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英文原文
2 Screw Compressor Geometry
x01 = (r1i ? r3) cos ψ2 + r3 cos t
y01 = ?(r1i ? r3) sinψ2 + r3 sin t (2.12)
Profile portion A1D1 is a circle of radius r2 on the main rotor, 0 < t < θ2.
x01 = r1e ? r2 cos t
y01 = r2 sin t (2.13)
Segment C1D1 emerges as a trochoid on the main rotor generated by the circle of radius r4 on the gate rotor, ?θ4 ? τ1 < t < ?π ? τ1. The trochoid is obtained from the gate rotor coordinates through the same meshing procedure.
The circle C2D2 is:
x02 = (r2e ? r4) cos τ1 + r4 cos t
y02 = (r2e ? r4) sinτ1 + r4 sin t (2.14)
Now, when all the segments of the main rotor are known, they are used as source curves. The gate rotor lobe can now be generated completely by the meshing procedure described in the previous section. Although essentially simple, the Demonstrator profile contains all the features which characterize modern screw rotor profiles. The pressure angles on both, the flat and the round profile lobes are not zero. This is essential for successful manufacturing. The profile is generated by the curves and not by points. This further enhances its manufacturability. By changing its parameters, C, r, r0, r2, r3 and r4, a variety of profiles can be generated, some with positive gate rotor torque, some suitable for low pressure ratios, and others for high pressure ratio compression. The profile is fully computerized and can be used for demonstration, teaching and development purposes.
2.4.2 SKBK Profile
Amosov’s 1977 SKBK profile is the first modern Russian profile to be published in the open literature and it is shown in Fig. 2.6. The profile has the same layout and sequence of segments as the Demonstrator profile apart frothe fact that the circles r2 and r3 the substituted by cycloids and the segments AB and AF are generated by point generation. This can be readily achieved if r2 and r3 in the Demonstrator profile tend to zero. Similarly to the Demonstrator profile, SKBK profile has an eccentric circle on the round lobe of the main rotor, which gives a pressure angle far different from zero in the pitch circle area. This further ensures both its ease of manufacture and the gate rotor torque stability. This characteristic of the SKBK profile was published at least five years prior the SRM “D” rotor patents which claimed the same feature. However, since the flat lobe sides on the main and gate rotors are generated by points E and A on the gate and main rotor respectively and since E is positioned on the gate rotor pitch circle, the pressure angle at the pitch circle on the flat side is zero. This does not allow manufacturing of this profile by milling or grinding unless the profile is modified.
Fig. 2.6. SKBK Profile
Fig. 2.7. Fu Sheng Profile
2.4.3 Fu Sheng Profile
The Fu Sheng profile, as shown in Fig. 2.7, is practically the same as the Demonstrator, but has one distinguishing feature. The segment AB is an ellipse.
2.4.4 “Hyper” Profile
The “Hyper” profile is virtually the same as the Fu Sheng profile, apart from the segment AB, which is a hyperbola on the main rotor instead of the ellipse of the original Fu Sheng profile. However, despite such a small difference, the “Hyper” is a better profile giving larger screw compressor displacement, a shorter sealing line and stronger gate rotor lobes. The Hitachi profile has the same layout as the “Hyper” profile.
2.4.5 “Sigma” Profile
The “Sigma” is a relatively old profile. It was developed in the late nineteen seventies as a response to SRM awarding an exclusive licence to Aerzener in Germany. Other German manufacturers, such as GHH and Kaeser, therefore, needed to develop their own profiles. The “Sigma”, shown in Fig. 2.8 is a beautiful and efficient profile. However, new and better profiles are now available. The flat side of the “Sigma” lobe is the same as that of the Demonstrator profile, but the round side of the profile is generated from the flat side by an envelope of circles, which touch both the flat and the round sides, the radii of which are given in advance. This is an acceptable method of profile generation if nothing more general is known, but seriously limits the generation procedure. There are several modifications of the “Sigma” profile. One of these, which is presented here, comprises a straight line BC2 on the round side of the gate rotor. This modification significantly improves the profile, which is less limited than the original.
Fig. 2.8. Sigma Profile
2.4.6 “Cyclon” Profile
The “Cyclon” shown in Fig. 2.9 is a profile developed by Compair. The layout and sequence of profile segments are not so different from the Demonstrator, but the “Cyclon” introduces parabolae instead of circles in segments BC, GH and JH. One of the interesting features of the “Cyclon” profile is the “negative” torque on the gate rotor which results in rotor contact on the flat side of the rotors.
Fig. 2.9. Cyclon Profile
2.4.7 Symmetric Profile
The Symmetric profile, shown in Fig. 2.10 is very simple and consists of three circles on the main rotor with centres positioned either on the rotor centre or on the pitch circle of the main rotor. Since the circles are on the main rotor with centres either at the rotor centre or on the pitch circle, they only generate circles on the gate rotor with centres either in the rotor centre, or on the rotor pitch circle. Is is therefore not surprising that this was the first screw rotor profile ever generated.
Segment D1E1 is a circle of radius r1w ? r0 with its centre on the rotor axis, while segment E1F1 is a circle of radius r0. Segment F1A1 is on a circle of radius r. Both, the last two segments have their centres on the rotor pitch circle. Further segments are symmetrically similar to the given ones.
Fig. 2.10. Symmetric Circular Profile
The Symmetric profile has a huge blow-hole area which excludes it from any compressor application where a high or even moderate pressure ratio is involved. However, the symmetric profile performs surprisingly well in low pressure compressor applications. More details about the circular profile can be found in Margolis, 1978.
2.4.8 SRM “A” Profile
The SRM “A” profile is shown in Fig. 2.11. It retains all the favourable features of the symmetric profile like its simplicity while avoiding its main disadvantage, namely, the large blow-hole area. The main goal of reducing the blow hole area was achieved by allowing the tip points of the main and gate rotors to generate their counterparts, trochoids on the gate and main rotor respectively. The “A” profile consists mainly of circles on the gate rotor and one line which passes through the gate rotor axis.
The set of primary curves consists of: D2C2, which is a circle on the gate rotor with the centre on the gate pitch circle, and C2B2, which is a circle on the gate rotor, the centre of which lies outside the pitch circle region. This was a new feature which imposed some problems in the generation of its main rotor counterpart, because the mathematics used for profile generation at that time was insufficient for general gearing. This eccentricity ensured that the pressure angles on the rotor pitches differ from zero, resulting in its ease of manufacture. Segment BA is a circle on the gate rotor with its centre on the pitch circle. The flat lobe sides on the main and gate rotors were generated as epi/hypocycloids by points G on the gate and H on the main rotor respectively. GF2 is a radial line at the gate rotor. This brought the same benefits to manufacturing as the previously mentioned circle eccentricity on the opposite lobe side. F2E2 is a circle with the centre on the gate pitch and finally, E2D2 is a circle with the centre on the gate axis.
More details on the “A” profile are published by Amosov et al., 1977 and by Rinder, 1979.
The “A” profile is a good example of how a good and simple idea evolved into a complicated result. Thus the “A” profile was continuously subjected to changes which resulted in the “C” profile. This was mainly generated to improve the profile manufacturability. Finally, a completely new profile, the “D” profile was generated to introduce a new development in profile gearing and to increase the gate rotor torque.
Despite the complexity of its final form the “A” profile emerged to be the most popular screw compressor profile, especially after its patent expired.
2.4.9 SRM “D” Profile
The SRM “D” profile, shown in Fig. 2.12, is generated exclusively by circles with the centres off the rotor pitch circles. Similar to the Demonstrator, C2D2 is an eccentric circle of radius r3 on the gate rotor. B1C1 is an eccentric circle of radius r1, which, together with the small circular arc A1J1 of radius r2, is positioned on the main rotor. G2H2 is a small circular arc on the gate rotor and E2F2 is a circular arc on the gate rotor. F2G2 is a relatively large circular arc on the gate rotor which produces a corresponding curve of the smallest possible curvature on the main rotor.
Both circular arc, B2C2 and F2G2 ensure a large radius of curvature in the pitch circle area. This avoids high stresses in the rotor contact region.
Fig. 2.11. SRM “A” Profile
Fig. 2.12. SRM “D” Profile
2.4.10 SRM “G” Profile
The “G” profile was introduced by SRM in the late nineteen nineties as a replacement for the “D” rotor and is shown in Fig. 2.13. Compared with the “D” rotor, the “G” rotor has the unique feature of two additional circles in the addendum area on both lobes of the main rotor, close to the pitch circle.
This feature improves the rotor contact and, additionally, generates shorter sealing lines. This can be seen in Fig. 2.13, where a rotor featuring “G” profile characteristics only on its flat side through segment H1I1 is presented.
Fig. 2.13. SRM “G” Profile
2.4.11 City “N” Rack Generated Rotor Profile
“N” rotors are calculated by a rack generation procedure. This distinguishes them from any others. In this case, the large blow-hole area, which is a characteristic of rack generated rotors, is overcome by generating the high pressure side of the rack by means of a rotor conjugate procedure. This undercuts the single appropriate curve on the rack. Such a rack is then used for profiling both the main and the gate rotors. The method and its extensions were used by the authors to create a number of different rotor profiles, some of them used by Stosic et al., 1986, and Hanjalic and Stosic, 1994. One of the applications of the rack generation procedure is described in Stosic, 1996.
The following is a brief description of a rack generated “N” rotor profile, typical of rotor profiles designed for the efficient compression of air, common refrigerants and a number of process gases. The rotors are generated by the combined rack-rotor generation procedure whose features are such that it may be readily modified further to optimize performance for any specific application.