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英文原文
Screw Compressors
The direction normal to the helicoids, can be used to calculate the coordinates of the rotor helicoids and from x and y to which the clearance is added as:
， ， （2.19）
where the denominator D is given as：
（2.20）
and serve to calculate new rotor end plane coordinates, x0n and y0n,with the clearances obtained for angles θ = /p and τ respectively. These and now serve to calculate the transverse clearance δ0 as the difference between them, as well as the original rotor coordinates and .
If by any means, the rotors change their relative position, the clearance distribution at one end of the rotors may be reduced to zero on the flat side of the rotor lobes. In such a case, rotor contact will be prohibitively long on the flat side of the profile, where the dominant relative rotor motion is sliding, as shown in Fig. 2.29. This indicates that rotor seizure will almost certainly occur in that region if the rotors come into contact with each other.
Fig. 2.29. Clearance distribution between the rotors: at suction, mid rotors, and discharge with possible rotor contact at the discharge
Fig. 2.30. Variable clearance distribution applied to the rotors
It follows that the clearance distribution should be non-uniform to avoid hard rotor contact in rotor areas where sliding motion between the rotors is dominant.
In Fig. 2.30, a reduced clearance of 65 μm is presented, which is now applied in rotor regions close to the rotor pitch circles, while in other regions it is kept at 85 μm, as was done by Edstroem, 1992. As can be seen in Fig. 2.31, the situation regarding rotor contact is now quite different. This is maintained along the rotor contact belt close to the rotor pitch circles and fully avoided at other locations. It follows that if contact occurred, it would be of a rolling character rather than a combination of rolling and sliding or even pure sliding. Such contact will not generate excessive heat and could therefore be maintained for a longer period without damaging the rotors until contact ceases or the compressor is stopped.
2.6 Tools for Rotor Manufacture
This section describes the generation of formed tools for screw compressor hobbing, milling and grinding based on the envelope gearing procedure.
2.6.1 Hobbing Tools
A screw compressor rotor and its formed hobbing tool are equivalent to a pair of meshing crossed helical gears with nonparallel and nonintersecting axes. Their general meshing condition is given in Appendix A. Apart from the gashes forming the cutter faces, the hob is simply a helical gear in which.
Fig. 2.31. Clearance distribution between the rotors: at suction, mid of rotor and discharge with a possible rotor contact at the discharge
Each referred to as a thread, Colburne, 1987. Owing to their axes not being parallel, there is only point contact between them whereas there is line contact between the screw machine rotors. The need to satisfy the meshing equation given in Appendix A, leads to the rotor – hob meshing requirement for the given rotor transverse coordinate points and and their first derivative.The hob transverse coordinate points and can then be calculated. These are sufficient to obtain the coordinate The axial coordinate , calculated directly, and are hob axial plane coordinates which define the hob geometry.
The transverse coordinates of the screw machine rotors, described in the previous section, are used as an example here to produce hob coordinates. he rotor unit leadsare 48.754mm for the main and ?58.504mm for the ate rotor. Single lobe hobs are generated for unit leads :6.291mm for the main rotor and ?6.291mm for the gate rotor. The corresponding hob helix angles ψ are 85? and 95?. The same rotor-to-hob centre distance C = 110mm and the shaft angle Σ = 50? are given for both rotors. Figure 2.32 contains a view to the hob.
Reverse calculation of the hob – screw rotor transformation, also given in Appendix A permits the determination of the transverse rotor profile coordinates which will be obtained as a result of the manufacturing process. These ay be compared with those originally specified to determine the effect of
Fig. 2.32. Rotor manufacturing: hobbing tool left, right milling tool
manufacturing errors such as imperfect tool setting or tool and rotor deformation upon the final rotor profile.
For the purpose of reverse transformation, the hob longitudinal plane coordinatesand andshould be given. The axial coordinate is used to calculate , which is then used to calculate the hob transverse coordinates:
， （2.21）
These are then used as the given coordinates to produce a meshing criterionand the transverse plane coordinates of the “manufactured” rotors.
A comparison between the original rotors and the manufactured rotors is given in Fig. 2.33 with the difference between them scaled 100 times. Two types of error are considered. The left gate rotor, is produced with 30um offset in the centre distance between the rotor and the tool, and the main rotor with
Fig. 2.33. Manufacturing imperfections
0.2? offset in the tool shaft angle Σ. Details of this particular meshing method are given by Stosic 1998.
2.6.2 Milling and Grinding Tools
Formed milling and grinding tools may also be generated by placing in the general meshing equation, given in Appendix A, and then following the procedure of this section. The resulting meshing condition now reads as:
(2.22)
However in this case, when one expects to obtain screw rotor coordinates from the tool coordinates, the singularity imposed does not permit the calculation of the tool transverse plane coordinates. The main meshing condition cannot therefore be applied. For this purpose another condition is derived for the reverse milling tool to rotor transformation from which the meshing angle τ is calculated:
(2.23)
Once obtained, τ will serve to calculate the rotor coordinates after the “manufacturing” process. The obtained rotor coordinates will contain all manufacturing imperfections, like mismatch of the rotor – tool centre distance, error in the rotor – tool shaft angle, axial shift of the tool or tool deformation during the process as they are input to the calculation process. A full account of this useful procedure is given by Stosic 1998.
2.6.3 Quantification of Manufacturing Imperfections
The rotor – tool transformation is used here for milling tool profile generation. The reverse procedure is used to calculate the “manufactured” rotors. The rack generated 5-6 128mm rotors described by Stosic, 1997a are used as given profiles: x(t) and y(t). Then a tool – rotor transformation is used to quantify the influence of manufacturing imperfections upon the quality of the produced rotor profile. Both, linear and angular offset were considered.
Figure 2.33 presents the rotors, the main manufactured with the shaft angle offset 0.5? and the gate with the centre distance offset 40 μm from that of the original rotors given by the dashed line on the left. On the right, the rotors are manufactured with imperfections, the main with a tool axial offset of 40 μm and the gate with a certain tool body deformation which resulted in 0.5? offset of the relative motion angle θ. The original rotors are given by the dashed line.
3Calculation of Screw Compressor Performance
Screw compressor performance is governed by the interactive effects of thermodynamic and fluid flow processes and the machine geometry and thus can be calculated reliably only by their simultaneous consideration. This may be chieved by mathematical modelling in one or more dimensions. For most applications, a one dimensional model is sufficient and this is described in full. 3-D modelling is more complex and is presented here only in outline. A more detailed presentation of this will be made in a separate publication.
3.1 One Dimensional Mathematical Model
The algorithm used to describe the thermodynamic and fluid flow processes in a screw compressor is based on a mathematical model. This defines the instantaneous volume of the working chamber and its change with rotational angle or time, to which the conservation equations of energy and mass continuity are applied, together with a set of algebraic relationships used to define various phenomena related to the suction, compression and discharge of the working fluid. These form a set of simultaneous non-linear differential equations which cannot be solved in closed form.
The solution of the equation set is performed numerically by means of the Runge-Kutta 4th order method, with appropriate initial and boundary conditions.
The model accounts for a number of “real-life” effects, which may significantly influence the performance of a real compressor. These make it suitable for a wide range of applications and include the following:
– The working fluid compressed can be any gas or liquid-gas mixture for which an equation of state and internal energy-enthalpy relation is known, i.e. any ideal or real gas or liquid-gas mixture of known properties.
– The model accounts for heat transfer between the gas and the compressor rotors or its casing in a form, which though approximate, reproduces the overall effect to a good first order level of accuracy.
– The model accounts for leakage of the working medium through the clearances between the two rotors and between the rotors and the stationary parts of the compressor.
– The process equations and the subroutines for their solution are independent of those which define the compressor geometry. Hence, the model can be readily adapted to estimate the performance of any geometry or type of positive displacement machine.
– The effects of liquid injection, including that of oil, water, or refrigerant can be accounted for during the suction, compression and discharge stages.
– A set of subroutines to estimate the thermodynamic properties and changes of state of the working fluid during the entire compressor cycle of operations completes the equation set and thereby enables it to be solved.
Certain assumptions had to be introduced to ensure efficient computation.These do not impose any limitations on the model nor cause significant departures from the real processes and are as follows:
– The fluid flow in the model is assumed to be quasi one-dimensional.
– Kinetic energy changes of the working fluid within the working chamber are negligible compared to internal energy changes.
– Gas or gas-liquid inflow to and outflow from the compressor ports is assumed to be isentropic.
– Leakage flow of the fluid through the clearances is assumed to be adiabatic.
3.1.1 Conservation Equations
For Control Volume and Auxiliary Relationships
The working chamber of a screw machine is the space within it that contains the working fluid. This is a typical example of an open thermodynamic system in which the mass flow varies with time. This, as well as the suction and discharge plenums, can be defined by a control volume for which the differential equations of the conservation laws for energy and mass are written. These are derived in Appendix B, using Reynolds Transport Theorem.
A feature of the model is the use of the non-steady flow energy equation to compute the thermodynamic and flow processes in a screw machine in terms of rotational angle or time and how these are affected by rotor profile modifications. Internal energy, rather than enthalpy, is then the derived variable. This is computationally more convenient than using enthalpy as the derived
Variable since, even in the case of real fluids, it may be derived, without reference to pressure. Computation is then carried out through a series of iterative cycles until the solution converges. Pressure, which is the desired output variable, can then be derived directly from it, together with the remaining required thermodynamic properties.
The following forms of the conservation equations have been employed in the model:
中文翻譯
螺桿式壓縮機
幾何的法線方向的螺旋，可以用來計算的坐標(biāo)轉(zhuǎn)子螺旋和的從x和y的間隙加入如：
， ， (2.19)
其中分母D被給定為：
(2.20)
，服務(wù)來計算新的轉(zhuǎn)子端的平面的坐標(biāo)， 和，得到的間隙角θ =鋅/ p和τ 。這些，現(xiàn)在的差額計算的橫向間隙δ0在它們之間，以及原來的轉(zhuǎn)子坐標(biāo)和 。
如果以任何方式，轉(zhuǎn)子的改變它們的相對位置，該間隙的平側(cè)面上分布在轉(zhuǎn)子的一端，也可以減少到零的轉(zhuǎn)子葉片。在這樣的情況下，轉(zhuǎn)子的接觸將是令人望而卻步長側(cè)扁的檔案中，其中占主導(dǎo)地位的相對滑動轉(zhuǎn)子運動，如示于圖。 2.29。這表明，轉(zhuǎn)子扣押幾乎肯定會
如果轉(zhuǎn)子進(jìn)入彼此接觸，發(fā)生在該區(qū)域。
圖.29。：吸力，中間轉(zhuǎn)子和轉(zhuǎn)子之間的間隙分布可能轉(zhuǎn)子接觸放電在放電
圖 2.30?？勺冮g隙分布應(yīng)用到轉(zhuǎn)子
如下的間隙分布應(yīng)該非均勻，以避免在轉(zhuǎn)子轉(zhuǎn)子之間的滑動運動的地方是硬轉(zhuǎn)子的接觸占主導(dǎo)地位。
另外，在圖2.30 ，清除率降低65微米，這是現(xiàn)在應(yīng)用在轉(zhuǎn)子靠近轉(zhuǎn)子節(jié)圓的區(qū)域，而在其他區(qū)域是保持在85微米所做的那樣， 1992年由Edstroem 。正如在圖中可以看出的。 2.31，現(xiàn)在的情況，轉(zhuǎn)子的接觸是完全不同的。這是保持靠近轉(zhuǎn)子的節(jié)圓沿轉(zhuǎn)子的接觸帶，并完全避免在其他位置。因此，如果發(fā)生接觸，這將是一個滾動字符，而不是相結(jié)合的滾動和滑動，甚至是純滑動。這樣的接觸不會產(chǎn)生過多的熱量，因此可以保持一段較長時間，而不會損壞轉(zhuǎn)子直到接觸終止或使壓縮機停止。
2.6 為轉(zhuǎn)子制造的工具
本節(jié)描述了一代形成的工具螺桿壓縮機滾齒，銑床和磨床的基礎(chǔ)上信封資產(chǎn)負(fù)債程序。
2.6.1 滾齒機工具
螺桿壓縮機轉(zhuǎn)子和其形成的滾齒機工具相當(dāng)于一個對相互嚙合的交錯軸斜齒輪與非平行不相交軸。他們的嚙合條件一般除了見附錄A。張裂縫形成刀具的面孔時，僅僅是一個螺旋齒輪的滾刀。
圖2.31。轉(zhuǎn)子之間的間隙分布：在抽吸，中期的轉(zhuǎn)子和可能轉(zhuǎn)子接觸放電在放電
每個齒被稱為作為一個線程， Colburne ，1987 。由于其自身的軸線不平行，它們之間的唯一的點接觸，而有行螺桿機轉(zhuǎn)子之間的接觸。需要滿足的嚙合在附錄A中給出的公式，導(dǎo)致轉(zhuǎn)子A “滾刀嚙合要求對于給定的轉(zhuǎn)子橫向坐標(biāo)點X01和Y01和他們的第一次衍生。滾刀橫向坐標(biāo)點X02和Y02計算出來的。這是足夠的，以獲得坐標(biāo)軸向坐標(biāo)z2的，直接計算，和R2是滾刀軸向平面坐標(biāo)它定義滾刀的幾何形狀。
軸向坐標(biāo)z2的，直接計算，和R2是滾刀軸向平面坐標(biāo)它定義滾刀的幾何形狀。螺桿機轉(zhuǎn)子的橫向坐標(biāo)，描述在前面的部分，被用來作為一個例子在這里產(chǎn)生滾刀坐標(biāo)。轉(zhuǎn)子單元導(dǎo)致p1的有48.754毫米的主和- 58.504毫米的門轉(zhuǎn)子。單葉爐產(chǎn)生單位領(lǐng)導(dǎo)P2 ： 6.291毫米的主轉(zhuǎn)子和閘轉(zhuǎn)子- 6.291毫米為。相應(yīng)的滾刀螺旋角ψ分別為85 ?和95 ? 。相同的轉(zhuǎn)子滾刀中心距離C = 110毫米和軸角Σ = 50 ?給出了兩個轉(zhuǎn)子。圖2.32中包含一個查看的爐灶。
反向計算的爐灶 - 螺桿轉(zhuǎn)子的改造，也給出了附錄A，允許確定轉(zhuǎn)子型線的橫向坐標(biāo)這將在制造過程中得到的結(jié)果。這些可能與最初指定的進(jìn)行比較，以確定影響
圖。 2.32。轉(zhuǎn)子制造業(yè)：滾齒刀具左，右銑刀
制造的錯誤，如完美的工具設(shè)置或工具，轉(zhuǎn)子變形后，最終的轉(zhuǎn)子型線。
如果在反向變換的目的，滾刀的縱向平面坐標(biāo)R2和Z2和應(yīng)給予。軸向坐標(biāo)z2的使用計算τ = z2/p2 ，然后將其用于計算滾刀橫向的坐標(biāo)：
， (2.21)
然后用這些作為給定的坐標(biāo)，以產(chǎn)生一個嚙合判據(jù)的橫向平面上的坐標(biāo)的“人造”轉(zhuǎn)子。
原來的轉(zhuǎn)子之間的比較和所制造的轉(zhuǎn)子給出圖。 2.33與它們之間的區(qū)別縮放100倍。二被認(rèn)為是類型的錯誤。左邊的門轉(zhuǎn)子，30微米抵消在該轉(zhuǎn)子與該工具，和主旋翼之間的中心距離0.2 ?偏移刀具軸角Σ 。這個特殊的網(wǎng)格劃分方法的詳情給出由Stosic 1998。
圖。 2.33。制造缺陷
2.6.2 銑削和磨削工具
形成銑削和磨削工具也可以通過放置p2= 0產(chǎn)生嚙合方程，一般在附錄A中，然后按照本節(jié)的程序的?，F(xiàn)在的嚙合條件內(nèi)容：
(2.22)
然而，在這種情況下，當(dāng)一個人希望獲得螺桿轉(zhuǎn)子坐標(biāo)從工具坐標(biāo)，所施加的奇異性，不允許計算該工具的橫向平面坐標(biāo)。主要的嚙合條件不能因此其應(yīng)用。為此目的，另一個條件推導(dǎo)了扭轉(zhuǎn)銑刀到轉(zhuǎn)子的變換從該嚙合角τ計算方法是:
(2.23)
一旦獲得，τ后，將用來計算轉(zhuǎn)子坐標(biāo)“制造”的過程。得到的轉(zhuǎn)子的坐標(biāo)將包含所有制造不完善的地方，如不匹配的轉(zhuǎn)子 - 刀具中心的距離，在轉(zhuǎn)子中的誤差 - 工具軸角度，軸向移位的工具或工具變形在這個過程中，因為它們是輸入到計算處理。一個完整的帳戶這個有用的程序是Stosic于 1998年定義的。
2.6.3 量化的制造缺陷
轉(zhuǎn)子 - 這里使用的工具轉(zhuǎn)換生成銑削刀具輪廓。相反的步驟是用來計算“制成品”轉(zhuǎn)子。機架產(chǎn)生的Stosic的5-6 128毫米轉(zhuǎn)子的描述， 1997年a使用給定的概況：X （t）和Y（T） 。然后一個工具 - 轉(zhuǎn)子的變換是用來量化后，所生成的質(zhì)量的影響的制造缺陷轉(zhuǎn)子型線。這兩種，線性和角度偏移量進(jìn)行了考慮。
圖2.33轉(zhuǎn)子，主要制造與軸角度偏移0.5 ?門與中心的距離偏移量40微米原來的轉(zhuǎn)子由在左邊的虛線給出。在屏幕的右側(cè)，轉(zhuǎn)子制造的缺陷，主要與刀具軸向偏移為40μm，具有一定的工具主體變形而導(dǎo)致的柵極0.5 ?偏移量的相對運動的角度θ 。原來的轉(zhuǎn)子由下式給出的虛線。
3螺桿壓縮機性能的計算
螺桿式壓縮機的性能是受熱力學(xué)的相互影響和流體流動過程和機器的幾何形狀，從而可以只能由他們同時考慮可靠地計量。這可能是實現(xiàn)在一個或多個維度的數(shù)學(xué)建模。對于大多數(shù)應(yīng)用，一個三維模型是足夠的，這充分說明。3-D模型比較復(fù)雜，而且只在這里介紹的輪廓。一個更詳細(xì)介紹了這個將在一個單獨的出版物。
3.1 一維數(shù)學(xué)模型
來描述的熱力學(xué)和流體的流動過程中所使用的算法的螺桿壓縮機的基礎(chǔ)上的數(shù)學(xué)模型。這個定義的瞬時工作腔的體積，其變化與旋轉(zhuǎn)角或時間，能量和質(zhì)量的連續(xù)性的守恒方程施加，連同一組用于定義各種代數(shù)關(guān)系的吸入，壓縮和排出的工作有關(guān)的現(xiàn)象流體。這些形成了一套同時非線性微分方程不能得到解決封閉形式。
方程組的溶液通過數(shù)值進(jìn)行龍格庫塔四階的方法，用適當(dāng)?shù)某跏紬l件和邊界條件。
模型考慮了一些“現(xiàn)實生活”的影響，這可能顯著一個真正的壓縮機的性能產(chǎn)生影響。這使得它適合一個廣泛的應(yīng)用范圍，包括以下內(nèi)容：
-工作流體壓縮可以是任何氣體或液 - 氣混合物的方程的狀態(tài)和內(nèi)能，焓關(guān)系是已知的，即任何理想的或真正的氣體或液 - 氣混合物的已知屬性。
- 的氣體之間的熱傳遞和壓縮機的模型占轉(zhuǎn)子或外殼的一種形式，雖然近似，再現(xiàn)一個良好的第一順序的精度水平的整體。
-工作介質(zhì)通過間隙泄漏的模型占兩個轉(zhuǎn)子之間的和之間的轉(zhuǎn)子和固定的壓縮機部分。
- 他們的解決方案的過程方程和子程序是獨立的的那些定義壓縮機的幾何形狀。因此，該模型可以估計很容易適應(yīng)任何幾何形狀的性能或類型容積式機器。
- 液體注入的影響，包括油，水，制冷劑或可以占的吸入，壓縮及排出階段期間。
- A組的子程序估計的熱力學(xué)性質(zhì)和變化的工作流體的狀態(tài)在整個壓縮機的操作循環(huán)期間完成方程組，從而使其能夠解決。
若干假設(shè)被引入，以確保有效的計算。這些不施加任何限制的模式，也沒有造成顯著偏離真實進(jìn)程如下：
-在模型中的流體流被假定為準(zhǔn)一維。
-內(nèi)的工作腔的工作流體的動能變化是相比微不足道內(nèi)部能量的變化。
-假定氣體或氣 - 液的流入和流出，從壓縮機端口等熵。
-流體通過間隙泄漏流被假定為絕熱。
3.1.1 守恒方程
控制音量和輔助的關(guān)系
工作腔的螺桿機內(nèi)的空間，它包含工作流體。這是一個開放的熱力學(xué)系統(tǒng)的一個典型例子在其中的質(zhì)量流量隨時間變化。這一點，以及吸入管和排出增壓室，可以定義由一個控制體積的差動方程，能量和質(zhì)量守恒定律被寫入。這些都是在附錄B中派生，使用雷諾運輸定理。
一個模型的特征是使用的非定常流動能量方程計算的螺桿機方面的熱力學(xué)和流動過程旋轉(zhuǎn)角度或時間，以及如何，這些受影響的的轉(zhuǎn)子配置文件修改。內(nèi)部的能量，而不是焓，然后導(dǎo)出變量。這是計算更方便，比焓派生變量，因為，即使在實際流體的情況下，它可以導(dǎo)出，而不參考的壓力。計算，然后通過一系列的迭代進(jìn)行周期，直到該溶液收斂。的壓力，這是所需的輸出變量，然后，可以直接從它派生，連同其余必需的熱力學(xué)性質(zhì)。
已使用了以下形式的守恒方程模型：
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