Abstract:
Rotary vane pumps include casings having asymmetrical cavities that accommodate a rotor. For a single pump chamber, one portion of the rotor abuts or nearly abuts the inner wall of the cavity at a single location while one portion of the rotor and the inner wall are not in contact with each other thereby defining a pump chamber. For a dual pump chamber embodiment, two diametrically opposed portions of the rotor abut or nearly abut the inner wall of the cavity while two portions of the rotor and inner wall are not in contact with each other thereby defining the two pump chambers. The two pump chambers are disposed on opposite sides of the minor axis of the cavity. The cavities of each pump are skewed so each pump chamber is larger in volume at the inlet end than at the outlet end.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This is a non-provisional application claiming priority under 35 U.S.C. 119(e) to U.S. Provisional Patent Application Ser. No. 61/560,245 filed on Nov. 15, 2011. 
     
    
     TECHNICAL FIELD 
       [0002]    This disclosure relates generally to a system and method for improving the performance of rotary vane pumps. 
       BACKGROUND 
       [0003]    A rotary vane pump is a positive-displacement pump that consists of vanes slidably mounted to a rotor that rotates inside of a cavity formed by a pump casing. In some cases, the vanes can be of variable length and/or spring-biased to maintain contact with the inner wall of the cavity as the rotor rotates. The simplest rotary vane pump includes a cylindrical rotor rotating inside of a larger cylindrical cavity. The axes of these two cylinders are offset, causing an eccentricity. Vanes are allowed to slide into and out of the rotor and seal against the inner wall of the cavity, creating rotating vane chambers disposed between two vanes. The rotor may engage or be disposed close to the inner wall at one point which creates a single pump chamber extending around the rotor and from a single inlet to a single outlet. 
         [0004]    However, other rotary vane pumps are designed with an elliptical cavity formed in the pump casing with a cylindrical rotor. The rotor may engage or be disposed close to the inner wall at two points along opposite ends of the minor axis of the ellipse, which creates two pump chambers (or “twin pump chambers”) on opposite sides of the minor axis and along the portions of the inner wall near the major axis. Such a design includes two inlet/outlet pairs, one for each pump chamber. 
         [0005]    At the inlet of each pump chamber, the inner wall that defines the cavity extends away from the rotor and causes the vanes to extend outward as the vane chambers increase in volume as the rotor and the vane chambers rotate away from the inlet towards the outlet. As the vane chambers pass the inlet, the vane chambers are filled with fluid drawn in through the inlet at inlet pressure, which may be atmospheric. At the outlet of the pump chamber, the inner wall extends towards the rotor, the vanes retract and the vane chambers therefore decrease in volume as the vane chambers rotate to the outlet, forcing the fluid out of the pump. For pumps with twin chambers, the above process is repeated twice for each rotation of the rotor. With a constant inlet pressure, the vane chambers deliver the same volume of fluid with each rotation. Multistage rotary vane vacuum pumps can attain pressures as low as 10 −3  mbar (0.1 Pa). 
         [0006]    The elliptical/twin chamber rotary vane pump design allows both sides of the rotor to generate pressure or vacuum, thus achieving greater flow in a smaller package size and because the pump chambers are disposed 180° from each other, side loading of the rotor is virtually eliminated. However because the cycle of each vane chamber of an elliptical vane pump is only 180° of rotation as opposed to 360° for a single chamber vane pump, at higher vacuum and pressure duties, there is not enough angular distance to effectively compress the fluid before it is exhausted without restricting the flow and increasing vane loading. The result is a pump having shorter vane life and that becomes louder, hotter, and less efficient as the pressure or vacuum is increased. 
         [0007]    Similarly, like the twin chamber rotary vane pump design, the internal compression of single chamber rotary vane pumps is limited by the angular distance between the inlet and exhaust ports. Therefore, achieving higher pressure duties in single chamber rotary vane pumps also adversely affects sound levels, efficiency, heat, and vane life. 
       SUMMARY OF THE DISCLOSURE 
       [0008]    By altering the shape of the elliptical cavity, the angle of maximum vane extension may be shifted away from the major axis of the ellipse to a position closer to the inlet. As a result, the angular distance between the inlet and outlet is increased allowing greater internal compression. Further adjustment of the inner wall curvature allows optimization of the vane acceleration where the average vane tip load can be reduced and vane “skipping” or loss of contact with the inner wall can be eliminated. The resulting pumps are quieter, cooler, more efficient and provide a longer vane life. 
         [0009]    In one example, a rotary vane pump is disclosed that comprises a casing comprising an asymmetrical cavity having a continuous inner wall. The cavity has a minor axis and a major axis for purposes of this description, even though it is asymmetrical. The pump also comprises a rotor disposed within the cavity for rotation within the cavity. During a rotation, two diametrically opposed portions of the rotor abut or nearly abut the inner wall of the cavity at or near the minor axis of the cavity while two portions of the rotor and inner wall are not in contact with each other thereby defining two pump chambers disposed on opposite sides of the minor axis. Each pump chamber comprises an inlet end and an outlet end. The inlet ends, disposed in different pump chambers, are on opposite sides of the minor axis and on opposite sides of the major axis. The outlet ends, disposed in different pump chambers, are on opposite sides of the minor axis and on opposite sides of the major axis. Each pump chamber is larger in volume between the inlet end and the major axis than between the major axis and the outlet end. In other words, the inlet portion of each pump chamber is larger than the outlet portion of each pump chamber. 
         [0010]    In a refinement, the rotor includes a plurality of slots ranging from about three to about 12 with each slot accommodating a vane. In a further refinement of this concept, the rotor comprises about eight slots and about eight vanes. 
         [0011]    In another refinement, the angle between each of the two diametrically opposed portions of the rotor abutting or nearly abutting the inner wall of the cavity at or near the minor axis of the cavity and the inner wall of the cavity in each chamber where each vane is fully extended as it engages the inner wall ranges from less than about 90° to about 50°, depending on the length of maximum vane extension. In a further refinement of this concept, the angle between each of the two diametrically opposed portions of the rotor abutting or nearly abutting the inner wall of the cavity at or near the minor axis of the cavity and the inner wall of the cavity in each chamber where each vane is fully extended as it engages the inner wall is about 80°. 
         [0012]    Another rotary vane pump is disclosed which comprises a casing comprising an asymmetrical cavity having a continuous inner wall. The pump also comprises a rotor disposed within the cavity for rotation within the cavity. During a rotation, the rotor abuts or nearly abuts the inner wall of the cavity at a single location with a remaining portion of the rotor not in contact with the inner wall of the cavity. The remaining portion of the rotor and the inner wall not in contact with each other defines a pump chamber. The pump chamber comprises an inlet end and an outlet end. The inlet end and outlet end are disposed on opposite sides of the portion of the rotor abutting or nearly abutting the inner wall of the cavity at a single location. The pump chamber side with the inlet end is larger in volume than the pump chamber side with the outlet end. In other words, the inlet portion of the pump chamber is larger than the outlet portion of the pump chamber. 
         [0013]    In a refinement, the rotor comprises a plurality of slots ranging from about three to about eight with each slot accommodating a vane. In a further refinement of this concept, the rotor comprises about four slots and about four vanes. 
         [0014]    In a refinement, the angle between the portion of the rotor abutting or nearly abutting the inner wall of the cavity at a single location and the inner wall of the cavity at the inlet end of the pump chamber where each vane is fully extended as it engages the inner wall ranges from less than 180° to about 100, depending on the length of maximum vane extension. In a further refinement of this concept, the angle between the portion of the rotor abutting or nearly abutting the inner wall of the cavity at a single location and the inner wall of the cavity at the inlet end of the pump chamber where each vane is fully extended as it engages the inner wall is about 125°. 
         [0015]    A method of increasing internal compression of a rotary vane pump is disclosed. The method providing a casing comprising an asymmetrical cavity having a continuous inner wall. The cavity has a minor axis and a major axis. The method also includes disposing a rotor within the cavity for rotation within the cavity so that, during a rotation, two diametrically opposed portions of the rotor abut or nearly abut the inner wall of the cavity at or near the minor axis of the cavity while two portions of the rotor and inner wall are not in contact with each other and thereby define two pump chambers disposed on opposite sides of the minor axis. The method also includes providing each pump chamber with an inlet end and an outlet end. The inlet ends are disposed in different pump chambers on opposite sides of the minor axis and on opposite sides of the major axis. The outlet ends are disposed in different pump chambers on opposite sides of the minor axis and on opposite sides of the major axis. Finally, the method comprises providing the asymmetrical cavity so that each pump chamber is larger in volume between the inlet end and the major axis than between the major axis and the outlet end. 
         [0016]    Another method for increasing the internal compression of a rotary vane pump is disclosed. The method comprises providing a casing comprising an asymmetrical cavity having a continuous inner wall. The method includes disposing a rotor within the cavity for rotation within the cavity, so that during a rotation, the rotor abuts or nearly abuts the inner wall of the cavity at a single location with a remaining portion of the rotor not in contact with the inner wall of the cavity. The remaining portion of the rotor and the inner wall not in contact with each other defines a pump chamber. The method also includes providing the pump chamber with an inlet end and an outlet end. The inlet end and outlet end are disposed on opposite sides of the portion of the rotor abutting or nearly abutting the inner wall of the cavity at a single location. The pump chamber side with the inlet end is larger in volume than the pump chamber side with the outlet end. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]      FIG. 1  is a perspective view of a combination suction and liquid separation apparatus that includes the disclosed rotary vane pump. 
           [0018]      FIG. 2  is an exploded view of the apparatus shown in  FIG. 1 . 
           [0019]      FIG. 3  is a partial perspective and sectional view of the apparatus illustrated in  FIGS. 1-2 , particularly illustrating the rotor and casing of the disclosed rotary vane pump. 
           [0020]      FIG. 4  is a top perspective view of the rotary vane pump and apparatus illustrated in  FIG. 3 , with the top cover or upper cap removed thereby exposing the rotor, vanes and cavity. 
           [0021]      FIG. 5  graphically illustrates the benefits of altering the shape of an ellipsoidal cavity for skewing the shape of an ellipsoidal cavity as the curves show that vanes of pumps with a skewed ellipsoidal cavity extend sooner and retract sooner during a pump cycle than the vanes of a conventional pump. 
           [0022]      FIG. 6  graphically illustrates the shapes of the three cavities referred to in  FIG. 5 . 
           [0023]      FIG. 7  is a top sectional view of a disclosed pump illustrating a skewed cylindrical rotor disposed within an elliptical cavity. 
           [0024]      FIG. 8  is a top schematic view of a conventional rotary vane pump with a cylindrical rotor disposed within an elliptical cavity. 
           [0025]      FIG. 9  is a top schematic view of a disclosed rotary vane pump with a skewed elliptical cavity. 
           [0026]      FIG. 10  graphically illustrates the vane tip force and average vane tip force for the conventional pump illustrated in  FIG. 8 . 
           [0027]      FIG. 11  graphically illustrates the vane tip force and average vane tip force for the disclosed pump of  FIG. 9 . 
           [0028]      FIG. 12  is a top schematic view of a conventional vane pump with a cylindrical rotor disposed within a cylindrical cavity. 
           [0029]      FIG. 13  is a top schematic view of a disclosed rotary vane pump with a cylindrical rotor disposed within a skewed cylindrical cavity. 
           [0030]      FIG. 14  illustrates, graphically, the vane tip force and average vane tip force for the pump illustrated in  FIG. 12 . 
           [0031]      FIG. 15  illustrates, graphically, the vane tip force and average vane tip force for the pump illustrated in  FIG. 13 . 
           [0032]      FIG. 16  is a front plan view of yet another disclosed rotary vane pump, particularly illustrating the outer casing. 
           [0033]      FIG. 17  is a perspective view of the pump illustrated in  FIG. 16 , schematically illustrating the direct or indirect coupling of the pump to a motor. 
           [0034]      FIG. 18  is a front plan view of a conventional vane pump with a cylindrical rotor disposed within a cylindrical cavity. 
           [0035]      FIG. 19  is a front plan view of the disclosed rotary vane pump with a cylindrical rotor disposed within a skewed cylindrical cavity. 
           [0036]      FIG. 20  illustrates, graphically, the vane tip force and average vane tip force for the pump illustrated in  FIG. 18 . 
           [0037]      FIG. 21  illustrates, graphically, the vane tip force and average vane tip force for the pump illustrated in  FIG. 19 . 
           [0038]      FIG. 22  is a front plan view of a conventional vane pump with a cylindrical rotor disposed within an elliptical cavity. 
           [0039]      FIG. 23  is a front plan view of the disclosed rotary vane pump with a cylindrical rotor disposed within a skewed elliptical cavity. 
           [0040]      FIG. 24  is a front schematic view of the conventional vane pump with a cylindrical rotor disposed within an elliptical cavity as shown in  FIG. 22 . 
           [0041]      FIG. 25  is a top schematic view of a disclosed rotary vane pump with a cylindrical rotor disposed within a skewed elliptical cavity as shown in  FIG. 23 . 
           [0042]      FIG. 26  illustrates, graphically, the vane tip force and average vane tip force for the conventional pump illustrated in  FIGS. 22 and 24 . 
           [0043]      FIG. 27  illustrates, graphically, the vane tip force and average vane tip force for the disclosed rotary vane pump illustrated in  FIGS. 23 and 25 . 
       
    
    
     DESCRIPTION 
       [0044]      FIG. 1  is a perspective view of a suction/liquid separator  10 , typically used in dental applications. As shown in  FIG. 1 , the combination suction/liquid separator  10  includes a pump  11 , which is one of the disclosed rotary vane pumps discussed in detail below, a liquid separator  12  and a motor  13  for operating the pump  11  and separator  12 . The pump  11  may include a pair of suction inlets  14  (see  FIG. 2 ) and a pair of outlets  15 . The outlet  15  may also be connected to an air discharge  16  via a hose or pipe  17 . The solids outlet is shown at  18 . The pump  11  includes a casing  21  enclosed by a cover  22 . The head plate  23  includes the inlets  14 , outlets  15  and also serves to enclose the casing  21 . 
         [0045]    Returning to  FIG. 2  the fasteners  25  connect the cover  22  to the pump casing  21 . The pump casing  21  is connected to the head plate  23  via the fasteners  33 . While the head plate  23  includes two inlets  14  and two outlets  15 , typically, only one inlet  14  and only one outlet  15  is used at a time. The rotor  26  includes a plurality of sliding vanes  27  and is disposed in the cavity  28  of the pump casing  21 . The bearing plate  29  disposed below the head plate  23  accommodates the bearing  31  and rotor shaft  32 . The rotor shaft  32  is frictionally coupled to the rotor  26  within the axial opening  34  in the rotor  26 . 
         [0046]    Still referring to  FIG. 2 , the motor  13  includes a motor housing  36  and a base plate  37  that is connected to the bearing plate  29  with the elongated fasteners or threaded rods  38 . The motor  13  also includes a drive shaft  41 . The lower end of the drive shaft  41  is coupled to the separator rotor  42  via a tongue-in-groove connection, splined connection or similar connection in the axial opening  43  of the separator rotor  42 . The separator  12  includes a housing  45  that is sandwiched between the separator base plate  46  and the motor base plate  37 . Sealing elements or O-rings are shown at  47 ,  48 . 
         [0047]      FIGS. 3-4  illustrate the position of rotor  26  within the pump casing  21  and between the cover  22  and head plate  23 . One of the vanes  27  is extended outward from the rotor  26  to engage the inner wall  51  (see also  FIG. 4 ) of the cavity  28 .  FIG. 3  also illustrates communication between the inlets  14 , outlets  15  and the cavity  28 , which may be defined by the cover  22 , the inner wall  51  of the cavity  28  and the head plate  23 . In  FIG. 4 , the rotor  26  includes four sliding vanes  27 . The number of vanes  27  may vary as will be apparent to those skilled in the art and as illustrated in  FIGS. 4 ,  8 - 9 ,  12 - 13 ,  18 - 19 , and  22 - 25 . 
         [0048]    Returning to  FIGS. 5-7 , certain advantages of the disclosed design are illustrated. Returning first to  FIG. 5 , the extension of a vane  27  from completely retracted (0%) to fully extended (100%) is plotted against the rotational angle of 0-360°. The three lines  55 ,  56 ,  57  relate to three differently shaped ellipsoidal cavities  28 . Specifically, referring to  FIG. 6 , the solid line  55  is indicative of an ellipsoidal cavity with regular major and minor axes. In other words, the cavity represented by the line  55  is not skewed. The cavities  28  presented by the lines  56 ,  57  are skewed or altered as shown in  FIG. 6 . 
         [0049]    Specifically, referring to the 0°-90° quadrant, it is clear that the ellipsoidal cavities represented by the lines  56 ,  57  are larger than the pure ellipsoidal cavity  55 . As this is the intake section of the pump  100 , more air, gas or fluid is collected at the inlet  14  using this design. Then, referring to the second quadrant 90°-180° of  FIG. 6 , the reader will note that the cavities represented by the lines  56 ,  57  are smaller than the regular ellipsoidal cavity  55 . As a result, the vane chambers are shrinking as the vanes  27  retract thereby increasing the pressure in the vane chambers as the vane chambers precede toward the outlet  15  at the bottom of the plot, near the 180° mark. In the third quadrant, 180°-270°, where the inlet  14  is disposed, the cavities represented by the lines  56 ,  57  are larger than the purely ellipsoidal cavity  55  and then the cavities represented by the lines  56 ,  57  shrink in the fourth quadrant 270°-0° as the vane chambers head toward the outlet  15 , at the top of the plot, near the 0° mark. Comparing  FIGS. 6 and 7 , it is clear that the vanes  27  used in the disclosed cavities represented by the lines  56 ,  57  rise more in the first quadrant 0°-90°, fall more in the second quadrant 90°-180°, rise more in the third quadrant 180°-270° and fall more in the fourth quadrant 270°-0°. 
         [0050]    This is further illustrated in  FIGS. 8-9 .  FIG. 8  illustrates a purely elliptical cavity  128  with a rotor  126  disposed therein. The pump  100  illustrated in  FIG. 8  includes two pump chambers  58 ,  59  disposed on either side of the rotor  126 . The rotor  126  engages the inner wall  51  of the cavity  128  at two points  62 ,  63 , or where the minor axis  64  intersects the inner wall  51 . Not only are the pump chambers  58 ,  59  of the same size, dividing each pump chamber  58 ,  59  into two using major axis  65  both “halves” of the pump chambers  58 ,  59  are of equal volume or each quadrant of each pump chamber  58 ,  59  is of equal volume. 
         [0051]    In contrast, turning to  FIG. 9 , the cavity  228  of the pump  200  is skewed. In other words, while the pump chambers  258 ,  259  are of the same size, the inlet portion  258   a  of the pump chamber  258  is larger than the outlet portion  258   b.  Similarly, the intake or inlet portion  259   a  of the pump chamber  259  is larger than the outlet portion  259   b.    
         [0052]    The effect of theses geometric changes can be seen in  FIGS. 10 and 11 . In  FIG. 10 , the vane tip force versus rotation is plotted for the conventional pump  100  of  FIG. 8 . In  FIG. 11 , the vane tip force versus rotation is plotted for the pump  200  of  FIG. 9 . Note that the average vane tip force represented by the line  71  in  FIG. 10  is greater than the average vane tip force represented by the line  72  of  FIG. 11 , thereby subjecting the vanes  27  of the pump  100  shown in  FIG. 8  to greater wear than the vanes of the pump  200  shown in  FIG. 9 . 
         [0053]    Similar results are achieved with single chamber pumps like those shown at  300 ,  400  in  FIGS. 12-13 . In  FIG. 12 , the pump  300  features a standard elliptical cavity  328 , a cylindrical rotor  226  with slots for four vanes, a single inlet  14  and a single outlet  15 . The single pump chamber  358  is of the same size on either side of the minor axis  64  and on either side of the major axis  65 . In contrast, turning to  FIG. 13 , the cavity  428  of the pump  400  is skewed. Even though there is a single pump chamber  458 , the pump chamber  458  is substantially larger on the left side of the minor axis  64 . Both the upper left and lower left quadrants of the pump chamber  458  are larger than the lower right and upper right quadrants of the pump chamber  458 . 
         [0054]    Turning to  FIGS. 14 and 15 , the average vane tip force represented by the line  371  in  FIG. 14  is about the same as the average vane tip force represented by the line  471  in  FIG. 15 . Thus, for single chamber pumps like those shown at  300 ,  400 , the modifications can be made without sacrificing vane life. 
         [0055]      FIGS. 16-17  illustrate a casing  121  that can be utilized for conventional pumps  500 ,  700  of  FIGS. 18 ,  22  and  24  or the disclosed pumps  600 ,  800  of  FIGS. 19 ,  23  and  25 .  FIG. 17  illustrates the versatility that can be employed in terms of providing a rotational power source. Specifically, a motor  113  is shown schematically that may be coupled directly or indirectly to the rotor shaft  132 . The casing  121  also includes a single inlet  14  and a single outlet  15 . 
         [0056]    Turning to  FIGS. 18-21 , similar results are again achieved with single chamber pumps like those shown at  500 ,  600  in  FIGS. 18-19 . In  FIG. 18 , the pump  500  features a standard cylindrical cavity  528 , a single inlet  14  and a single outlet  15 . The single pump chamber  558  is of the same size on either side of the axis  164  and on either side of the axis  165 . In contrast, turning to  FIG. 19 , the asymmetrical cylindrical cavity  628  of the pump  600  is skewed. Even though there is a single pump chamber  658 , the pump chamber  658  is substantially larger on the left side of the axis  264 . Both the upper left and lower left quadrants of the pump chamber  658  are larger than the lower right and upper right quadrants of the pump chamber  658 . 
         [0057]    Turning to  FIGS. 20 and 21 , the average vane tip force represented by the line  571  in  FIG. 20  is about the same as the average vane tip force represented by the line  671  in  FIG. 21 . Thus, for single chamber pumps like those shown at  500 ,  600  in  FIGS. 18-19  and like those shown at  300 ,  400  in  FIGS. 12 ,  13 , the modifications can be made without sacrificing vane life. Further, the reader will also note that vane tip bouncing or skipping indicated at  173  in  FIG. 20  has been eliminated by the pump  600  as shown in  FIG. 21 . 
         [0058]      FIGS. 22 and 24  illustrates a purely elliptical cavity  728  with a rotor  326  disposed therein. The pump  700  illustrated in  FIG. 22  includes two pump chambers  758 ,  759  disposed on either side of the rotor  326 . The rotor  326  engages the inner wall  751  of the cavity  728  at two points  762 ,  763 , or where the minor axis  764  intersects the inner wall  751 . Not only are the pump chambers  758 ,  759  of the same size, dividing each pump chamber  758 ,  759  into two using major axis  765  both “halves” of the pump chambers  758 ,  759  are of equal volume or each quadrant of each pump chamber  758 ,  759  is of equal volume. 
         [0059]    In contrast, turning to  FIGS. 23 and 25 , the cavity  828  of the pump  800  is skewed. In other words, while the pump chambers  858 ,  859  are of the same size, the inlet portion  858   a  of the pump chamber  858  is larger than the outlet portion  858   b.  Similarly, the intake or inlet portion  859   a  of the pump chamber  859  is larger than the outlet portion  859   b . The effect of theses geometric changes can be seen in  FIGS. 26 and 27 . In  FIG. 26 , the vane tip force versus rotation is plotted for the conventional pump  700  of  FIGS. 22 and 24 . In  FIG. 27 , the vane tip force versus rotation is plotted for the pump  800  of  FIGS. 23 and 25 . Note that the average vane tip force represented by the line  771  in  FIG. 26  is about the same as the average vane tip force represented by the line  871  of  FIG. 27 . 
       INDUSTRIAL APPLICABILITY 
       [0060]    By shifting the angle of maximum vane extension closer to the intake (skewing the circular single chamber or ellipsoidal or oval twin chamber), the angular distance between the intake and exhaust is increased allowing greater internal compression. Further adjustment of the chamber curvature allows optimization of the vane acceleration where the average vane tip load can be reduced and vane “skipping” can be eliminated. The resulting pump is more quiet, cool, and efficient and that has longer vane life. 
         [0061]    Referring back to  FIGS. 5-6 , by way of example only, following methodology may be used in generating the skewed ellipsoidal cavities represented by the lines  56 ,  57 . If,
   L=distance from center;   D minor =minor diameter;   d=maximum radial extension;   A=angle at maximum radial extension (less than or equal to 90°);   a=angle (0 to 360°);   b=skewed angle; and   g=curvature factor (the curvature factor may be varied along the curve as needed to refine the bore shape), then the following asymmetrical bore shape equations apply:   
 
         [0000]        b   a=0 to A°   =a* 90/ A    
         [0000]        b   a=A to 180°   =a +(90− A )*(180− a )/(180− A )
 
         [0000]        b   a=180 to 180+A° =180+( a− 180)*90/ A    
         [0000]        b   a=180+A to 360°   =a +(90− A )*(360− a )/(180− A )
 
         [0000]        L=d *|SIN( b )| g   D   minor /2 
         [0069]    The rise and fall of a vane  27  for an elliptical cavity is d*SIN(a) 2  while the rise and fall of a vane  27  in a skewed elliptical body is d*|SIN(b)| g . Other mathematical techniques for generating various skewed ellipsoidal cavity shapes will be apparent to those skilled in the art. Further, the ellipsoidal cavities shown in the figures may be varied without departing from the scope of this disclosure.