Screw assembly for a three-axis screw pump and screw pump including the assembly

JP7920202B2Active Publication Date: 2026-09-14SETTIMA FLOW MECHANISMS SRL
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Patent Information

Application Number
JP2023574750
Authority / Receiving Office
JP · JP
Patent Type
Patents
Current Assignee / Owner
Priority Date
2021-02-23
Filing Date
2021-12-28
Publication Date
2026-09-14
Estimated Expiration
2041-12-28

AI Technical Summary

Benefits of technology

【0059】 本発明によるポンプのその他の利点は、その性能に関するものである。特に、ポンプの容積効率は同じだが、圧力リップルが改善され、騒音が減少し、正味吸込ヘッド(NPSH)が低くなる。

✦ Generated by Eureka AI based on patent content.

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Abstract

The screw assembly (1) of the three-shaft screw pump (10) includes a central screw (2) and a central screw shaft (z c ) and the transverse screw axis (z l and at least one transverse screw (3) configured to mesh with the central screw (2) having an outer diameter (Φ ce ) is the outer diameter of the lateral screw (Φ le ) and the inner diameter of the central screw (Φ ci ) is the outer diameter of the horizontal screw (Φ le ) is smaller than
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Description

[Technical Field]

[0001] The present invention relates to a screw assembly for positive displacement gear pumps, particularly for three-screw pumps. The present invention also relates to a three-screw pump including the above-mentioned screw assembly.

[0002] This invention can find useful applications in various industrial fields where gear pumps, particularly three-screw pumps, have conventionally been used.

[0003] While lifting systems are a typical application where three-screw pumps are highly valued, they are also widely used in a variety of other fields, including power hydraulics, lubrication, cooling, filtration, and transfer. As a non-limiting example, other industrial sectors where three-screw pumps are applied, in addition to lifting systems, include oil and gas, chemicals, navy, transportation, agrofood, power generation and alternative energy, paper industry, and pharmaceutical industry. [Background technology]

[0004] Designed in 1923 by Swedish engineer Carl Montelius, the three-screw pump is a positive displacement pump now widely used in various industrial sectors. In fact, this pump boasts remarkable overall efficiency, good reliability, a reasonable price, and low levels of acoustic emission and vibration in fluid transmission.

[0005] A three-screw pump has a set of three screws: a central lead screw and two lateral driven screws. Preferably, the screws, each having two helical threads, are mounted in parallel within the casing and mesh with each other to create a sealed volume between the body and the casing. The number of closed chambers thus formed is directly proportional to the length of the screw (also called the rotor) and inversely proportional to the pitch of the helical threads. The closed chambers are occupied by the working fluid that moves continuously from the suction port to the discharge port as the screw rotates.

[0006] The profiles of the three screw sets are designed such that only the driving screw transmits pressure. Due to the configuration of the pump, this screw is not exposed to radial forces, and thus exhibits the excellent overall efficiency mentioned above. As previously stated, the two driven screws idle and are guided by the pressurized fluid. The only obstacles to their rotation are the viscous friction with the working fluid, and the sliding friction between the central screw and the casing that accommodates them. Therefore, the tooth flanks of the screws experience almost no wear even after long periods of operation.

[0007] Even now, nearly one century after its invention, the three-shaft screw pump still retains the characteristic appearance conceived by its creator, and is characterized by the typical ratio of the diameters of the surface profiles of the central driving screw and the lateral driven screws. Φ li and Φ le represent the inner diameter and outer diameter of the lateral screw respectively, Φ ci and Φ c represent the inner diameter and outer diameter of the central screw respectively, and the dimensions Φ li :Φ le :Φ ci :Φ ce actually follow the ratio of 1:3:3:5, which is considered optimal because it represents the most optimal possible ratio between the area occupied by the fluid and the area of the solid defined by the screw material.

[0008] With this ratio, it should be noted that the outer diameter Φ of the lateral screw le and the inner diameter Φ of the central screw ci are always strictly equal. This equality of these diameters is regarded as an absolute axiom in the prior art, and all designs of three-shaft screw pumps are based on it.

[0009] The circles identified by the aforementioned diameter represent the pitch diameter used to create the curves that make up the ideal profile, i.e., the original profile before modification, which is commonly employed to eliminate sharp edges. The rationale behind choosing this design is the consideration that two base cylinders of equal diameter, having equal tangential velocities and rotating at opposite angular velocities, will roll over each other without slipping, resulting in less heat and energy dispersion.

[0010] Furthermore, the inner diameter of the horizontal screw Φ ci The outer diameter of the horizontal screw is Φ le Reducing the Φ of the inner diameter of the lateral screw offers no advantages, as it causes rolling with wear due to sliding and reduced efficiency, as well as a reduction in the empty cross-sectional area where the working fluid is trapped, i.e., a reduction in pump capacity. Conversely, reducing the Φ of the inner diameter of the lateral screw ci The outer diameter of the lateral screw relative to Φ le The increase in this factor would lead to the mutual penetration of helicoid profiles generated during the rotation of the screw, which seemed impossible in prior art studies.

[0011] Therefore, if the above equal diameters are selected, in the prior art, the tooth surfaces of the central screw and transverse screw are obtained by applying the epitrochoid equation, where the epitrochoid is defined as a fixed point at a distance p from the center of a circle of radius r, and the radius r b This is a roulette curve obtained by connecting points described in space by rolling the aforementioned circle outside another circle. The epitrochoid defines the cross-sectional shape of the tooth surface of each screw, and as it advances along the axis of rotation of the screw, the shape rotates continuously and defines a helix.

[0012] The known parametric equations for epitrochoids are as follows: TIFF0007920202000001.tif17150 TIFF0007920202000002.tif9150

[0013] The polar equation is as follows: TIFF0007920202000003.tif9150

[0014] In FIG. 5, with respect to the prior art, R1 and d1 are beside parameters related to the tooth profile structure of the screw, and R2 and d2 are center parameters related to the tooth profile structure of the screw. From the above discussion, the base radius r is the same for both structures. As can be seen, the radii R1 and R2 of the rotation circle are both equal to r in either structure. Furthermore, in the configuration of the tooth surface of the central screw, the trace point is located at the end of the radius r1 of the rotation circle, and the resulting curve is called an epicycloid.

[0015] Therefore, the design parameter that should be set is only the distance from the center of point d2 that defines the tooth profile of the transverse screw, and respectively determines the outer diameter of the central screw and the interior of the transverse screw. The selection of this parameter aims to optimize the displacement, that is, the volume of the screw occupied by the fluid, without affecting the mechanical strength of the screw.

[0016] Specifically, the general ratio 1:3:3:5 between the diameters of the screws can be obtained by selecting a value of d2 equal to 5 / 3d1, for example, by selecting the following parameters. R1=R2=1.5 r=1.5 d2=2.5 (generates tooth profile of driven screw) d1=1.5 (generates tooth profile of lead screw or drive screw)

[0017] Since these profiles are similar, profiles of any size can be obtained by using a simple scaling effect to create this basic relationship.

[0018] It should be noted that the ideal profile generated by the epitrochoid equation has sharp edges. These edges are easily deformed. If the edges are deformed, there is a risk of noise and abnormal vibrations occurring during pump operation, or irreparable damage to the pump itself. Furthermore, it is difficult to manufacture the edges with tooling precision, and as a result, localized shape errors can cause undesirable difficulties in screw meshing.

[0019] For the reasons stated above, in prior art, the ideal profile is generally modified by chamfering the aforementioned sharp edges, especially those of the driven screw that are sharper and potentially more critical. Chamfering can be done in a simple way by cutting the edges in a straight line, or in a more sophisticated way by using an arc-shaped or elliptical arc-shaped connecting profile. The latter solution minimizes leakage and volume loss.

[0020] Clearly, introducing the above geometric corrections results in the loss of complete conjugation along the screw tooth surface lines, requiring a complete recalculation of the profiles of both the driven and driving screws.

[0021] Prior art three-screw pumps are disclosed, for example, in references US3,814,557A and US2012 / 258000A1.

[0022] As discussed in this chapter regarding prior art, it should be noted that the three-screw pump originated in the early 1900s, and the screw profile has remained largely unchanged to this day. Improvements introduced to date have always involved structural or material modifications.

[0023] On the other hand, such widely used machines always have a need for improvement, particularly in terms of increasing capacity and reducing radial and axial dimensions.

[0024] Therefore, the technical problem of the present invention is to provide a screw assembly and a corresponding three-screw pump having a significantly larger flow rate than conventional pumps of similar size. [Overview of the project]

[0025] The underlying solution idea of ​​this invention is the outer diameter of the lateral screw and center The objective is to provide a screw assembly and a corresponding three-screw pump by relaxing the condition that the inner diameters of the screws must be the same.

[0026] From a purely theoretical standpoint, the above conditions are necessary to ensure there is no slippage between the two cylinders, which consist of the inner diameter of the transverse screw and the outer diameter of the transverse screw, although in practice it has been confirmed that the diameters do not touch. On the one hand, the clearance that inevitably exists to enable the operation of the pump must be taken into consideration. On the other hand, the force exchanged between the screws during the transmission of motion is the resultant force, as specified, for example, in Figure 14. F res This has been confirmed to generate a reaction force that theoretically pushes two contacting surfaces apart.

[0027] On the other hand, there is a second reason why the aforementioned diameters are kept equal in the prior art. As explained in the previous chapter, the only advantageous change in diameter from the perspective of overall capacity is the Φ of the inner diameter of the transverse screw. ci The outer diameter of the lateral screw relative to Φ le This would increase the screw profile. However, this change was considered impossible because it would lead to penetration between the screw profiles.

[0028] However, this theoretical perspective failed to adequately consider the actual profile changes introduced during machining, particularly the chamfering applied to the sharp edges of the profile. These geometric corrections are precisely performed where mutual penetration would occur and within the range of the pitch diameter that generated the edge. Therefore, theoretical mutual penetration can be avoided in practice by utilizing the technical necessity of edge chamfering.

[0029] Relaxing the equivalent conditions brought about entirely new design perspectives, allowing for a rethinking of the fixed profiles of prior art. As a result, a larger fluid capture area could be obtained, ultimately enabling an increase in flow rate with a screw of the same outer diameter.

[0030] In this invention, various parameters affecting the definition of the profile were considered, and a relaxation of the equivalence conditions between diameters was achieved. First, the clearance required for a three-axis screw set to rotate correctly was examined. Therefore, the maximum range of corrections that can be applied to the theoretical formula was verified to prevent imbalances in the profile that would cause fluid leakage. Finally, the minimum inner diameter achievable with a transverse screw was evaluated from a technical standpoint, taking into account the machinability by the machine tool and the mechanical strength of the part itself.

[0031] Therefore, the technical problems revealed above are solved by the screw assembly according to claim 1 and by the respective three-screw pumps according to claim 16.

[0032] Therefore, a screw assembly for a three-screw pump, comprising a central screw and at least one transverse screw arranged such that its transverse screw axis is parallel to the central screw axis and meshes with the central screw, the central screw outside The diameter is larger than the outer diameter of the lateral screw. center A screw assembly characterized in that the inner diameter of the screw is smaller than the outer diameter of the transverse screw. Both screws contain one or more helical threads having a constant pitch.

[0033] center The inner diameter of the screw is preferably included in 60% to 99% of the outer diameter of the transverse screw, more preferably in 68% to 98%, and even more preferably in 85% to 92%.

[0034] Preferably, center The inner diameter of the screw is smaller than the diameter of each pitch circle, while the outer diameter of the transverse screw is larger than the diameter of each pitch circle.

[0035] Preferably, the outer diameter of the transverse screw is between 1 and 1.3 times the diameter of each pitch circle, more preferably between 1 and 1.2 times, and even more preferably, the outer diameter of the transverse screw is equal to 1.1 times the diameter of each pitch circle.

[0036] Preferably, the distance between the axes of the central screw and the transverse screws is greater than 1 / 2 of the outer diameter of the central screw and less than 3 / 5 of the outer diameter.

[0037] The distance between the axes of the central screw and the transverse screws is preferably set to 52% to 56% of the outer diameter of the central screw, and more preferably equal to 54%.

[0038] The features and advantages of the gears and apparatus of the present invention will become apparent from the following description of its embodiments, which are given as non-limiting examples with reference to the accompanying drawings. [Brief explanation of the drawing]

[0039] [Figure 1] Figure 1 schematically shows a three-screw pump that can characterize the screw assembly according to the present invention. [Figure 2] Figure 2 schematically shows a portion of the central screw of the screw assembly according to the present invention in a side view. [Figure 3]Figure 3 schematically shows a portion of the central screw of the screw assembly according to the present invention in a side view. [Figure 4] Figure 4 shows a cross-section of the screw assembly according to the present invention in its operating configuration, where the fluid capture region is identified by the meshing portion. [Figure 5] Figure 5 shows the generation of the screw profile in a prior art three-screw pump. [Figure 6] Figure 6 shows the first step of a conceptual procedure for generating a tooth surface profile in a screw assembly according to the present invention. [Figure 7] Figure 7 shows the second step of a conceptual procedure for generating a tooth surface profile in a screw assembly according to the present invention. [Figure 8] Figure 8 shows the third step of a conceptual procedure for generating a tooth surface profile in a screw assembly according to the present invention. [Figure 9] Figure 9 shows the fourth step of a conceptual procedure for generating a tooth surface profile in a screw assembly according to the present invention. [Figure 10] Figure 10 compares the profile of the central screw according to the present invention with the profile of the central screw according to the prior art. [Figure 11] Figure 11 compares the profile of the transverse screw according to the present invention with the profile of the transverse screw according to the prior art. [Figure 12] Figure 12 compares the profile of the central screw according to the present invention with the profile of the central screw according to the prior art, with the auxiliary fluid capture region indicated by the shaded area. [Figure 13] Figure 13 compares the profile of the transverse screw according to the present invention with the profile of a transverse screw according to the prior art, with the auxiliary fluid capture region identified by the hatched area. [Figure 14] Figure 14 shows the forces acting on a rotor driven by a typical three-screw pump. [Modes for carrying out the invention]

[0040] Referring to Figure 1 above, the three-screw pump is shown as a whole by reference no. 10, and reference no. 1 shows the screw assemblies 2 and 3 assembled on it. As previously stated, the present invention relates in particular to the screw profiles 20 and 30, 2 and 3, which in Figures 10-13 face the corresponding profiles 20' and 30' of the prior art. The new profiles 20 and 30 define, in cross-section, the capture volume V in which the fluid pumped is captured, relative to the corresponding profiles 20' and 30' of the prior art.

[0041] It should be noted that the figures are schematic diagrams and are not drawn to a fixed scale, but rather to highlight key features of the invention. Furthermore, the figures show elements schematically, as their shapes may vary depending on the desired application. It should also be noted that in the drawings, the same reference numerals refer to elements with the same shape or function.

[0042] In known embodiments, a three-screw pump 10 includes a pump body 5 having a suction port S and a discharge port D. Inside the pump body is a screw assembly 1 having a lead central screw 2 integrated with the drive shaft 4 and two driven lateral screws 3. The axis z of the lateral screws 3 l and the axis z of the central screw 2 c The screws are parallel to each other and mesh with each other. The rotational motion of the central screw 2 thus moves the two transverse screws 3, and as shown in Figure 4, the fluid F is transported from the suction port S to the discharge port D in the space enclosed between the opposing screw threads.

[0043] The central screw 2 has a fixed pitch p c It has two threads 21 and 22, and the lateral screw 3 also has the same pitch p as the central screw 2. l It has two screw threads.

[0044] Therefore, the profile 20 of the central screw 2 has, in cross-section, two circular apex portions connected to a cylindrical base by a convex tooth surface.

[0045] The profile 30 of the lateral screw 3 also has, in cross-section, two circular apex portions connected to a cylindrical base by a significantly recessed tooth surface.

[0046] Note that, by known methods, the two transverse screws 3 are equivalent to each other or have the same profile 30.

[0047] As described above, the present invention relates to specific shapes of the tooth surface profiles 20 and 30 of screws 2 and 3.

[0048] Preferred embodiments described herein illustrate preferred shapes of the profiles and how they are obtained from prior art profiles.

[0049] As described in the corresponding paragraph of this disclosure, the prior art profile is created from equivalent conditions between the inner diameter and outer diameter of the transverse screw. Thus, as shown in Figure 5, center There exists an interaxial distance s' equal to the inner diameter of screw 2, i.e., the outer diameter of the transverse screw 3. Furthermore, in the prior art, the inner diameter of the transverse screws corresponds to 1 / 3 of their respective outer diameters, and the outer diameter of the central screw corresponds to 5 / 3 of its inner diameter. Therefore, the typical diameter ratio is 1:3:3:5.

[0050] To obtain a new profile, first modify the above ratios and identify new parameter settings that can increase the pump capacity without compromising the mechanical resistance of the screw. The new ratio between diameters Φ' is shown in Figure 6. li :Φ' le :Φ' ci :Φ' ceThe ratios were conveniently selected as 0.4:2.7:2.7:5, which allows for an increase of approximately 7% in the suction section. Following the proposed parameter setting for diameter, the inter-axis distance S was reduced from 3 to 2.7 compared to the prior art.

[0051] Starting with new parameters, the ideal profiles of the two screws are generated using the epitrochoid equation described in the prior art analysis. As previously mentioned, an epitrochoid is a curve obtained by connecting points described in space by rolling a circle outside another circle, starting from a fixed point at a constant distance from the center of the radius circle. In this case, the distance from the circle and the radius of the circle are determined by the inner and outer diameters selected for the two screws. The epitrochoid is circumscribed and inscribed in the circle defined by the inner and outer diameters selected for both screws, determining the ideal profile shown in Figure 7.

[0052] Another parameter to be determined is the starting points p, p', and p'' of the epitrochoid formation. In fact, the parameter characterizing the screw is the angle α determined by the chord connecting the two consecutive starting points p' and p'' of the epitrochoid that generate the profile of the central screw 2. This value is uniquely related to the corresponding angle β of the other screw 3. The angles, defined as the tooth opening angle α and the tooth surface opening angle β, define the length of the arc connecting the tooth surfaces on the outer shape of the central screw 2, and the length of the arc between two consecutive teeth of the transverse screw 3. On the one hand, they determine the cylindrical surface that slides in contact with the screw housing, and on the other hand, they determine the mechanical strength of the helix defined by the screw. The applicant has determined through geometric analysis that the volume effective in capturing the working fluid is invariant with respect to the selection of the tooth opening angle α and the tooth surface opening angle β. Thus, the angles can be arbitrarily selected based on tribological and mechanical considerations without affecting the pump's capacity.

[0053] Next, an additional geometric shape g is applied to the ideal profile of the driven lateral screw 3. The additional geometric shape g shown in Figure 8 extends outward from the pitch diameter and onto the tooth surface f defined by the epitrochoid equation, which is the previously set outer diameter Φ' of the lateral screw. le A diameter larger than that, i.e., the pitch circle diameter C pl Larger diameter Truncated circle (truncation (circle) C t The additional geometric shape g defines a face c of the screw profile, which connects to the tooth surface at the previously identified point p. The connection point p between the face c defined by the epitrochoid and the tooth surface f defined by the additional geometric shape is preferably an inflection point rather than an angle point (an angle point being a point of non-differentiability of the first kind). The additional geometric shape g can be appropriately selected according to design choices; for example, it can be an elliptic curve or a spline function.

[0054] Once the final profile of the transverse screw 3 is obtained, the profile of the central screw 2 is determined by interpolation. The two final profiles are shown in Figure 9. As you can see, at the base of the plane c' of the central screw 2 defined by the epitrochoid, there is a pitch circle C pc A new connecting tooth surface f' is formed for the inner circle. Therefore, redefining the profile will change the inner and outer diameters of the two screws. In particular, center Screw inner diameter Φ ci The outer diameter of the horizontal screw is Φ le It became smaller. Using the parameter settings mentioned above, the final diameter Φ li :Φ le :Φ ci :Φ ce The ratio becomes 0.4:2.97:2.43:5.

[0055] The modifications made to the ideal profile shown in Figure 7 result in an additional approximately 10% increase in pump capacity for the same screw diameter. Therefore, compared to the prior art, the overall capacity increase is approximately 17%. Furthermore, the reduced distance between the screw axes leads to a reduction in the radial dimensions of the pump.

[0056] The improvements mentioned above can be clearly seen in Figures 12 and 13. In fact, the hatched areas indicate that even with the same outer diameter of the screw, the free frontal volume that the fluid pumped up by the pump can occupy has increased, resulting in an increased capacity.

[0057] The advantages of the pump according to the present invention stem particularly from its compact radial dimensions, but for the same flow rate, the axial dimensions also become more compact due to the shorter screw pitch.

[0058] Furthermore, because the pump's structure requires less material, it has the advantage of lower manufacturing costs.

[0059] Other advantages of the pump according to the present invention relate to its performance. In particular, while the volumetric efficiency of the pump remains the same, the pressure ripple is improved, noise is reduced, and the net suction head (NPSH) is lower.

[0060] Clearly, those skilled in the art can make several modifications and alterations to the gears and apparatus described above to meet incidental and specific needs, all of which fall within the scope of protection of the present invention as defined by the following claims.

Claims

1. A central screw (2) having one or more threads with a constant pitch, Each is configured to mesh with the central screw (2), and each is connected to the central screw axis (Z c ) Horizontal screw axis (Z l It includes two transverse screws (3) having ), The outer diameter (Φ) of the central screw (2) ce ) is the outer diameter (Φ) of the horizontal screw (3). le Larger than ) The inner diameter (Φ) of the central screw (2) ci ) is the outer diameter (Φ) of the horizontal screw (3). le Smaller than ) The distance (S) between the axes of the central screw (2) and the transverse screw (3) is the outer diameter (Φ) of the central screw (2). ce It is characterized by being greater than half and less than 3 / 5 of ) Screw assembly (1) of a three-screw pump (10).

2. The inner diameter Φ of said central screw (2) ci ) is smaller than the diameter of the pitch circle C of said central screw (2) pc ), and the outer diameter Φ of said lateral screws (3) le ) is larger than the diameter of the respective pitch circle C of said lateral screws (3) pl ) The screw assembly (1) according to claim 1.

3. The outer diameter (Φ) of the aforementioned horizontal screw (3) le ) is the respective pitch circle (C pl ) is included in a range of 1 to 1.3 times the diameter of the aforementioned. The screw assembly (1) according to claim 2.

4. The outer diameter (Φ) of the aforementioned horizontal screw (3) le ) is the respective pitch circle (C pl ) is equal to 1.1 times the diameter of the aforementioned The screw assembly (1) according to claim 3.

5. The cross-sectional profile of the transverse screw (3) has a tooth surface (f) that follows the epitrochoid equation, and the tooth surface (f) is connected to the pitch circle (C) of the transverse screw (3) via a surface (c) defined by the epitrochoid. pl The truncated circle (C) has a diameter larger than the diameter of ). t ) Connected A screw assembly (1) according to any one of claims 1 to 4.

6. The connection point between the tooth surface (f) and the surface (c) is the inflection point of the cross-sectional profile of the transverse screw (3). The screw assembly (1) according to claim 5.

7. The surface (c) of the transverse screw (3) is curved and does not pass through an angle point which is a non-differential point of the first kind, and the tooth surface (f) and the truncation circle (C t ) Connected The screw assembly (1) according to claim 5 or claim 6.

8. The cross-sectional profile of the central screw (2) has a plane (c') defined by the epitrochoid equation, The aforementioned surface (c') is formed by the tooth surface (f') of the base circle (C) of the central screw (2). b ) is connected A screw assembly (1) according to any one of claims 5 to 7.

9. The distance (S) between the axes of the central screw (2) and the transverse screw (3) is the outer diameter (Φ) of the central screw (2). ce ) is included in 52% to 56% A screw assembly (1) according to any one of claims 1 to 8.

10. The distance (S) between the axes of the central screw (2) and the transverse screw (3) is the outer diameter (Φ) of the central screw (2). ce ) is equal to 54% The screw assembly (1) according to claim 9.

11. The two horizontal screws (3) are equal to each other, and each is on the central screw axis (Z c The horizontal screw axis (Z) parallel to ) l ) has and is configured to mesh on both sides of the central screw (2) A screw assembly (1) according to any one of claims 1 to 10.

12. The central screw (2) has an equal pitch (p c It includes a first thread (21) and a second thread (22) having ), The two transverse screws (3) have equal pitch (p l It includes a first thread (31) and a second thread (32) having ), The pitch (p) of the threads of the central screw (2) c ) is the pitch (p) of the threads of the horizontal screw (3). l ) is the same as The screw assembly (1) according to claim 11.

13. Pump body (5) and Suction port (S), Outlet (D), The screw assembly (1) according to claim 11 or 12, The central screw (2) and the lateral screw (3) are rotatably arranged within the pump body (5) and are meshed together. The rotation of the central screw (2) and the lateral screw (3) moves the fluid (F) from the suction port (S) to the discharge port (D). Three-screw pump (10).

Citation Information

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