The present invention relates to plural screw positive displacement machines comprising a housing having at least two intersecting bores the axes of which are coplanar in pairs, and usually parallel, and male and female rotors mounted for rotation about their axes which coincide one with each of the housing bore axes, the rotors each having helical lands which mesh with helical grooves between the lands of at least one other rotor, the or each male rotor having as seen in cross section a set of lobes corresponding to the lands and projecting outwardly from its pitch circle, the or each female rotor having as seen in cross section a set of depressions extending inwardly of its pitch circle and corresponding to the grooves of the female rotor(s), the number of lands and grooves of the male rotor(s) being different to the number of lands and grooves of the female rotor(s) .

Examples of such machines, which may be used as compressors or expanders are disclosed in GB 1,197,432, GB 1,503,488 and GB 2,092,676.

A plural screw positive displacement machine according to the invention is characterised in that, as seen in cross section, the profiles of at least those parts of the lobes projecting outwardly of the pitch circle of the male rotor(s) and the profiles of at least the depressions extending inwardly of the pitch circle of the female rotor(s) are generated by the same rack formation, the latter being curved in one direction about the axis of the male rotor (s) and in the opposite direction about the axis of the female rotor(s) , the portion of the rack which

generates the higher pressure flanks of the rotors being generated by rotor conjugate action between the rotors.

Advantageously, a portion of the rack, preferably that portion which forms the higher pressure flanks of the rotor lobes, has the shape of a cycloid. Alternatively, this portion may be shaped as a generalized parabola, for example of the form: ax+by ^<3 =l.

Normally, the bottoms of the grooves of the male rotor(s) lie inwardly of the pitch circle as "dedendum" portion and the tips of the lands of the female rotor (s) extend outwardly of its pitch circle as "addendum" portions. Preferably, these dedendum and addendum portions are also generated by the rack formation.

The invention will now be described by way of example with reference to the drawings, in which:

Figure 1 is a diagrammatic cross section of a twin screw machine;

Figure 2 shows one unit of a rack for generating the profiles of the rotors shown in Figure 1 ; Figure 3 shows the relationship of the rack formation of Figure 2 to the rotors shown in Figure 1, and Figure 4 shows the outlines of the rotors shown in Figure 3 superimposed on a prior art rotor pair by way of comparison.

The main or male rotor 1 and gate or female rotor 2 shown in Figure 1 roll on their pitch circles, P _x , P ₂ about their centres 0 ₁ and 0 ₂ through respective angles ψ and τ=Z ₁ /Z ₂ ψ=ψ/i

The pitch circles P have radii proportional to the number of lands and grooves on the respective rotors.

If an arc is defined on either main or gate rotor as an arbitrary function of an angular parameter φ and denoted by subscript d:

*,,=*/Φ ⁾ (1)

y _d ⁼ y, ,(Φ) (2)

the corresponding arc on the other rotor is a function of both φ and ψ : -

x =x(φ,ψ) = -αcos— +x ,cos£ψ +y ,sin£ψ

ψ is the rotation angle of the main rotor for which the primary and secondary arcs have a contact point. This angle meets the conjugate condition described by Sakun in: "Vintovie kompressori" , Mashgiz Leningrad 1960

δφ δψ δψ δφ ^{5)

which is the differential equation of an envelope of all "d" curves. Its expanded form is:

This can be expressed as a quadratic equation of sin ψ. Although it can be solved analytically, its numerical solution is recommended due to its mixed roots. Once determined, ψ is inserted in (3) and (4) to obtain conjugate curves on the opposite rotor. This procedure requires the definition of only one given arc. The other arc is always found by a general procedure.

These equations are valid even if their coordinate system is defined independently of the rotors. Thus, it is possible to specify all "d" curves without reference to the rotors . Such an arrangement enables some curves to be expressed in a more simple mathematical form and, in addition, can simplify the curve generating procedure.

A special coordinate system of this type is a rack (rotor of infinite radius) coordinate system, indicated at R in Fig. 2. An arc on the rack is then defined as an arbitrary function of a parameter φ:

=*ΛΦ) (7;

⁽⁸⁾

Secondary arcs on the rotors are derived from this as a function of both, φ and ψ.

y +x sinψ+(y -r ψ)cosψ (10)

ψ represents a rotation angle of the rotor where a given arc is projected, defining a contact point. This angle satisfies the condition (5) which is:

-^■(rj -y _d ) -(r _w -χ _d ) =0 (ii! dx _J

The explicit solution ψ is then inserted into (9) and (10) to find conjugate arcs on rotors. Fig. 3 shows the rack and rotors generated by the rack.

Wherever curves are given, their convenient form may be:

ax§ + Jby§ » 1, (12)

which is a "general circle" curve. For p = q = 2 and a = b = 1/r it is a circle, unequal a and b will give ellipses, a and b of opposite sign, hyperbolae, p = 1 and q = 2 will give parabolae.

In addition to the convenience of defining all given curves with one coordinate system, rack generation offers

two advantages compared with rotor coordinate systems: a) a rack profile represents the shortest contact path in comparison with other rotors. This means that points from the rack will be projected onto the rotors without any overlaps or other imperfections, b) a straight line on the rack will be projected onto the rotors as involutes.

In order to minimize the blow hole area on the high pressure side of a rotor profile, the profile is usually produced by a conjugate action of both rotors, which undercuts the high pressure side of them. The practice is widely used; thus in GB-A-1197432, singular points on main and gate rotors were used, in GB-A-2092676 and 2112460 circles, in GB-A-2106186 ellipses were used and in EP- 0166531 parabolae were used. An appropriate undercut has not hitherto been achievable directly from a rack. In arriving at the invention, it has been found that there exists only one analytical curve on a a rack which can exactly replace the conjugate action of rotors. In accordance with a preferred aspect of the present invention, this is a cycloid, which is undercut as an epicycloid on the main rotor and as a hypocycloid on the gate rotor. This is in contrast to the undercut produced by singular points which produces epicycloids on both rotors. The deficiency of this is usually minimized by a considerable reduction in the outer diameter of the gate rotor within its pitch circle. This reduces the blow-hole area, but also reduces the throughput.

A conjugate action is a process when a point (or points on a curve) on one rotor during a rotation cuts its (their) path(s) on another rotor. An undercut occurs if there exists two or more common contact points at the same time,

which produces "pockets" in the profile. It usually happens if small curve portions (or a point) generate long curve portions, when a considerable sliding occurs.

This invention overcomes this deficiency by generating the high pressure part of a rack by a rotor conjugate action which undercuts an appropriate curve on the rack. This rack is later used for the profiling of both the main and gate rotors by the usual rack generation procedure.

The following is a detailed description of a simple rotor lobe shape of a rack generated profile family designed for the efficient compression of air, common refrigerants and a number of process gases, obtained by the combined procedure. This profile contains almost all the elements of modern screw rotor profiles given in the open literature, but its features offer a sound basis for additional refinement and optimisation.

The coordinates of all primary arcs on the rack are summarised here relative to the rack coordinate system.

The lobe of this profile is divided into several arcs. The divisions between the profile arcs are denoted by capital letters and each arc is defined separately, as shown in Fig. 2.

Segment A-B is a general arc of the type ax§ + Jby§ = 1 on the rack with p = 0.43 and g = 1.

Segment B-C is a straight line on the rack, p = q = 1.

Segment C-D is a circular arc on the rack, p = q = 2, a = b .

Segment D-E is a straight line on the rack.

Segment E-F is a circular arc on the rack, p = q = 2, a = b .

Segment F-G is a straight line.

Segment G-H is an undercut of the arc G ₂ -H ₂ which is a general arc of the type ax| + by% ^' = 1, p = l, g = 0.75 on the main rotor.

Segment H-A on the rack is an undercut of the arc A ₁ -H ₁ which is a general arc of the type ax§ + byg = 1, p = l, q = 0.25 on the gate rotor.

At each junction A, ....H, the adjacent segments have a common tangent .

The rack coordinates are obtained through the procedure inverse to equations (7) -(11) .

As a result, the rack curve E-H-A is obtained and shown in Fig. 3.

Figure 4 shows the profiles of main and gate rotors 11,12 generated by this rack procedure superimposed on the well- known profiles 21,22 of corresponding rotors generated in accordance with GB-A-2 092 676, in 5/7 configuration.

With the same distance between centres and the same rotor diameters, the rack-generated profiles give an increase in displacement of 2.7% while the lobes of the female rotor are thicker and thus stronger.

In a modification of the rack shown m Fig. 3, the segments GH and HA are formed by a contiuous segment GHA of a cycloid of the form: y=R _σ cosτ-R _p , y=R ₀ sιnτ-R _p τ, where R ₀ is the outer radius of the main rotor (and thus of its bore) and R _p is the pitch circle radius of the main rotor.

The segments AB, BC, CD, DE, EF and FG are all generated by equation (12)above. For AB, a=b,p=0.43, q=l. For the other segments, a=b=l/r,and p=q=2. The values of p and q may vary by +10%. For the segments BC, DE and FG r is greater than the pitch circle radius of the mam rotor, and is preferably infinite so that each such segment is a straight line. The segments CD and EF are cicular arcs when p=q=2, of curvature a=b.