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Screw Compressors Mathematical
2.4 Review of Most Popular Rotor Profiles 37
Fig. 2.21. “N” Rotors in 5-6 configuration
Fig. 2.22. “N” Rotors in 5-7 configuration
38 2 Screw Compressor Geometry
Fig. 2.23. “N” rotors in 6/7 configuration
sealing lines, small confined volumes, involute rotor contact and proper gate rotor torque distribution together with high rotor mechanical rigidity.
The number of lobes required varies according to the designated compressor duty. The 3/5 arrangement is most suited for dry air compression, the4/5 and 5/6 for oil flooded compressors with a moderate pressure difference and the 6/7 for high pressure and large built-in volume ratio refrigeration applications.
Although the full evaluation of a rotor profile requires more than just a geometric assessment, some of the key features of the “N” profile may be readily appreciated by comparing it with three of the most popular screw rotor profiles already described here, (a) The “Sigma” profile by Hammertoe,1979, (b) the SRM “D” profile by Absterge 1982, and (c) the “Cyclin” profile by Hough and Morris, 1984. All these rotors are shown in Fig. 2.20 where it can be seen that the “N” profiles have a greater throughput and a stiffer gate rotor for all cases when other characteristics such as the blow-hole area,confined volume and high pressure sealing line lengths are identical.
Also, the low pressure sealing lines are shorter, but this is less important because the corresponding clearance can be kept small.
The blow-hole area may be controlled by adjustment of the tip radii on both the main and gate rotors and also by making the gate outer diameter equal to or less than the pitch diameter. Also the sealing lines can be kept very short by constructing most of the rotor profile from circles whose cen tres are close to the pitch circle. But, any decrease in the blow-hole area will increase
the length of the sealing line on the flat rotor side. A compromise between these trends is therefore required to obtain the best result.
2.4 Review of Most Popular Rotor Profiles 39
Rotor instability is often caused by the torque distribution in the gate rotor changing direction during a complete cycle. The profile generation procedure described in this paper makes it possible to control the torque on the gate rotor and thus avoid such effects. Furthermore, full involute contact between the “N” rotors enables any additional contact load to be absorbed more easily than with any other type of rotor. Two rotor pairs are shown in Fig. 2.24 the first exhibits what is described as “negative” gate rotor torque while the second shows the more usual “positive” torque.
Fig. 2.24. “N” with negative torque, left and positive torque, right
2.4.13 Blower Rotor Profile
The blower profile, shown in Fig. 2.25 is symmetrical. Therefore only one quarter of it needs to be specified in order to define the whole rotor. It consists of two segments, a very small circle on the rotor lobe tip and a straight line. The circle slides and generates cycloids, while the straight line generates involutes.
40 2 Screw Compressor Geometry
Fig. 2.25. Blower profile
2.5 Identification of Rotor Position in Compressor Bearings
The rotor axial and radial forces are transferred to the housing by the bearings. Rolling element bearings are normally chosen for small and medium screw compressors and these must be carefully selected to obtain a satisfactory design. Usually, two bearings are employed on the discharge end of each of the rotor shafts in order to absorb the radial and axial loads separately.
Also, the distance between the rotor center lines is in part determined by the bearing size and internal clearance. Any manufacturing imperfection in the bearing housing, like displacement or eccentricity, will change the rotor position and thereby influence the compressor behaviour.
The system of rotors in screw compressor bearings is presented in Fig. 2.26.The rotor shafts are parallel and their positions are defined by axes and .
The bearings are labeled 1 to 4, and their clearances, as well as the manufacturing tolerances of the bearing bores, and in the x and y directions respectively, are presented in the same figure. The rotor center distance is and the axial span between the bearings is a.
All imperfections in the manufacture of screw compressor rotors should fall within and be accounted for by production tolerances. These are the wrong position of the bearing bores, eccentricity of the rotor shafts, bearing clearances and imperfections and rotor misalignment. Together, they account for the rotor shafts not being parallel. Let rotor movement in the y direction contain all displacements, which are presented in Fig. 2.27, and cause virtual rotation of the rotors around the , and axes, as shown in Fig. 2.27. Let
2.5 Identification of Rotor Position in Compressor Bearings 41
Fig. 2.26. Rotor shafts in the compressor housing and displacement in bearings
Fig. 2.27. Rotors with intersecting shafts and their coordinate systems
rotor movement in the x direction cause rotation around the , and axes, as shown in Fig. 2.28. The movement can cause the rotor shafts to intersect. However, the movement causes the shafts to become non-parallel and non-intersecting. These both change the nature of the rotor position so that the shafts can no longer be regarded as parallel. The following analytic-al
approach enables the rotor movement to be calculated and accounts for these changes.
Vectors and ,now represent the helicoid surfaces of the main and gate rotors on intersecting shafts. The shaft angle is the rotation about.
42 2 Screw Compressor Geometry
Fig. 2.28. Rotors with non-parallel and non-intersecting shafts and their coordinate systems
(2.15)
(2.16)
Since this rotation angle is usually very small, the relationship (2.16) can be assumed. Equation (2.15) can then be simplified for further analysis.
The rotationwill result in a displacement in the x direction and a displacement in the z direction, while there is no displacement in the y direction. The displacement vector becomes:
In the majority of practical cases, is small compared with and only displacement in the x direction need be considered. This means that rotation around the Y axis will, effectively, only change the rotor center distance. Displacement in the z direction may be significant for the dynamic behaviour of the rotors. Displacement in the z direction will be adjusted by the rotor relative rotation around the Z axis, which can be accompanied by significant angular acceleration. This may cause the rotors to lose contact at certain stages of the compressor cycle and thus create rattling, which may increase the compressor noise.
Since the rotation angle , caused by displacement within the tolerance limits, is very small, a two-dimensional analysis in the rotor end plane can be applied, as is done in the next section.
2.5 Identification of Rotor Position in Compressor Bearings 43
As shown in Fig. 2.28, where the rotors on the nonparallel and nonintersecting axes are presented, vectors r1= [x1,y1,z1] and r2, given by (2.10) now represent the helicoid surfaces of the main and gate rotors on the intersecting shafts. Σ is the rotation angle around the X axes given by (2.11).
(2.17)
(2.18)
Since angle Σ is very small, it can be expressed in simplified form as in (2.18).Further analysis is then facilitated by writing (2.17) as:
The rotation Σ will result in displacement in the y direction and dis-placement in the z direction, while there is no displacement in the x direction. The displacement vector can be written as:
[]
Although, in the majority of practical cases, displacement in the z direction is very small and therefore unimportant for consideration of rotor interference,it may play a role in the dynamic behaviour of the rotors. The displacement in the z direction will be fully compensated by regular rotation of the rotors around the Z axis. However, the angular acceleration involved in this process
may cause the rotors to lose contact at some stages of the compressor cycle.
Rotation about the X axis is effectively the same as if the main or gate rotor rotated relatively through angles or respectively and the rotor backlash will be reduced by . Such an approach substantially simplifies the analysis and allows the problem to be presented in two dimensions in the rotor end plane.
Although the rotor movements, described here are entirely three-dimension-al, their two-dimensional presentation in the rotor end plane section can be used for analysis.
Equation (2.2) serves to calculate both the coordinates of the rotor meshing points ,on the rotor helicoids and ,in the end plane from the given rotor coordinates points and . It may also be used to determine the contact line coordinates and paths of contact between the rotors. The sealing line of screw compressor rotors is somewhat similar to the rotor contact line. Since there is a clearance gap between rotors, sealing is effected at the points of the most proximate rotor position. A convenient practice to obtain the clearance gap between the rotors is to consider the gap as the shortest distance between the rotors in a section normal to the rotor helicoids. The end plane clearance gap can then be obtained from the normal clearance by appropriate transformation.
If is the normal clearance between the rotor helicoid surfaces, the cross product of the r derivatives, given in the left hand side of (2.5), which defines.
螺桿壓縮機
2.4審查最流行的轉子型線 37
圖.2.21.“N”轉子在5-6配置
圖.2.22. “N”轉子在5-7配置
38 2螺桿式壓縮機幾何
圖.2.23. “N”轉子6/7的配置
密封線，小局限于卷，漸開線轉子的接觸和正確的門轉子與轉子的機械剛性高扭矩分配。
所需的波瓣的數目，根據指定的壓縮機的工作而變化。3/5的安排是最適合于干燥的空氣壓縮，4/5和5/6的石油淹沒具有適度的壓力差的壓縮機6/ 7內置的體積比制冷的高壓和大應用程序。
雖然全面評估的轉子型線，需要的不僅僅是一個幾何評估，一些關鍵功能的“ N”配置文件可能它有三個最流行的螺絲比較容易理解在這里已經描述了轉子型線， （一） “西格瑪”配置文件 Bammert1979年， （二）， （三） SRM“ D” Astberg1982年的檔案，并在“ CYCLON ”個人資料霍夫和莫里斯，1984年。所有這些轉子的示于圖中. 2.20地方
可以看出，在“N”公司有一個更大的吞吐量和一個更硬的閘轉子可用于所有情況下，當其他特性，如吹孔區(qū)域，密閉體積和高壓力的密封線的長度是相同的。
此外，在低壓力密封線短，但，這是不太重要的因為相應的間隙可以保持很小。
可以控制的吹塑孔區(qū)域的尖端半徑調整的兩個主轉子和閘轉子，并通過使柵極的外徑等于或小于的節(jié)圓直徑。此外，密封線可以保持非常短的轉子型線，圈，其中心是通過構建距離的節(jié)圓。但是，吹孔區(qū)域的任何減少會增加轉子側上的平坦的密封線的長度。之間的折衷因此，這些趨勢要求，以獲得最佳的結果。
2.4 審查最流行的轉子型線 39
在閘轉子的扭矩分配通常是由轉子失穩(wěn)一個完整的周期過程中改變方向。該配置文件的生成過程本文中描述的，使得它能夠控制柵極上的扭矩轉子，從而避免這種影響。此外，完整的漸開線之間的聯系的“N”的轉子允許任何額外的觸點負載更容易被人體吸收比與任何其他類型的轉子。兩個轉子對示于圖.2.24什么被描述為“負”的閘轉子轉矩的第一展品而第二更常見的“積極的”扭矩。
圖.2.24.“N”負轉矩，左側和正面的扭矩，對
2.4.13鼓風機轉子型線
鼓風機的檔案中，示于圖. 2.25是對稱的。因此，只有一個季它需要被指定，以便定義整個轉子。它由兩個分部，在轉子上的葉尖端的一個非常小的圓和一個直線。擺線圈滑動產生，而直線生成漸開線。
40 2螺桿式壓縮機幾何
圖. 2.25. 吹風機配置文件
2.5 轉子位置的識別
在壓縮機軸承
轉子的軸向力和徑向力被傳遞到殼體由軸承 。滾動元件軸承通常選擇為中小型螺桿壓縮機，這些都必須精心挑選，以獲得滿意保守黨的設計。一般，兩個軸承中采用的每個的排出端為了吸收在轉子軸的徑向和軸向負荷分開。此外，轉子的中心線之間的距離是確定的部分軸承的尺寸和內部游隙。任何制造缺陷軸承箱，如位移或偏心，將改變轉子位置和從而影響壓縮機行為。
轉子的螺桿式壓縮機軸承的是，該系統示于圖中.2.26.轉子軸是平行的，它們的位置由軸和定義。
軸承被標記為1至4，和他們的間隙，以及在和方向上的軸承孔的制造公差，和分別在同一圖中。轉子中心的距離為和軸承之間的軸向跨度是一個。
螺桿壓縮機轉子的制造中的所有缺陷應該落在內，占生產公差。這些都是錯誤的軸承位置的軸承孔，轉子軸的偏心度，明確差和不完善之處，轉子不對?？傊?，他們占轉子軸不平行。讓在方向上的轉子運動包含所有的位移，這被示于圖.2.27，并導致虛擬周圍的和軸的轉子的旋轉，如圖所示.2.27.讓
2.5 鑒定壓縮機軸承轉子的位置 41
圖.2.26.在壓縮機殼體和位移在軸承的轉子軸
圖.2.27.轉子與相交軸和坐標系
轉子運動在的方向的原因左右旋轉的，和軸，如圖所示.2.28.的運動可能導致轉子軸相交。然而，運動使軸成為非平行和非相交。這些都改變了性質的轉子位置，所以軸可以不再被視為平行。以下分析方法使轉子的運動來計算，這些帳戶的變化。
矢量和，現在代表的螺旋面的表面在交叉軸的主轉子和閘轉子。軸角，是旋轉關于。
42 2螺桿式壓縮機幾何
圖.2.28.轉子與非平行的和非相交的軸和它們的坐標系統
(2.15)
(2.16)
由于該旋轉角的關系（2.16 ）通常非常小，可以假定。方程（2.15），然后，可以簡化用于進一步分析。
旋轉將導致位移在方向和方向的位移，而在沒有位移的方向發(fā)展。位移矢量變?yōu)椋?
在大多數實際情況下， 是小比和只在方向上的位移，需要加以考慮。這意味著，旋轉繞軸的，有效的，只有改變轉子的中心的距離。在方向上的位移可能是顯著的動態(tài)行為的轉子。在方向上的位移將調整由轉子繞軸的相對旋轉，它可以伴隨著顯著的角加速度。這可能會導致轉子失去在一定的接觸壓縮機循環(huán)階段霍霍，這可能會增加，從而創(chuàng)造壓縮機的噪聲。
由于旋轉角時，所造成的公差范圍內的位移限制，是非常小的，在轉子端面上可以是一個兩維的分析應用，如在下一節(jié)中完成。
2.5 鑒定壓縮機軸承轉子的位置 43
如圖中所示.2.28，其中的轉子對非平行和不相交軸，矢量[]和,（2.10）現在給出代表螺旋面的主轉子和閘轉子的表面上的交叉軸。是（2.11 ）給出的繞X軸的旋轉角度。
(2.17)
(2.18)
由于角是非常小的，它可以以簡化的形式表示，如在（2.18）。然后促進進一步的分析，以書面形式（ 2.17）：
旋轉將導致顯示投放在方向和位移在方向上，而沒有位移在的的方向展。的位移矢量可以被寫為：
[]
雖然，在大多數實際情況下，在方向上的位移是不重要的考慮轉子的干擾非常小，因此，它可能發(fā)揮的作用在轉子的動態(tài)行為。位移在方向上，將被充分通過定期的轉子的旋轉補償繞軸的。然而，參與在這個過程中的角加速度可能會導致轉子在壓縮機循環(huán)的某些階段，失去接觸。
繞軸的旋轉實際上是一樣的，如果主要或門轉子相對旋轉通過的角度 或再分別與轉子的齒隙將減少 。這樣的做法大大簡化了分析，并允許將呈現的問題在轉子端面上的兩個維度。
雖然轉子的運動，這里描述的是完全的三維人，其二維地列在轉子端部的平面部，可以用于分析。
等式（2.2），用于計算轉子網格的坐標是點，在轉子的螺旋和，在的端面給定的轉子的坐標點和。它也可以被用來確定轉子之間的接觸的接觸線的坐標和路徑。該密封螺桿壓縮機轉子的線是有點類似的轉子接觸線。因為有一個之間的間隙的轉子，封裝是在點的最接近的轉子位置。獲得一個方便的做法在轉子之間的間隙是要考慮的差距，在最短的之間的距離中的轉子的截面垂直于轉子螺旋?！岸嗣骈g隙，然后可以正常清關適當的改造。
如果是轉子的螺旋表面的正常間隙，交叉產品的的衍生物,（2.5）的左手側，它定義中給出
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