value of (L/T) ev  at the periphery and the maximum relative thickness for the exit vane profiles along the exit vane span are determined from structural considerations. 
   The exit stay vane profiles are subsets of the axisymmetric stream surfaces bounding the elementary turbines. The profile contours are the lines of intersection of said axisymmetric stream surfaces with exit vane bounding surface. 
   Let β i  and β e  denote respectively the angles of inlet and exit profile elements relative to the turbine circumference. Along its leading edge each profile the inlet angle β i  is given by 
               tan   ⁢           ⁢     β   i       =         (     V   m     )     opt         (     V   u     )     opt               (   7   )             
 
where (V m ) opt  and (V u ) opt  are meridional and circumferential components of velocity along the leading edge in the optimum operating regime of the turbine. The exit stay vane exit angle along its trailing edge is β e =90°.
 
   The geometrical shape of the exit stay vanes, described above, enables the exit stay apparatus to substantially decrease the loss of turbine efficiency in operational regimes other than optimum. In the optimum regime the inlet shock losses for all profiles along the exit vane span vanish. There are small losses due to friction at the stay vane cascades. These fraction losses are barely noticeable for the elementary turbines near the turbine axis, where (L/T) ev ≈1.0, and are practically zero for the midstream and peripheral elementary trubines. Since the predominant portion of the flow passes through the midstream and peripheral elementary turbines, the impact of the additional friction losses caused by the exit stay apparatus on the maximum efficiency is not noticeable. 
   The solidity of the exit vane crown elementary turbine cascade is (L/T) ev &gt;1.1, therefore, for all operational regimes other than optimum the exit vane crown elementary turbine cascade redirects the flow to become meridional and eliminates the axial circular vortex. The loss of efficiency caused by this redirection constitutes a small fraction of the loss due to the axial circular vortex, since the redirection occurs at the inlet to the crown exit vane profile relatively big value of radius. 
   The solidity of the remaining exit vane apparatus elementary turbines is (L/T) ev &lt;&lt;1.0. Partial redirection of the flow by the remaining exit vane apparatus elementary turbines cause a loss of efficiency much smaller than the recovery of efficiency due to the decrease in the value of whirl at the entrance of the draft tube cone. The partial redirection of the flow by the rest of the exit vane apparatus elementary turbines, having (L/T) ev &lt;&lt;1.0, causes the loss of efficiency smaller than the recovery of efficiency cause by the decrease in the whirl at entrance to draft tube cone, since this redirection for each elementary turbine is done at bigger radius than its radius at draft tube cone entrance. 

   
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       FIG. 1  is an elevation view, partially in cross-section, of a radial intake turbine with a a mixed flow runner having a periphery rim and with an exit stay apparatus having an exit stay flange; 
       FIG. 2  is an elevation view, partially in cross-section, of an exit stay apparatus with an exit stay flange; 
       FIG. 3  is a straight cascade of profiles with absolute flow velocity diagrams for a turbine with a mixed-flow runner being a conformal mapping of the cascade of exit stay vane profiles at the exit stay crown, which is an intersection of the crown stream surface with the exit stay vanes in  FIG. 2 ; 
       FIG. 4  is a straight cascade of profiles with absolute flow velocity diagrams for a conventional turbine with a mixed flow runner being a conformal mapping of the cascade of the exit stay vane profiles at the periphery, which is an intersection of the periphery stream surface with the exit stay vanes in  FIG. 2 ; 
       FIG. 5  is a straight cascade of profiles with absolute flow velocity diagrams for a Potential Flow turbine with a mixed flow runner being a conformal mapping of the cascade of exit stay vane profiles at the periphery, which is an intersection of the peripheral stream surface with the exit stay vanes in  FIG. 2 ; 
       FIG. 6  is an elevation view of water passages of a radial intake turbine with a mixed flow runner without an exit stay apparatus showing leading and trailing edges of the runner blades, the elementary turbine at the runner crown and the axial circular vortex core with its radius, also the radius of the draft tube cone inlet; and 
       FIG. 7  is an elevation view of water passages of a radial intake turbine with a mixed flow runner and an exit stay apparatus showing leading and trailing edges of the runner blades and of the exit stay vanes, also showing the elementary turbine at the runner crown with radius of inlet to the exit stay vane, and the radius of the draft tube cone inlet. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Referring now to  FIG. 1 , a radial intake turbine installation is shown. The installation comprises a spiral casing  1  with radial stay vanes  2 , upper head cover  3  and a discharge ring  4  both secured to the spiral casing  1 , a guide gate apparatus  12  with radial wicket gates  5  pivotally secured to the head cover  3  and the discharge ring  4 , a mixed flow runner  6  with a runner crown  7  secured to the turbine shaft  8 , exit stay apparatus  9 , and a draft tube  10  with draft tube cone  11  and a draft tube horizontal diffuser not shown in FIG.  1 . Mixed-flow runner  6  together with shaft  8  rotates around the central axis X—X. 
   The power output of the turbine is regulated by radial wicket gates  5  which can be pivoted from a maximum open position to a closed position. The mixed flow runner  6  comprises a runner crown  7 , turbine blades  13 , and rim  14 . Turbine blades  13  are secured to the runner crown  7  and to the rim  14 . Rim  14  forms turbine water passages at the periphery (in the case of a propeller mixed flow runner the rim  14  is absent and turbine water passages at the periphery are formed by the discharge ring  4 ). The runner crown  7  is truncated by a plane perpendicular to central axis X—X. Exit stay apparatus  9  comprises exit stay crown  15 , exit stay vanes  16 , and exit stay flange  17 . Plurality of exit stay vanes  16  are arranged in a circular array around the central axis X—X. 
   Exit stay vanes  16  are secured to the exit stay crown  15  and to the exit stay flange  17 . Exit stay crown  15  is installed immediately under the truncated runner crown  7  and together with runner crown  7  forms the water passages, which in prior art turbines are formed solely the runner crown  7 . Exit stay flange  17  is secured to the discharge ring  4  and to the draft tube cone  11 .  FIG. 1  also shows a stream surface of meridional flow XI—X. 
   To those skilled in the art it is clear that the turbine installation shown in  FIG. 1  differs from prior art turbines by the presence of the exit stay apparatus  9 . 
   The exit stay apparatus  9  is there to eliminate the loss of efficiency due to an axial circular vortex formed in the flow after runner crown of prior art turbines in operational regimes other than optimum. 
     FIG. 2  shows an exit stay apparatus  9  with exit stay crown  15 , exit stay vanes  16 , and exit stay flange  17 . Exit stay vanes  16 , secured to exit stay crown  15  and to exit stay flange  17 , are identical in shape and profiled along their spans with maximal profile length at exit stay crown  15  and with minimal profile length at exit stay flange  17 . Exit stay crown  15  has the shape of a cup. The outer part  23  of exit stay crown  15  is secured to crown profiles  24  of exit stay vanes  16  and forms the water passages after the runner crown  7  of  FIG. 1  when the exit stay apparatus  9  is installed in the turbine. Exit stay flange  17  comprises the side wall  18 , upper ring  19 , and lower ring  20 . The inner part  21  of the side wall  18  is secured to peripheral profiles  22  of exit stay vanes  16  and forms the water passage at the periphery when the exit stay apparatus  9  is installed in the turbine. Upper ring  19  is secured to the discharging ring  4  of FIG.  1  and lower ring  20  is secured to the draft tube cone  11  of  FIG. 1  when the exit stay apparatus  9  is installed in the turbine.  FIG. 2  also shows inlet radii of crown profiles  24  and of peripheral profiles  22 , denoted by (R evi ) cr  and (R evi ) pe  respectively. 
   The intersection of the stream surface XI—XI of  FIG. 1  with exit stay vanes  16  gives a cascade of profiles. 
     FIGS. 3-5  show planar images of such cascades under a conformal mapping given by
 Ξ=ΦR 0   (8)                 H   =       ∫   0     l   s       ⁢       Rdl   s       R   0           ⁢                   (   9   )                 V   ξ     =       R     R   0       ⁢     V   u               (   10   )                 V   η     =       R     R   0       ⁢     V   m               (   11   )               
where:
         (Ξ,H) are Cartesian coordinates in the image plane,   (R,Φ) are cylindrical coordinates of a point on the stream surface XI—XI,   l s  is length along a streamline of the stream surface XI—XI,   R 0  is the radial coordinate of points on the stream surface XI—XI with l s =0, and   V ξ  and V η  are absolute velocity components in the image plane.       
     FIGS. 3-5  show conformal images of cascades of profiles of the exit stay apparatuses  9  of  FIGS. 1 and 2  designed for a conventional turbine with a mixed flow runner and for a Potential Flow turbine with a mixed flow runner (U.S. Pat. No. 5,441,384, Aug. 1995). In the optimum operating regime both turbines have unit flow rate: (Q 11 ) opt =1.00 m 3 /sec and unit rotation rate: (N 11 ) opt =66.00 prm. The particular Potential Flow turbine which exit stay vane profiles shown in  FIG. 3 and 5  was designed as a replacement of runner blades and wicket gates for the Boundary hydroelectric plant. It has the following design parameters: turbine flow rate Q t =225.3 m 3 /sec, turbine head H t =76.2 m, and runner diameter D r =5.2 m. For both turbines the number of exit stay vanes  16  in the exit stay apparatus  9  of  FIGS. 1 and 2  is Z ev =6. An exit stay apparatus  9  must be designed with the following constrains along the spans of exit stay vanes  16 : the whirl at the inlet to the profiles of exit stay apparatus  0  (V u R) evi  must equal the whirl at the blade trailing edge (V u R) 2 ; the whirl at the exit of the profiles of the exit stay apparatus  9 , (V u R) eve , must vanish. Thus, the exit elements of the profiles of the exit stay apparatus  9  must be meridional. 
   When the exit stay apparatus is retrofitted into Francis turbine with badly designed draft tube elbow the presence of the whirl at the periphery at inlet to the draft tube cone  10  of  FIG. 1  is required in order to prevent the flow separation from the elbow wall. In that case the exit discharge angle from stay vane  16  of  FIG. 2  gradually changes along vane span from 90° at the stay crown  15  to the value required by the whirl at periphery. 
     FIGS. 3-5  also show velocity vectors V i  at the inlet to the profiles with components (V ξ ) i  and (V η ) i , and the angle β i  between V i  and OΞ. The subscripts “opt”, “sop”, and “bop” to V i , (V ξ ) i , (V η ) i , and β i  mean respectively: for optimal power (P t ) opt , for P 1 &lt;(P t ) opt , and for P t &gt;(P t ) opt . 
     FIG. 3  shows a planar cascade of profiles  25  which is a conformal image of the cascade formed by crown profiles  24  of  FIG. 2  under the mapping ( 8 - 11 ) with R 0 =(R evi ) cr . The profiles  25  are the same for conventional and Potential Flow turbines with middle lines  26  being straight line segments. This is because in the optimum operating regime the whirl at the inlet to the crown profiles  24  of  FIG. 2 , (V u R) 2cr , vanishes for both types of turbine. In order to direct the flow along the meridional profile exit element and to completely eliminate whirl after the exit stay apparatus  9  at the exit stay crown  15 , the solidity of planar cascade of profiles  25  must satisfy (L/T) ev &gt;1.1, where L is the length of the middle segment  26  and T=(R 0 π)/Z ev . As can be seen in FIG.  3 : (V ξ ) opt =0, (V ξ ) bop &lt;0, (V ξ ) sop &gt;0, (β i ) opt =90°, (β i ) bop &gt;90°, and (β i ) sop &lt;90°. 
     FIG. 4  shows a planar cascade of profiles  27  which is a conformal image of the cascade formed by crown profiles  24  of  FIG. 2  under the mapping ( 8 - 11 ) with R 0 =(R evi ) pe . The middle lines  28  of profiles  27  are not straight line segments and at the inlet they have an angle relative to OΞ equal to (β i ) opt . The solidity of the planar cascade of profiles  26  satisfies (L/T) ev &lt;0.1. For a conventional turbine in the optimum operational regime the whirl leaving the runner at the periphery satisfies [(V u R) 2 ] pe &gt;&gt;0, since [(V u R) 1 ] pe =kΔ(V u R) with 2≦k≦4. For a conventional turbine the whirl [(V u R) 2 ] pe &gt;0 for all values of power P t . As can be seen in FIG.  4 : (V ξ ) opt &gt;0, (V ξ ) bop &gt;0 (V 86 ) sop &gt;0, (β i ) opt &lt;90°, (β i ) bop &lt;90°, (β i ) sop &lt;90°, and (β i ) sop &lt;(β i ) opt &lt;(β i ) bop . 
     FIG. 5  shows a planar cascade of profiles  27  for the Potential Flow turbine, which is a conformal image of the cascade formed by peripheral profiles  22  of  FIG. 2  under the mapping ( 8 - 11 ) with R 0 =(R evi ) pe . The middle lines  28  of profiles  27  are straight line segments, since for the Potential Flow turbine in optimum operating regime the whirl leaving the runner at the peripheral [(V u R) 2 ] pe  vanishes. The solidity of planar cascade of profiles  27  (L/T) ev &lt;0.1, just as for the conventional turbine in FIG.  4 . As can be seen in FIG.  5 : (V ξ ) opt =0, (V ξ ) bop &lt;0, (V ξ ) sop &gt;0, (V ξ ) opt =90°, (β i ) bop &lt;90°, (β i ) sop &lt;90°. 
     FIG. 6  shows an elevation view of water passages of a radial intake turbine with a mixed flow runner and without an exit stay apparatus.  FIG. 6  shows a stream surface XII—XII. The stream surface XII—XII together with the crown stream surface bounds the crown elementary turbine  29 . In operating regimes other than optimum the crown elementary turbine  29  forms an axial circular vortex core  30  trailing at the tip of the crown  7  of  FIGS. 1 and 6  (in this case not truncated).  FIG. 6  also shows the radius of the axial circular vortex core  30  and the radius at the inlet to the draft tube cone  11  of  FIGS. 1 and 6 , denoted R cv  and R dt  respectively. 
     FIG. 7  shows an elevation view of water passages of a radial intake turbine with a mixed flow runner and an exit stay apparatus  9 .  FIG. 7  shows a stream surface XII—XII. The stream surface XII—XII together with the crown stream surface bounds the crown elementary turbine  29 . For a turbine without an exit stay apparatus an axial circular vortex core  30  of  FIG. 6  is formed in the crown elementary turbine  29 .  FIG. 7  also shows the radius (R evi ) cr  of the inlet exit stay vanes  16  inside the crown elementary turbine  29  and the radius R dt  at the inlet to the draft tube cone  11  of  FIGS. 1 and 7 . Further,  FIG. 7  also shows an i-th elementary turbine  31  inside the turbine water passages with its radii at the inlet to the exit stay vanes  16  and at the inlet to the draft tube cone  11 , denoted (R ev ) i  and (R dt ) i  respectively. 
   The exit stay vanes  16  of  FIGS. 1 and 2  are evenly distributed in space, are identical in shape and all are secured to exit stay flange  17 . However in a case of fish friendly turbine the exit stay vanes form an additional obstacle for the passing fish. In this case it is better to have some stay vanes to be secured only to exit stay crown  15  and passing only after the crown part of blades, therefore, creating the obstacle for the fish only at the at the crown part of turbine flow where velocity is substantially smaller than at periphery. These shortened exit stay vanes will help to the cascades after the crown elementary turbine to reach solidity, (L/T) ev &gt;1.1, and to decrease the number of exit stay vanes at periphery where the major part of the flow is passing and velocities are higher. The tips of the shortened exit stay vanes must be of the shape similar to the ship screw tips in order to reduce losses caused by vortices leaving the tips. 
   In the following analysis two turbines are compared: one with and one without an exit stay apparatus  9  of  FIGS. 1 and 7 . The two turbines under consideration are identical from a fluid-mechanical point of view, except that one has exit stay vanes  16  of  FIGS. 1 and 7  and the other does not. Specifically, both turbines have the same geometry of the water passages, wicket gates  5 , turbine blades  13 , and draft tube  10  of FIG.  1 . The particular parameters, shared by both turbines, are those of the Potential Flow turbine designed for the Boundary plant (without an exit stay apparatus). 
   Total relative head losses Δζ t  can be computed by combining the relative head losses for elementary turbines Δζ et  using the following formula: 
               Δ   ⁢           ⁢     ζ   t       =       1     Q   t       ⁢       ∑     i   =   1     I     ⁢         (     Δζ   et     )     i     ⁢   Δ   ⁢           ⁢     q   i                   (   12   )             
 
where:
         (Δζ et ) i  denotes the relative head losses for the i-th elementary turbine,   Δ i  is the flow rate through the i-th elementary turbine, and   I is the total number of elementary turbines.       

   There are two types of head losses due the presence of exit stay vanes  16 : shock losses ΔH si  at the inlet to the vane and friction losses ΔH ft  along vane surface. Relative friction losses along a profile of length L pr  are given by the following formulae (see T. Schlichting,  boundary Layer Theory,  McGraw-Hill [1979]): 
               Δζ   fl     =         λ   R     ⁢     L   pr     ⁢     V     a   ⁢           ⁢   υ     2         2   ⁢     gD   r     ⁢     H   t                 (   13   )             
 λ R   −0.5 =2.0 In (Reλ R   0.5 )−0.8  (14)
 
where:
         V av  is the average velocity along the profile, and   Re=V av D r /λ R  is the Reynolds number.       

   For the Boundary turbine relative friction losses for the crown profiles  24  and the peripheral profiles  22  of  FIG. 2  have the following values;
 
(Δζ fl ) cr =0.00044
 
(Δζ fl ) pe =0.00007
 
Noting that most of the flow passes through peripheral elementary turbines, one can see from (12) that relative friction losses of turbine head caused by the exit stay vanes  16  are negligible (&lt;&lt;0.1%). Thus, for the purposes of this analysis it is sufficient to compare shock losses of head at the inlet to the stay vanes  16  of  FIG. 7  to losses caused by the axial circular vortex core  30  of FIG.  6 .
 
   Relative shock losses of head Δζ si  at the inlet to the exit stay vanes  16  for the elementary turbine  31  are given by the following formula (see G. I. Topazh,  Computation of Integral Hydraulic Indicators for Hydromachines,  Leningrad [1989]): 
               Δζ   sl     =           K   s     ⁡     (       V   u   +     -     V   u   -       )       2       2   ⁢     gH   t                 (   15   )             
 
where V u   +  and V u   −  are the values of V u  before and after the inlet to the exit stay vanes  16  and K s  is a constant, which depends on the solidity (L/T) ev  of the exit stay vanes  16 . If (L/T) ev &gt;1.1, then K s &lt;1.0 and K s ≈1.0, while if (L/T) ev &lt;0.1, then K s ≈0.0.
 
   In the optimum operating regime the shock loss of head (ΔH sl ) et  at the inlet to the exit stay vanes  16  vanishes for all elementary turbines along the span of exit stay vanes  16 , because V u   + =V u   − =(V u R) 2 /R evi . Therefore, in the optimum operating regime both turbines, which and without the exit stay apparatus  9 , have practically the same efficiency. 
   In operating regimes other than optimum the turbine of  FIG. 6  without the exit stay apparatus  9  of  FIG. 7  has an axial circular vortex core  30 . This causes a loss of efficiency given by (6). In addition, there are loses of kinetic energy due to the ultimate dissipation of the exit whirl (V u R) 2  at the inlet to the draft cone  11 . The turbine with the exit stay apparatus  9  of  FIG. 7  does not have an axial circular vortex core and a corresponding loss of efficiency. However, in contrast to the turbine without the exit stay apparatus  9 , there are relative head shock losses at the inlet to the exit stay vanes  16  inside the crown elementary turbine  29 . Using (15) and taking into account that V u   + −V u   − =(V u R) 2cr /(R evi ) cr , one sees that these losses are given by 
                   (     Δζ   sl     )       c   ⁢           ⁢   τ       =           K     sc   ⁢           ⁢   τ       ⁡     (       V   u     ⁢   R     )         2   ⁢   c   ⁢           ⁢   τ     2       2   ⁢       g   ⁡     (     R     e   ⁢           ⁢   υ   ⁢           ⁢   i       )         c   ⁢           ⁢   τ     2     ⁢     H   t           ,           (   16   )             
 
There are also shock losses of head at the inlet to exit stay vanes  16  for each i-th elementary turbine  31  in FIG.  7 .
 
   Here is a comparison of inlet shock losses with losses of kinetic energy. For each i-th elementary turbine in  FIG. 7  V u   −  vanishes, so 
                 (     Δζ   sl     )     i     =             K   si     ⁡     (       V   u     ⁢   R     )         2   ⁢   i     2       2   ⁢       g   ⁡     (     R     e   ⁢           ⁢   υ       )       i   2     ⁢     H   t         .             (   17   )             
 
Meanwhile, the kinetic energy losses at the inlet to the draft cone  11  in  FIG. 6  are 
                 (     Δζ   ke     )     i     =           (     1   -     K   si       )     ⁢       (       V   u     ⁢   R     )       2   ⁢   i     2         2   ⁢       g   ⁡     (     R   dt     )       i   2     ⁢     H   t         .             (   18   )             
 
As can be seen from  FIGS. 1 ,  6 , and  7 , for all elementary turbines other than several elementary turbines near the periphery, (R dt ) i &lt;(R ev ) i . Therefore, (Δζ sl ) i &lt;(Δζ ke ) i . Elementary turbines near the periphery may be neglected, because the value of solidity there is (L/T) ev ≈0.1, so there is very little interference with the flow.
 
   The above comparison shows that the exit stay apparatus  9  in  FIG. 7  partially recovers losses of kinetic energy due to whirl leaving the runner, with the sole exception of the crown elementary turbine  29 . Thus, for the purposes of the present analysis, it suffices to compare losses in the crown elementary turbine  29  of  FIG. 7  (i=1) to losses caused by the axial circular vortex core  30  of FIG.  6 . Using (12) and (16) and assuming (Δζ sl ) i =0 for i=2, . . . I, one obtains the loss of efficiency caused by exit stay vanes  16  of FIG.  7 : 
               Δη     e   ⁢           ⁢   υ       =             K     sc   ⁢           ⁢   τ       ⁡     (       V   u     ⁢   R     )         2   ⁢   c   ⁢           ⁢   τ     2     ⁢   Δ   ⁢           ⁢     q     c   ⁢           ⁢   τ           2   ⁢       gQt   ⁡     (     R     e   ⁢           ⁢   υ   ⁢           ⁢   i       )         c   ⁢           ⁢   τ     2     ⁢     H   t                 (   19   )             
 
Assuming (K s ) cr =1.0 and taking into account Q t =π(R dt ) 2 (V z ) dt  and Δq cr =π(R cv ) 2 (V z ) dt , where (V z ) dt  is the axial component of velocity at the inlet to draft tube cone  11  of  FIG. 1 , gives 
               Δη     e   ⁢           ⁢   υ       =           (       V   u     ⁢   R     )       2   ⁢   c   ⁢           ⁢   τ     2     ⁢     R     c   ⁢           ⁢   υ     2         2   ⁢     gR   dt   2     ⁢         H   t     ⁡     (     R     e   ⁢           ⁢   υ   ⁢           ⁢   i       )         c   ⁢           ⁢   τ     2                 (   20   )             
 
Using (6) and (20) one can compare losses Δη cv  in  FIG. 6  to Δη ev  in FIG.  7 : 
               Δη     e   ⁢           ⁢   υ       =       Δη     c   ⁢           ⁢   υ       ⁢         R     c   ⁢           ⁢   υ     2         (     R     e   ⁢           ⁢   υ   ⁢           ⁢   i       )       c   ⁢           ⁢   τ     2       .               (   21   )             
 
   If the inlet to the crown elementary turbine  29  of  FIG. 7  is not far from the trailing edge of turbine blades  13  of  FIG. 1 , then it can be safely assumed that R cv &lt;0.1 (R evi ) cr . Thus, in operating regimes other than optimum, the exit stay apparatus recovers more than 99.0% of efficiency loss due to the axial circular vortex. 
   In the particular example of Boundary potential flow turbine the recovery of efficiency by the exit stay apparatus amounts to Δ(η) t =8.0% for Q 11 =0.50 (Q 11 ) opt  and N 11 =1.10 (N 11 ) opt . 
   For a conventional turbine with a mixed flow runner and properly designed draft tube elbow the exit stay apparatus will increase efficiency even in the optimum regime and will recover a larger value of Δ(η) t  in regimes other than optimum. In the optimum regime the exit stay vanes do not cause shock losses at the inlet and recover the losses of kinetic energy in all elementary turbines other than the crown elementary turbine given by 
                 (     Δζ   ke     )     i     =             (     1   -     K   si       )     ⁡     [       (       V   u     ⁢   R     )       2   ⁢   i       ]       opt   2       2   ⁢       g   ⁡     (     R   dt     )       i   2     ⁢     H   t         .             (   22   )             
 
In regimes other than optimum, for the i-th elementary turbine of  FIG. 7  the shock losses at the inlet to the exit stay vanes  16  are 
                 (     Δζ   sl     )     i     =           K   si     ⁢       {         (       V   u     ⁢   R     )       2   ⁢   i       -       [       (       V   u     ⁢   R     )       2   ⁢   i       ]     opt       }     2         2   ⁢       g   ⁡     (     R     e   ⁢           ⁢   υ       )       i   2     ⁢     H   t         .             (   23   )             
 
Meanwhile, the losses of kinetic energy for every elementary #i at the inlet to draft cone  11  in  FIG. 6  are defined by the formula (18). Comparing (23) to the formula (18) for the losses of kinetic energy for each i-th elementary turbine at the inlet to the draft cone  11  in  FIG. 6  one sees that for a conventional turbine the value of (Δζ sl ) i  is much smaller than value of (Δζ ke ) i .