Abstract:
Disclosed is a progressive cavity device. In some embodiments, the device includes a stator with an inner surface having a number of lobes and a rotor disposed within the stator and having a different number of lobes. The stator lobes define a major diameter and a minor diameter, where the major diameter circumscribes the stator lobes and the minor diameter inscribes the stator lobes. A rotor-stator, defined as the major diameter divided by the minor diameter, is selected from the group consisting of 1.350 or less for a progressive cavity device with a stator having two lobes, 1.263 or less for three lobes, 1.300 or less for four lobes, 1.250 or less for five lobes, 1.180 or less for six lobes, 1.175 or less for seven lobes, 1.150 or for eight lobes, 1.125 or less for nine lobes, and 1.120 or less for ten lobes.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of 35 U.S.C. 111(b) provisional application Ser. No. 60/762,599 filed Jan. 26, 2006, and entitled “Positive Displacement Motor/Progressive Cavity Pump With Novel Stator Design”, which is hereby incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
     FIELD OF THE INVENTION 
     The present invention relates generally to positive displacement motors and progressive cavity pumps. More particularly, the present invention relates to a rotor, a stator, and a rotor-stator assembly for a progressive cavity pump and/or positive displacement motor. 
     BACKGROUND 
     A progressive cavity pump, comprising a rotor and a stator, transfers fluid by means of a sequence of discrete cavities that move through the pump as the rotor is turned within the stator. Transfer of fluid in this manner results in a volumetric flow rate proportional to the rotational speed of the rotor within the stator, and relatively low levels of shearing applied to the fluid. Hence, progressive cavity pumps have typically been used in fluid metering and pumping of viscous or shear sensitive fluids. 
     A progressive cavity pump (PCP) may be used in reverse as a positive displacement motor to convert the hydraulic energy of a high pressure fluid into mechanical energy in the form of speed and torque output, which may be harnessed for a variety of applications, including downhole drilling. A positive displacement motor (PDM) comprises a power section including a rotor disposed within a stator, a bearing assembly, and a driveshaft. The driveshaft is coupled to the rotor of the power section and supported by the bearing assembly. Fluid is pumped under pressure through the power section, causing the rotor to rotate relative to the stator, thereby rotating the coupled driveshaft. In general, the rotor has a rotational speed proportional to the volumetric flow rate of fluid passing through the power-section. Another component, for example, a drill bit for downhole drilling, may be attached to the driveshaft. As high pressure fluid is pumped through the power section, rotary motion is transferred from the rotor to the drill bit through the bearing assembly and driveshaft, permitting the rotor to turn the drill bit. 
     A PCP or power section of a PDM generally includes a helical-shaped rotor, typically made of steel that may be chrome-plated or coated for wear and/or corrosion resistance, and a stator, typically a heat-treated steel tube lined with a helical-shaped elastomeric insert.  FIG. 1  illustrates a perspective, cut-away view of a conventional rotor-stator assembly  5  comprising a rotor  10  disposed within a stator  20 . This rotor-stator assembly  5  may be employed as a PCP or the power section of a PDM.  FIG. 2  illustrates a cross-sectional view of the conventional rotor-stator assembly  5  depicted in  FIG. 1 . As shown in this figure, the rotor  10  has one fewer lobe  15  than the stator  20 . When the two components are assembled, a series of cavities  25  are formed between the outer surface  30  of the rotor  10  and the inner surface  35  of the stator  20 . Each cavity  25  is sealed from adjacent cavities by seal lines formed along the contact line between the rotor  10  and the stator  20 . The center  40  of the rotor  10  is offset from the center  45  of the stator  20  by a fixed value known as the “eccentricity” of the rotor-stator assembly  5 . 
     During operation of a PDM, high pressure fluid is pumped into one end of the power section where it fills the first set of open cavities. The pressure differential across the two adjacent cavities forces the rotor to turn. As previously stated, a PCP may be described as operating in reverse of a PDM, meaning the application of speed and torque to the PCP rotor causes the rotor to rotate within the stator, resulting in fluid flow through the length of the PCP, whereas fluid flow through the power section of a PDM causes the rotor to turn. In both types of assemblies, adjacent cavities are opened and filled with fluid as the rotor turns. As this rotation and filling process repeats in a continuous manner, fluid flows progressively down the length of the PCP or the power section of the PDM. Moreover, as the rotor turns inside the stator, the rotor&#39;s center moves in a circular motion about the stator&#39;s center. Because the rotor center is offset from the stator center, out of balance forces are generated by the rotation or nutation of the rotor within the stator. Without being limited by theory, it is believed that the greater the eccentricity of the PCP or power section of the PDM, the higher these out of balance or centrifugal forces. 
     Rotor-stator assembly failures may occur due to the destruction of the stator elastomer. Mechanical failure of the elastomer occurs when it is overloaded beyond its stress and strain limits, such as may be caused by a high compression fit between the rotor and stator. Thermal failure of the elastomer occurs when the temperature of the elastomer exceeds its rated temperature for a prolonged period. Even for shorter periods of time, increasing elastomer temperature causes elastomer physical properties to weaken, resulting in a shortened elastomer life. 
     There are several mechanisms or modes of heat generation that may elevate the elastomer temperature above its rated temperature as follows: interference, hysteresis, centrifugal forces, and downhole sources. Interference between the rotor and the stator is necessary to seal the discrete cavities. Centrifugal forces are exerted on the elastomer by the rotor as the rotor nutates within the stator. The combined effects of interference, centrifugal forces, and sliding or rubbing of the rotor within the stator generate heat within the stator elastomer, causing the temperature of the elastomer to rise. Also, as the rotor nutates within the stator, the elastomer compresses and expands repeatedly. Heat is generated by internal viscous friction of the elastomer molecules, a phenomenon known as hysteresis. Furthermore, heat may be generated by other downhole sources. Heat from these mechanisms—interference, centrifugal forces, hysteresis, and other downhole sources—may cause the elastomer temperature to rise above its rated temperature, resulting in shortened elastomer life or its failure. 
       FIG. 3  illustrates a conventional rotor-stator assembly  50  that includes a rotor  55  inside a stator  60 . The stator  60  further includes an elastomeric liner  62  inside an outer housing  65 . This conventional rotor-stator design and others similar to it are prone to high centrifugal forces as the rotor  55  turns within the stator  60  due to the high eccentricity of the rotor-stator assembly  50 . As described above, these forces generate heat causing the elastomer temperature to rise during operation of the rotor-stator assembly  50 . Additionally, the elastomer design itself inhibits the ability of the elastomer  62  to dissipate heat due to the liner thickness and its relatively low thermal conductivity. Assuming all other factors remain constant, the greater the thickness of the elastomer and the lower its thermal conductivity, the greater the capacity of the elastomer to retain heat. 
     Attempts have been made to modify the conventional design of the stator elastomer in an effort to reduce heat retention by the elastomer.  FIG. 4  illustrates a modified stator  70 , referred to as a constant wall stator, comprising an elastomeric liner  75  with a reduced, as compared to elastomeric liner  62  illustrated in  FIG. 3 , uniform thickness inside an outer housing  80 . By reducing the thickness of the elastomeric liner  75 , its ability to retain heat is also reduced. However, this design modification does not directly address the sources of that heat—the centrifugal forces resulting from nutation of the rotor within the stator and the eccentricity of the rotor-stator assembly. Moreover, this design configuration adds manufacturing complexity, and therefore expense, due to the non-cylindrical inner surface or shape of the stator housing  80 . Still further, this design configuration also limits the range of applications for which the housing  80  may be used. With a housing having a cylindrical inner shape or surface, the lobe configuration in the rotor-stator assembly (e.g., the number of lobes) is commonly changed simply by replacing the elastomeric liner in the stator, whereas the stator housing design illustrated in  FIG. 4  is limited to the lobe configuration shown (i.e., three lobed stator configuration). 
     Due to the shortcomings of conventional rotor-stator assemblies described above, there remains a need for an improved rotor and stator for use in a PCP or power section of a PDM. Such an improved rotor and stator would be particularly well received if it offered the potential to reduce heat generation from centrifugal forces, heat retention by elastomeric components (e.g., the elastomeric stator liner), if present, and/or manufacturing costs while retaining design configuration flexibility. 
     SUMMARY OF THE DISCLOSURE 
     A rotor-stator assembly for a progressive cavity pump and/or positive displacement motor is disclosed, wherein the rotor-stator assembly permits reduced heat generation due to centrifugal forces caused by nutation of the rotor within the stator, heat retention by the stator&#39;s elastomeric liner, if present, and manufacturing costs for the stator housing while retaining the ability of the stator to assume various lobe configurations. 
     In some embodiments, the stator includes a housing having a through bore defining an inner surface, where the inner surface has a plurality of lobes. The plurality of lobes defines a major diameter circumscribing the plurality of lobes and a minor diameter inscribing the plurality of lobes. A stator ratio is equal to the major diameter divided by the minor diameter. The stator ratio is selected from the group consisting of 1.350 or less for a stator with two lobes, 1.263 or less for a stator with three lobes, 1.300 or less for a stator with four lobes, 1.250 or less for a stator with five lobes, 1.180 or less for a stator with six lobes, 1.175 or less for a stator with seven lobes, 1.150 or less for a stator with eight lobes, 1.125 or less for a stator with nine lobes, and 1.120 or less for a stator with ten lobes. 
     In some embodiments, the rotor includes an outer surface having at least one lobe. The at least one lobe defines a major diameter circumscribing the at least one lobe and a minor diameter inscribing the at least one lobe. A rotor ratio is equal to the major diameter divided by the minor diameter. The rotor ratio is selected from the group consisting of 1.350 or less for a rotor with one lobe, 1.263 or less for a rotor with two lobes, 1.300 or less for a rotor with three lobes, 1.250 or less for a rotor with four lobes, 1.180 or less for a rotor with five lobes, 1.175 or less for a rotor with six lobes, 1.150 or less for a rotor with seven lobes, 1.125 or less for a rotor with eight lobes, and 1.120 or less for a rotor with nine lobes. 
     In some embodiments, the progressive cavity device includes a stator and a rotor. The stator has an inner surface with a first number of lobes, where the lobes define a major diameter circumscribing the lobes and a minor diameter inscribing the lobes. The rotor is disposed within the stator and has a second number of lobes different from the first number of lobes. A rotor-stator ratio equals the major diameter divided by the minor diameter. The rotor-stator ratio is selected from the group consisting of 1.350 or less for a progressive cavity device with a stator having two lobes, 1.263 or less for a progressive cavity device with a stator having three lobes, 1.300 or less for a progressive cavity device with a stator having four lobes, 1.250 or less for a progressive cavity device with a stator having five lobes, 1.180 or less for a progressive cavity device with a stator having six lobes, 1.175 or less for a progressive cavity device with a stator having seven lobes, 1.150 or less for a progressive cavity device with a stator having eight lobes, 1.125 or less for a progressive cavity device with a stator having nine lobes, and 1.120 or less for a progressive cavity device with a stator having ten lobes 
     The various characteristics described above, as well as other features of the disclosed apparatus, will be readily apparent to those skilled in the art upon reading the following detailed description and by referring to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more detailed description of the preferred embodiments, reference will now be made to the accompanying drawings, wherein: 
         FIG. 1  depicts a perspective, partial cut-away view of a conventional rotor-stator assembly; 
         FIG. 2  depicts a cross-sectional view of a typical, conventional rotor-stator assembly; 
         FIG. 3  depicts a cross-sectional view of another typical, conventional rotor-stator assembly; 
         FIG. 4  depicts a cross-sectional view of a modified stator, also referred to as a constant wall stator; 
         FIG. 5  depicts an embodiment of a rotor-stator assembly with a two in three lobe configuration made in accordance with the principles described herein; 
         FIG. 6  depicts one illustrative embodiment of a stator with a five lobe configuration made in accordance with the principles described herein; 
         FIG. 7  is a line plot showing the maximum ratio of the stator major diameter to the stator minor diameter as a function of the number of stator lobes for stators made in accordance with the principles described herein as compared to particular known prior art stators; 
         FIG. 8  depicts one illustrative embodiment of a stator with a five lobe configuration but no elastomeric liner in accordance with the principles described herein; and 
         FIG. 9  depicts one illustrative embodiment of a rotor with a four lobe configuration in accordance with the principles described herein. 
     
    
    
     NOTATION AND NOMENCLATURE 
     Certain terms are used throughout the following description and claims to refer to particular assembly components. This document does not intend to distinguish between components that differ in name but not function. In the following discussion and in the claims, the terms “including” and “comprising” are used in an open-ended fashion, and thus should be interpreted to mean “including, but not limited to . . . ”. 
     As used herein, and in the claims that follow, the term “progressive cavity device” refers collectively to a stator with a rotor disposed within. 
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Various embodiments of a rotor-stator assembly for a positive displacement motor and/or a progressive cavity pump that offer the potential to reduce heat generation caused by centrifugal forces resulting from nutation of the rotor within the stator, heat retention by the stator elastomeric liner, if present, and manufacturing costs while retaining design configuration flexibility, will now be described with reference to the accompanying drawings. Like reference numerals are used for like features throughout the several views. There are shown in the drawings, and herein will be described in detail, specific embodiments of the rotor-stator assembly with the understanding that this disclosure is representative only and is not intended to limit the invention to those embodiments illustrated and described herein. The embodiments of the rotor-stator assembly disclosed herein may be used in any type of positive displacement motor (PDM) or progressive cavity pump (PCP). It is to be fully recognized that the different teachings of the embodiments disclosed herein may be employed separately or in any suitable combination to produce desired results. 
       FIG. 5  depicts a cross-sectional, end view of an embodiment of a rotor-stator assembly  100 , including a rotor  102  within a stator  104 . Assembly  100  may be a PCP or a power section of a PDM. Collectively, the rotor  102  and stator  104 , as well as all other rotor-stator assemblies according to the present disclosure, are referred to herein as “progressive cavity devices”. The stator  104  includes a relatively thin liner  105  disposed within, and surrounded by, an outer housing  110 . The outer housing  110  includes a substantially cylindrical inner surface  115  that engages the outer surface  120  of the liner  105 . Specifically, the shape and size (e.g., radius) of the inner surface  115  of housing  110  corresponds to the shape and size (e.g., radius) of the outer surface  120  of liner  105  such that the outer surface  120  of the elastomeric liner  105  statically engages the inner surface  120  of the housing  110 . For instance, an interference fit may be formed between the liner  105  and the housing  110 . In addition to, or as an alternative, the liner  105  may be bonded to the inner surface  115  of the housing  110 . Although this exemplary configuration of the rotor-stator assembly  100  shown in  FIG. 5  has a two in three lobe configuration, meaning a two lobe rotor  102  disposed within a three lobe stator  104 , it should be appreciated that other embodiments may include other lobe numbers and combinations. 
     In general, the stator housing  110  may comprise any suitable material(s) including, without limitation, metals and metal alloys (e.g., stainless steel, titanium, etc.), non-metals (e.g., polymers), composite(s) (e.g., carbon fiber and epoxy composite), or combinations thereof. In one embodiment, stator housing  110  is preferably constructed of a heat-treated carbon steel alloy. Similarly, liner  105  may comprise any suitable materials including, without limitation, metals and metal alloys, non-metals, composites, or combinations thereof. In this embodiment, liner  105  is preferably constructed of an elastomer or synthetic rubber. Thus, liner  105  may be referred to herein as an “elastomeric liner”. 
     The stator  104  depicted in  FIG. 5  may be described in terms of a major diameter (SD) and a minor diameter (Sd). Major diameter (SD) is defined by the dashed circle circumscribing the radially outermost points or surfaces of lobes  125 . Minor diameter (Sd) is defined by the dashed circle inscribing the innermost radial points or surfaces of the elastomeric liner  105 . In general, the eccentricity of a rotor-stator assembly, including rotor-stator assembly  100  depicted in  FIG. 5 , is a function of the major diameter SD and the minor diameter Sd. For a rotor-stator assembly comprising a stator with more than one lobe (e.g., stator  104 ), the eccentricity, as used herein, equals (SD−Sd)/4. Without being limited by this or any particular theory, for a rotor-stator assembly comprising a stator with a single lobe, the eccentricity equals (SD−Sd)/2. 
     As described previously, centrifugal forces caused by nutation of a rotor inside a stator result in heat generation due to friction between the rotor and stator. In some conventional rotor-stator assemblies that include a stator with an elastomeric liner, the heat generation may cause the elastomer temperature to exceed its rated temperature. Without being limited by this or any particular theory, it is believed that the greater the eccentricity of the rotor-stator assembly, the greater the centrifugal forces and resulting heat generation, and the greater the potential for damage, breakdown, and/or failure of the elastomeric liner. Thus, it is desirable to reduce the eccentricity of the rotor-stator assembly. 
     According to the eccentricity equations described above, the eccentricity of a rotor-stator assembly may be decreased by reducing the difference between the major diameter SD and the minor diameter Sd of the stator. In other words, the eccentricity of a rotor-stator assembly may be decreased by reducing the ratio SD/Sd. 
     Embodiments described herein have a maximum SD/Sd ratio of 1.263 for a rotor-stator assembly comprising a three-lobe stator, such as the three-lobe stator  100  depicted in  FIG. 4 . Stated differently, embodiments described herein have an SD/Sd ratio no more than 1.263 for a rotor-stator assembly comprising a three-lobe stator. For comparison purposes, a commonly employed conventional rotor-stator assembly having a three-lobe stator and a two-lobe rotor has an SD/Sd ratio near 1.65, significantly higher than 1.263. Further, another conventional prior art rotor-stator with a three-lobe stator and a two-lobe rotor has a SD/Sd ratio of 1.367, still higher than 1.263. As previously described, and without being limited by this or any particular theory, the lower the eccentricity of a rotor-stator assembly, the lower the centrifugal forces and resulting heat generation. Consequently, embodiments of rotor-stator assemblies including the stator  100  having a maximum SD/Sd ratio of 1.263 offer the potential to reduce centrifugal forces and heat generation within the rotor-stator assembly as compared to many conventional rotor-stator assemblies having a three-lobed stator. 
     In addition, and still referring to  FIG. 5 , it should be appreciated that the inner surface  115  of the stator housing  110  is cylindrical, unlike the cross-section of the prior art stator depicted in  FIG. 4 . In general, a stator housing with a cylindrical inner surface (e.g., inner surface  115  of stator housing  110 ) yields reduced manufacturing costs as compared to the prior art stator  70  depicted in  FIG. 4  and other similarly designed stators having inner surfaces of more complex shape (e.g., a tri-oval surface generally similar to the shape of the desired liner internal profile). Further, a stator housing with a cylindrical inner surface offers the potential for greater versatility than a stator with a non-cylindrical inner surface. In particular, a stator with a cylindrical inner surface may be used with various lobe configurations. For example, the liner  105  of stator  104  shown in  FIG. 5  may be removed and replaced with another liner having a different lobe configuration (e.g., a liner having a four lobed configuration). In contrast, the non-cylindrical inner surface of the prior art stator  70  depicted in  FIG. 4 , and other similar stator configurations, are limited to a particular lobe configuration. Specifically, any liner  75  inserted into the prior art stator  70  depicted in  FIG. 4  can only accommodate a rotor with no more than two lobes. 
     Although the inner surface  115  of the stator housing  100  shown in  FIG. 5  is substantially cylindrical and the liner  105  has a non-uniform wall thickness, thereby enabling the lobed-configuration, in other embodiments, the liner (e.g., liner  105 ) has a substantially uniform wall thickness, yet still enable a lobed-configuration satisfying the preferred maximum SD/Sd ratios described above. In such an embodiment, the housing includes a non-cylindrical outer surface that engages a non-cylindrical outer surface of the liner. 
     Finally, the elastomeric liner  105  of the stator  104  depicted in  FIG. 5  may be made significantly thinner than that of the prior art stators depicted in  FIGS. 2 and 3 . Given that the thermal conductivity of elastomeric materials is relatively low (i.e., relatively high resistance to heat transfer), the amount of heat retained by an elastomeric liner generally increases as the thickness of liner increases. Thus, the thinner the elastomeric liner, the less thermal energy retained by the elastomer. Therefore, providing a thinner elastomeric liner  105 , as compared to the liners of the prior art stators typified by the stators depicted in  FIGS. 2 and 3 , offers the potential to reduce heat retention by the elastomeric liner  105 , and thereby increase the life of the liner. 
     While the embodiment of stator  104  illustrated in  FIG. 5  includes three lobes, other lobe configurations are also possible. For example,  FIG. 6  depicts a cross-sectional, end view of another embodiment of a stator  200  including five lobes  205 . Stator  200  has a maximum SD/Sd ratio of 1.25. Many conventional rotor-stator assemblies including a five-lobed stator configuration have SD/Sd ratios generally in the range 1.4 to 1.45. As compared to such conventional five-lobe designs, embodiments of stator  200  have a reduced SD/Sd ratio, and thus, for similar reasons as described above, offer the potential for lower centrifugal forces and associated thermal energy, reduced elastomeric liner thickness and heat retention in those embodiments including an elastomeric liner, and reduced manufacturing costs while retaining design configuration flexibility for those embodiments having a stator with a liner disposed within a housing. 
     Other embodiments with different lobe configurations (e.g., 6 lobe stator, 8 lobe stator, etc.) made in accordance with the principles described herein offer the potential for similar benefits and advantages. Specifically, Table 1 below lists maximum SD/Sd ratios for a variety of rotor-stator configurations made in accordance with the principles described herein. As the SD/Sd ratios listed are the maximum SD/Sd ratios, it should be understood that some embodiments may comprise SD/Sd ratios lower than those listed. For example, a rotor-stator assembly with a four in five lobe configuration, meaning a four-lobe rotor inside a five-lobe stator, may have an SD/Sd ratio equal to 1.100, which is less than the maximum value permitted, or 1.250. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 No. of Rotor Lobes 
                 No. of Stator Lobes 
                 SD/Sd Ratio 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 1 
                 2 
                 1.350 
               
               
                 2 
                 3 
                 1.263 
               
               
                 3 
                 4 
                 1.300 
               
               
                 4 
                 5 
                 1.250 
               
               
                 5 
                 6 
                 1.180 
               
               
                 6 
                 7 
                 1.175 
               
               
                 7 
                 8 
                 1.150 
               
               
                 8 
                 9 
                 1.125 
               
               
                 9 
                 10 
                 1.120 
               
               
                   
               
             
          
         
       
     
     Referring now to  FIG. 7 , there is shown a line plot of the maximum SD/Sd ratio  300  for a rotor-stator assembly in accordance with the principles described herein as a function of the stator lobe configuration of Table 1. For purposes of comparison, SD/Sd ratios for certain conventional prior art rotor-stator assemblies are plotted as a function of their stator lobe configuration. SD/Sd ratio  310  is relatively low, while SD/Sd ratio  320  is substantially higher. As seen in  FIG. 7 , rotor-stator assemblies constructed in accordance with the principles described herein have lower SD/Sd ratios as compared to these common prior art rotor-stator assemblies. Thus, embodiments of rotor-stator assemblies that satisfy the design criteria specified in Table 1 above share a common design feature, relatively low eccentricity (e.g., relatively low SD/Sd ratio). As previously discussed, rotor-stator assemblies exhibiting reduced eccentricity offer the potential for lower centrifugal forces resulting in lower out of balance forces and reduced heat generation. Further, for those embodiments including an elastomeric liner (e.g.,  FIG. 5 ), a reduced eccentricity enables a thinner wall elastomeric liner, which in turn offers the potential for lower heat retention and a longer life elastomeric liner. 
     It should be appreciated that that rotor-stator assemblies constructed in accordance with the principles described herein may have a variety of suitable configurations (e.g., with a liner, without a liner, having a housing with a cylindrical inner surface, etc.), but are preferably constructed in accordance with the SD/Sd ratios disclosed in Table 1 above. Assuming the preferred SD/Sd ratio criteria is satisfied, additional benefits potentially may be obtained, as previously described, by utilizing a thinner stator elastomeric liner, a stator housing with a cylindrical inner surface, etc. In some applications, however, it may be advantageous for the rotor-stator assembly to be configured such that it does not have one or more of these additional design features. 
     For example, a common failure mode in conventional rotor-stator assemblies is damage or destruction of the stator elastomer. To eliminate that as a potential failure mode, certain embodiments of the rotor-stator assembly designed in accordance with Table 1 are constructed such that the stator is free of (or constructed without) an elastomeric liner within the stator. In such embodiments, the stator is a solid, integral stator. For example,  FIG. 8  depicts a cross-sectional, end view of one representative liner-less stator  400  according to the present disclosure, wherein the stator  400  comprises a housing or shell  405  with five lobes  410  defined along its inner surface. Stator  400  includes no elastomeric liner. By eliminating the elastomeric liner, such embodiments also eliminate the component most likely to fail. In the absence of an elastomeric liner, the inner surface of the stator defines the stator lobe configuration and is the surface contacted by the rotor as it nutates within the stator. Otherwise, the rotor-stator assembly functions the same as previously discussed embodiments. Embodiments constructed in accordance with the preferred maximum SD/Sd ratios described herein and shown in Table 1 enable a reduced eccentricity, and reduced centrifugal forces, regardless of whether the stator includes an elastomeric liner. 
       FIGS. 6 and 8  depict representative embodiments of stators constructed in accordance with the principles described herein. While these figures do not also depict a rotor, it is to be understood that in operation, a rotor will be disposed within each stator constructed in accordance with the principles disclosed herein, including those depicted in  FIGS. 6 and 8 , to form a PCP or power section of a PDM. Each such rotor will also be constructed generally in accordance with the SD/Sd ratios disclosed in Table 1 above, meaning the ratio of the rotor major diameter to the rotor minor diameter will satisfy the maximum SD/Sd values listed in this table with slight differences to provide an interference fit between the rotor and the stator within which the rotor will be disposed. The interference fit creates the seal lines between the inner surface of the stator and the outer surface of the rotor. For example,  FIG. 9  depicts a four lobe rotor  500  constructed in accordance with the principles disclosed herein. In operation, it will preferably be assembled inside a five-lobe stator also constructed in accordance with the principles disclosed herein, such as the stator  200  depicted in  FIG. 6  and/or the stator  400  depicted in  FIG. 8 , to form a PCP or power section of a PDM. The four-lobe rotor  500  depicted in  FIG. 9  is constructed to also satisfy the SD/Sd ratio criteria disclosed in Table 1, meaning the rotor  500  is constructed such that the ratio of its major diameter  505  to its minor diameter  510  will be less than or equal to 1.263. 
     While various embodiments of a low eccentricity rotor-stator assembly for a positive displacement pump and/or progressive cavity pump have been shown and described herein, modifications may be made by one skilled in the art without departing from the spirit and the teachings herein. The embodiments described are representative only, and are not intended to be limiting. Many variations, combinations, and modifications of the applications disclosed herein are possible and are within the scope of the invention. Accordingly, the scope of protection is not limited by the description set out above, but is defined by the claims which follow, that scope including all equivalents of the subject matter of the claims.