U.S. patent application number 15/618459 was filed with the patent office on 2018-04-12 for rotary device and a method of designing and making a rotary device.
The applicant listed for this patent is EPICAM LIMITED. Invention is credited to Marcus ARDRON, Anthony Osborne DYE, Gunnar MOELLER.
Application Number | 20180100502 15/618459 |
Document ID | / |
Family ID | 41717036 |
Filed Date | 2018-04-12 |
United States Patent
Application |
20180100502 |
Kind Code |
A1 |
MOELLER; Gunnar ; et
al. |
April 12, 2018 |
ROTARY DEVICE AND A METHOD OF DESIGNING AND MAKING A ROTARY
DEVICE
Abstract
The invention provides a rotary device comprising a first rotor
rotatable about a first axis and having at its periphery a recess
bounded by a curved surface, and a second rotor counter-rotatable
to said first rotor about a second axis, parallel to said first
axis, and having a radial lobe bounded by a curved surface, the
first and second rotors being coupled for intermeshing rotation,
wherein the first and second rotors of each section intermesh in
such a manner that on rotation thereof, a transient chamber of
variable volume is defined, the transient chamber having a
progressively increasing or decreasing volume between the recess
and lobe surfaces, the transient chamber being at least in part
defined by the surfaces of the lobe and the recess; the ratio of
the maximum radius of the lobe rotor and the maximum radius of the
recess rotor being greater than 1.
Inventors: |
MOELLER; Gunnar; (Hagen,
DE) ; ARDRON; Marcus; (Edinburgh, GB) ; DYE;
Anthony Osborne; (Girton, GB) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
EPICAM LIMITED |
Linton |
|
GB |
|
|
Family ID: |
41717036 |
Appl. No.: |
15/618459 |
Filed: |
June 9, 2017 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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13516121 |
Dec 20, 2012 |
9714655 |
|
|
PCT/GB2010/052128 |
Dec 17, 2010 |
|
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15618459 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
F01C 1/123 20130101;
F01C 1/084 20130101; F04C 2240/20 20130101; F04C 2/02 20130101;
F01C 1/20 20130101; F04C 2250/301 20130101 |
International
Class: |
F04C 2/02 20060101
F04C002/02; F01C 1/08 20060101 F01C001/08; F01C 1/12 20060101
F01C001/12; F01C 1/20 20060101 F01C001/20 |
Foreign Application Data
Date |
Code |
Application Number |
Dec 17, 2009 |
GB |
0921968.4 |
Claims
1.-15. (canceled)
16. A rotary device comprising a first rotor rotatable about a
first axis and having at its periphery a recess bounded by a curved
surface, and a second rotor counter-rotatable to said first rotor
about a second axis, parallel to said first axis, and having a
radial lobe bounded by a curved surface, the first and second
rotors being coupled for intermeshing rotation, wherein the first
and second rotors of each section intermesh in such a manner that
on rotation thereof, a transient chamber of variable volume is
defined for acting on working fluid in order to reduce its volume
in the case of compression or for being acted on by working fluid
to allow increase in its volume in the case of expansion, the
transient chamber being defined between interacting surfaces of the
lobe and recess rotor, the transient chamber having a progressively
decreasing or increasing volume between the recess and lobe
surfaces, the transient chamber being at least in part defined by
the surfaces of the lobe and the recess; a ratio of the maximum
radius of the lobe rotor and the maximum radius of the recess rotor
being greater than 1 thereby increasing the swept volume per cycle
of interaction as compared to if the ratio was less than or equal
to one.
17. The rotary device according to claim 16, wherein the ratio of
the maximum radius of the lobe rotor and the maximum radius of the
recess rotor is between 1.1 and 1.5.
18. The rotary device according to claim 17, wherein the ratio of
the maximum radius of the lobe rotor and the maximum radius of the
recess rotor is about 1.3.
19. The rotary device according to claim 16, wherein a housing is
provided to enclose the rotors.
20. The rotary device according to claim 19, wherein the housing
includes a moveable containment wall, said wall being moveable so
as to vary the maximum possible volume of the transient chamber of
variable volume.
21. The rotary device according to claim 16, wherein the rotors
extend axially in a helical configuration.
22. The rotary device according to claim 16, wherein the geometry
of the or each lobe is determined by an inner radius of the lobe
.rho..sub.Li, an outer rotor radius at a tip of the lobe
.rho..sub.Lo, an outer radius of the recess rotor .rho..sub.Po, and
a circular arc segment A.sub.l of radius R.sub.l defining a bulk of
the lobe.
23. The rotary device according to claim 22, wherein the geometry
of the or each lobe is, in addition, determined by a circular arc
segment A.sub.c of radius R.sub.c wherein the arc segment A.sub.l
defines the bulk of the lobe from its tip to an inflection point
and the circular arc segment A.sub.c defines a base of the lobe
connecting between the arc segment A.sub.l and a core of the
lobe.
24. The rotary device according to claim 22, wherein a position of
the centre of the circular arc segment A.sub.l is defined in
dependence on the separation of the centre of the circular arc
segment A.sub.l from the centre of the lobe rotor.
25. The rotary device according to claim 16, wherein the lobe
profile comprises plural arc segments.
26. One or more of an engine, a compressor, an expander, and a
supercharger each comprising a rotary device according to claim
16.
27. A method of designing the rotors for a rotary device having a
lobe rotor and a recess rotor coupled for intermeshing rotation,
wherein the lobe and recess rotors intermesh in such a manner that
on rotation thereof, a transient chamber of variable volume is
defined, the transient chamber having a progressively increasing or
decreasing volume between the recess and lobe surfaces, the method
comprising: determining the geometry of the or each lobe in
dependence on an inner radius of the lobe .rho..sub.Li, an outer
rotor radius at the tip of the lobe .rho..sub.Lo, and a circular
arc segment A.sub.l of radius R.sub.l defining a bulk of the lobe
and an outer radius of the recess rotor .rho..sub.Po.
28. The method according to claim 27, wherein the geometry of the
or each lobe is, in addition, determined by a circular arc segment
A.sub.c of radius R.sub.c wherein the arc segment A.sub.l defines
the bulk of the lobe from its tip to an inflection point and the
circular arc segment A.sub.c defines a base of the lobe connecting
between the arc segment A.sub.l and a core of the lobe.
29. The method according to claim 27, comprising making a lobe
rotor having the determined geometry.
30. The method according to claim 29, comprising making the recess
rotor to correspond with the lobe rotor.
Description
[0001] The present invention relates to a rotary device and to a
method of designing and making a rotary device. Typically the
rotary device might be an engine, a compressor, an expander or a
supercharger. When used herein, the term "rotary device" includes
but is not limited to any or all of the above.
[0002] Rotary engines are known that use a pair of rotors to
achieve compression or expansion by displacement. The engines
typically utilise the interaction between pairs of lobed and
recessed rotors, in which the volume change applied to a
compressible working fluid is achieved in a manner determined by
the cross-sectional shape of the rotor pairs.
[0003] In WO-A-91/06747, the entire contents of which are hereby
incorporated by reference, there is disclosed an internal
combustion engine comprising separate rotary compression and
expansion sections. Each of the compression and expansion sections
is a rotary device comprising a first rotor rotatable about a first
axis and having at its periphery a recess bounded by a curved
surface, and a second rotor counter-rotatable to said first rotor
about a second axis, parallel to said first axis, and having a
radial lobe bounded by a curved surface. The first and second
rotors are coupled for intermeshing rotation. The first and second
rotors of each section intermesh in such a manner that on rotation
thereof, a transient chamber of variable volume is defined. The
transient chamber has a progressively increasing (expansion
section) or decreasing (compression section) volume between the
recess and lobe surfaces.
[0004] The manner of the interaction relies on the fact that the
surfaces are contoured such that during passage of said lobe
through the recess, the recess surface is continuously swept, by
both a tip of said lobe and a movable location on the lobe. The
moving tip and location on the lobe can each be said to define a
locus. The location on the lobe progresses along both the lobe
surface and the recess surface, to define the transient chamber.
Thus, in such devices the form of the rotors is important and it is
necessary that they should conform with the requirement to provide
a sweep of the lobe through the recess, in which two minimum
clearance points (at the tip and the movable location on the lobe)
are maintained for the duration of the volume change cycle from a
maximum volume at the start of the cycle to an effectively zero
volume at the end of the cycle (for a compressor) and an effective
zero volume increasing to a capacity limited maximum volume in the
case of an expander.
[0005] These devices work well in that the low friction function
means they are comparatively efficient as compared to other known
rotary devices or indeed other engines. They are "low friction" in
that the rotors do not actually contact each other but instead
there is a minimum clearance between the rotors at the two points
mentioned above.
[0006] Subsequent improvements and modifications to the basic form
of such devices added new features. In WO-A-98/35136, the entire
contents of which are hereby incorporated by reference, there is
disclosed the use of helical forms of the rotors in the axial
direction and a variable maximum possible volume for the transient
chamber. Furthermore, in WO-A-2005/108745, the entire contents of
which are hereby incorporated by reference, there is disclosed a
method and apparatus by which the port flow area of such devices is
increased. Indeed, in WO-A-2005/108745, an endplate was provided at
the axial end of the recess rotor that enclosed the transient
chamber of variable volume. A valve was provided in the endplate
and an opening was provided in the surrounding housing. As the
recess rotor rotates, the valving action between the endplate and
the housing serves to control the flow of working fluid into and
out of the transient chamber during an operating cycle. The sizing
and positioning of the valve in the endplate and the opening in the
housing enables accurate control of the rotary device.
[0007] The modifications and additions of WO-A-98/35136 and
WO-A-2005/108745 did not change the form of the rotors nor their
manner of interaction.
[0008] Rules were established which governed the distance apart of
the central axes of rotation of the rotors and the magnitude of the
outer radius of both rotors. In the case of the distance between
axes of rotation, it was known that if this was reduced beyond a
certain extent, then rotor forms could not be devised which would
complete the interaction without either fouling or creating
unavoidable leakage areas. Where this limit precisely lay in
geometrical terms which could be related to other geometrical rotor
parameters however, was unknown. It was therefore considered unsafe
to reduce it arbitrarily below one and one third times the outer
radius of the lobe. Once this parameter was fixed, then any
reduction in the outer radius of the lobe rotor alone, without
change in the radius of the recess rotor, would necessarily reduce
the maximum 2-dimensional area swept by the lobe and therefore
would reduce the swept volume of the machine. Subsequent models
were therefore developed in which equality of outer radius
dimensions for each of the recess and lobe rotors was retained.
[0009] This limitation, together with the limited value of the
distance between the rotor axes, necessarily constrained the inner
radius of the lobe rotor, i.e. the radius of the lobe rotor core,
in order to provide rolling contact with the segments of the
circumference of the recess rotor between the recesses. It also
determined the maximum penetration of the lobe into the recess
rotor.
[0010] According to a first aspect of the present invention, there
is provided a rotary device comprising a first rotor rotatable
about a first axis and having at its periphery a recess bounded by
a curved surface, and a second rotor counter-rotatable to said
first rotor about a second axis, parallel to said first axis, and
having a radial lobe bounded by a curved surface, the first and
second rotors being coupled for intermeshing rotation, wherein the
first and second rotors of each section intermesh in such a manner
that on rotation thereof, a transient chamber of variable volume is
defined, the transient chamber having a progressively increasing or
decreasing volume between the recess and lobe surfaces, the
transient chamber being at least in part defined by the surfaces of
the lobe and the recess; the ratio of the maximum radius of the
lobe rotor and the maximum radius of the recess being greater than
1.
[0011] Thus, the present rotary device provides for a radius of the
lobe rotor to be larger than that of the recess rotor and therefore
enables the working volume of the device to be increased on a per
cycle basis. This change to the previously established form of the
rotor geometry increases the 2-dimensional sweep area through the
interaction cycle. When translated into a 3-dimensional design,
this change allows the maximum swept delivery volume per revolution
to be increased by more than 100 percent when compared with rotors
of the same overall dimensions but following the previously
established rules.
[0012] Previously, if the radial length of the lobe were to be
increased by making the outer radius of the lobe greater than the
radius of the recess rotor, then there could be no certainty that
an effective interaction between the rotors could be achieved. In
the present case, it has been recognised that the outer radius of
the lobe can be greater than the radius of the recess rotor whilst
still providing a functioning rotary device. Furthermore, the
increased radius of the lobe provides for a greater swept area
during each cycle.
[0013] There is a desire to generate a means by which the
interaction of the rotors could be modelled and then to use the
generated means to provide a new engine having optimised rotors
such that swept volume and therefore power-per-cycle can be
maximised.
[0014] The nature of the constraints discussed above emphasizes the
lack of a clear mathematical model by which the interaction between
the rotors could be understood or by which rules could be
established to distinguish rotor forms with characteristics capable
of supporting effective gas displacement without leakage and
without fouling.
[0015] Preferably, the geometry of the or each lobe is determined
by the inner radius of the lobe .SIGMA..sub.Li, the outer rotor
radius at the tip of the lobe .rho..sub.Lo, the outer radius of the
recess rotor .rho..sub.Po and a circular arc segment A.sub.l of
radius R.sub.l defining a bulk of the lobe.
[0016] In one embodiment, the geometry of the or each lobe is, in
addition, determined by a circular arc segment A.sub.c of radius
R.sub.c wherein the arc segment A.sub.l defines the bulk of the
lobe from its tip to an inflection point and the circular arc
segment A.sub.c defines a base of the lobe connecting between the
arc segment A.sub.l and the core of the lobe.
[0017] In one embodiment, the position of the centre of the
circular arc segment A.sub.l is defined in dependence on the
separation of the centre of the circular arc segment A.sub.l from
the centre of the lobe rotor.
[0018] Thus, in the absence previously of a basis for determining
rotor shape, new physical models developed for practical
applications could only be reasonably assured of success provided
that they conformed to the parametric relationships of the
geometrical entities which defined their predecessors, i.e. by
ensuring equality of outer radius dimensions for both rotors.
[0019] According to a second aspect of the present invention there
is provided a method of designing the rotors for a rotary device
having a lobe rotor and a recess rotor coupled for intermeshing
rotation, wherein the lobe and recess rotors intermesh in such a
manner that on rotation thereof, a transient chamber of variable
volume is defined, the transient chamber having a progressively
increasing or decreasing volume between the recess and lobe
surfaces, the method comprising: determining the geometry of the or
each lobe in dependence on the inner radius of the lobe
.rho..sub.Li, the outer rotor radius at the tip of the lobe
.rho..sub.Lo, the outer radius of the recess rotor .rho..sub.Po and
a circular arc segment A.sub.l of radius R.sub.l defining a bulk of
the lobe. Preferably the method also comprises making a lobe rotor
having the determined geometry.
[0020] In a preferred embodiment, the geometry of the or each lobe
is, in addition, determined by a circular arc segment A.sub.c of
radius R.sub.c wherein the arc segment A.sub.l defines the bulk of
the lobe from its tip to an inflection point and the circular arc
segment A.sub.c defines a base of the lobe connecting between the
arc segment A.sub.l and the core of the lobe.
[0021] A method and device is provided by which rotors can be
designed and built so as to provide a functioning engine capable of
improved performance as compared to previous engines. A means is
provided to realise designs of the rotor interaction which conform
to the characterisation requirement established in the aforesaid
prior art but which are not necessarily constrained by the
arbitrary limits to which the prior art was subject.
[0022] In the present case, the search for improved performance
from rotors of given overall size, has led to an exploration of the
general rules which limit the size and shape of lobe and recess
rotors which are capable of interaction in the manner defined as
acceptable in the prior art cited above. A 2-dimensional
mathematical model is hereby provided, in which the geometrical
form of the pair of interacting rotors is represented by a minimum
number of key parameters whose relative magnitudes determine the
properties of an effective pair of interacting rotors.
[0023] Use of this mathematical model to explore the potential for
improved performance has led to the recognition that effectively
interacting rotor forms are possible in which the maximum radius of
the lobe rotor can be advantageously extended to a value
substantially greater than that of the recess rotor. This change to
the previously established form of the rotor geometry increases the
2-dimensional sweep area through the interaction cycle. When
translated into a 3-dimensional design, this change allows the
maximum swept delivery volume per revolution to be increased by
more than 100 percent when compared with rotors of the same overall
dimensions but following the previously established rules.
[0024] The mathematical model that is preferably used to determine
parameters for the rotors to enable the present rotary device to
operate is set out in detail in the Appendix forming part of the
description of this patent application.
[0025] According to a third aspect of the present invention, there
is provided a rotary device having a lobe rotor and a recess rotor
in which the lobe rotor has an outer radius and an inner radius and
the inner radius is minimised so as to maximise swept area or
volume of the lobe.
[0026] Preferably, the swept area is maximised in accordance with
the equation:
.rho. Po + .rho. Li .ltoreq. 1 q ( 1 + q ) 2 .rho. M l 2 + 1 27 ( 1
+ 2 q ) 3 R l 2 ##EQU00001##
[0027] in which
[0028] .rho..sub.po is the outer radius of the recess rotor;
[0029] .rho..sub.Li is the inner radius of the lobe rotor;
[0030] .rho..sub.Ml is the separation between the centre of the
lobe rotor and the centre of the circle from which the arc that at
least in part defines the shape of the lobe is taken;
[0031] q is the ratio of angular velocities of the recess and lobe
rotor; and
[0032] R.sub.l is the radius of the arc defining at least in part
the shape of the lobe.
[0033] This may be thought of as a condition on the curvature of
the main lobe segment A.sub.l.
[0034] Embodiments of the present invention will now be described
in detail with reference to the accompanying drawings, in
which:
[0035] FIG. 1 shows a schematic representation of the rotary device
of WO-A-91/06747 (it is the same as FIG. 8 of WO-A-91/06747);
[0036] FIGS. 2A to 2D show schematic representations of rotor pairs
in which the radius of the lobe rotor is greater than that of the
recess rotor;
[0037] FIG. 3 is a schematic representation of a rotary device
including a lobe rotor and a recess rotor used in the derivation of
a mathematical model to develop and design new rotors;
[0038] FIGS. 4A to 4F, 5A to 5F, 6A to 6F, 7A to 7F, 8A to 8F, 9A
to 9F and 10A to 10F are schematic representations of rotor pairs
for use in rotary devices.
[0039] FIG. 1 shows a schematic representation of the rotary device
of WO-A-91/06747 and, as mentioned above is the same as FIG. 8 of
WO-A-91/06747. The rotary device 2 comprises a lobe rotor 4 and a
recess rotor 6 contained within a housing 8. A transient chamber of
variable volume 10 is defined at least in part by the surfaces of
the lobe 12 and recess 14 of the respective rotors 4 and 6. In this
particular example, a curved containment wall 16 is provided as
part of the housing 8 and this also serves to form the transient
chamber of variable volume 10 together with the lobe 12 and recess
14. As explained above, in this rotary device 2 the separation 18
between the axes of rotation of the lobe rotor 4 and recess rotor 6
is fixed and the outer radius dimension is the same for both
rotors. In other words, the radius 20 of the lobe rotor 4 is fixed
at the same value as the radius 22 of the recess rotor 6. The
rotors each have a core, e.g. such as a central cylindrical
component, on which the recesses and/or lobes are formed.
[0040] In contrast, in the present rotary device the radius of the
lobe rotor and the radius of the recess rotor are different such
that an increased swept area (in 2D) and consequently, volume (in
3D) can be achieved without increasing the overall size of the
rotary device.
[0041] In an example, the two rotors are sized and configured in
such a way that it is possible to increase the outer radius of the
lobe rotor so that it is larger than that of the recess rotor.
Comparing this with the previous arrangement using a pair of
intermeshing rotors of given equal outer radius and given distance
between the rotor axes, then the change is seen only as an increase
in the tip radius of the lobe rotor. Thus, the arc described by the
lobe tip describes a larger circular area than the recess rotor. It
has been recognised by the inventors that it is possible that the
close contact point remote from the tip of the lobe, i.e. near to
the base of the lobe, is able to maintain close proximity to
successive points on the surface of the recess to enable the
familiar displacement of 2 dimensional area between the lobe and
recess to be executed in the same manner as was previously
possible.
[0042] The result of making this change in geometry is significant.
The result is to effect a substantially increased swept volume from
the paired rotor device on each cycle of operation. As an example,
when comparing the new geometry with a previous design, it is shown
that the swept volume delivery per revolution of the lobe rotor is
twice that of the previous design for rotors having the same shaft
centre distance.
[0043] In a previous design with shaft centre distance set at a
value such that the maximum possible volume of the transient
chamber of variable volume was 125 cc, the lobe rotor had four
lobes and the recess rotor had six recesses, each interaction
yielding a swept volume of 120 cc, thus making a total of 480 cc.
per revolution of the lobe rotor.
[0044] Using the geometry of embodiments of the present invention
in which the ratio of the maximum radius of the lobe and the
maximum radius of the recess is greater than 1, the increased
penetration of the lobe also increases the length of the arc
traversed by the lobe tip from the start of the cycle. Thus, in
this particular example, it is only possible to accommodate two
lobes which requires a matching three-recessed complementing rotor.
Nevertheless, the cycle swept volume for the new geometry is 500
cc. per lobe which means that the new design can deliver 1 Litre
per revolution of the lobe rotor.
[0045] Rotor lengths are preferably kept constant between previous
and new geometries in this comparison.
[0046] FIGS. 2A to 2D show schematic representations of rotor pairs
in which the radius of the lobe rotor is greater than that of the
recess rotor. As can be seen, the radius 20 of the lobe rotor 4 is
greater than that 22 of the recess rotor 6. A single pair of rotors
is shown in four different stages of a cycle of rotation. In FIG.
2A the tip of the lobe rotor (rotating anticlockwise) first engages
with the curved containment wall. Together with the recess in the
recess rotor a transient chamber of variable volume is first
defined within this cycle. In FIG. 2B the rotors have rotated
further, the lobe rotor rotating anticlockwise and the recess rotor
rotating clockwise. The transient chamber of variable volume has
decreased in size and so any working fluid trapped in the chamber
at the start of the cycle i.e. when the chamber is first formed,
will have been correspondingly compressed. In FIGS. 2C and 2D the
compression cycle continues. Despite the difference in radius of
the lobe rotor and the recess rotor the transient chamber is still
suitably defined and the required clearance between the two rotors
is maintained. Thus, the increased lobe rotor radius leads to an
increase in swept volume per cycle.
[0047] FIG. 3 is a schematic representation of a basic geometry of
the displacement engine or rotary device, as used to determine a
mathematical model for use in rotor design and manufacture. The
rotary device includes a lobe rotor 4 arranged in this example to
rotate in a clockwise direction, and a recess rotor 6 arranged in
this example to rotate in an anti-clockwise direction. The rotary
device is shown in the state of rotation where the tip of the lobe
T first penetrates the recess rotor. In other words although the
more forward part of the lobe profile is already within the outer
perimeter of the recess rotor at the instant shown in FIG. 3, the
tip T is just about to penetrate the outer perimeter. It will be
appreciated that it is the "front" surface of the lobe that
determines its interaction with the recess. The following or rear
surface can be any convenient or desired shape. The lobe may be
shaped in its rear surface or body so as to minimise the amount of
material required to make it to minimise weight of the rotors.
[0048] In the example shown, the y-axis of the co-rotating
coordinate system (x, y) in the recess frame is chosen such that it
pierces T at this instant. The shape of the lobe is, in this
example, defined by the two circles of Radius R.sub.l and R.sub.c
for the bulk of the lobe and its base. As shown there are various
angles near centres of the lobe O.sub.L and the centre of the
recess O.sub.P.
[0049] These angles are defined by triangles of named points,
namely
.phi..sub.Ml=/(M.sub.l,O.sub.L,T),
.phi..sub.Mc=/(M.sub.c,O.sub.L,T),
.phi..sub.lc=/(M.sub.l,O.sub.L,M.sub.c),
.alpha..sub.L=/(O.sub.P,O.sub.L,T), and
.alpha..sub.P=/(O.sub.L,O.sub.P,T),
[0050] where the angle defined is near the second of each triple of
points.
[0051] FIGS. 4A to 4F and 10A to 10F are schematic representations
of rotor pairs for use in rotary devices.
[0052] FIGS. 4A to 4F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. The lobe rotor is shown rotating clockwise and the
recess rotor is shown rotating anti-clockwise. In this example
there are two lobes and two recesses. The views shown in the figure
are the start of the cycle (FIG. 4A) when a compression chamber can
first be formed with the housing (represented by a bolder dark line
at the upper boundary), the rotation when the base of the lobe
first penetrates the recess rotor (FIG. 4B), the time about half
way between the base of the lobe and the tip of the lobe entering
the recess rotor (FIG. 4C), the time when the tip of the lobe first
penetrates the recess rotor (FIG. 4D). Next, in FIG. 4E, there is
shown the position when the lobe and recess rotors have rotated
further such that the transient chamber of variable volume is
formed entirely between the curved lobe surface and recess and can
be seen to have significantly reduced in volume as compared to the
previous figure (FIG. 4D). Last, in FIG. 4F, the end of the cycle
is shown when the inner and outer locus meet.
[0053] With reference to the parameters defined above with respect
to FIG. 3, the values for the parameters chosen for this
configuration are .rho..sub.Li=30 mm, .rho..sub.Lo=99 mm,
R.sub.l=64 mm, R.sub.c=37 mm, .rho..sub.Ml=37.5 mm, .rho..sub.Po=75
mm. Although the values, in this example and the examples below
shown in and described with reference to FIGS. 5A to 5F and 9A to
9F are given in units of mm, it will be understood that these can
equally be thought of as an arbitrary basic unit of length in that
the dimensions of the rotary device are fully scalable. For
simplicity, the trailing or following edge of the lobe is drawn
simply as a straight line. Any appropriate or desired shape may be
used for this trailing edge. What is important is the leading edge
of the lobe that interacts with the surface of the recess. In
practice the trailing edge is preferably shaped so as to avoid
sharp corners. For example, it might be continuously contoured or
curved.
[0054] FIGS. 5A to 5F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. The lobe rotor is shown rotating clockwise and the
recess rotor is shown rotating anti-clockwise. In this example
there is a single lobe interacting with a single recess. In
addition, this example illustrates a case where the lobe consists
of a single circle segment. As in FIGS. 4A to 4F, the snapshots
shown represent the start of the cycle (FIG. 5A) and the points of
penetration at the base (FIG. 5B) and at the tip (FIG. 5D), as well
as the cycle end (FIG. 5F) and intermediate positions (FIGS. 5C and
5E). In this particular case of a single lobe, the earliest useful
start of the cycle is given when the enclosed volumes within the
lobe and recess rotors first communicate with each other, leading
to a large total sweep angle and volume per cycle. The values for
the parameters chosen for this configuration are .rho..sub.Li=35
mm, .rho..sub.Lo=100 mm, R.sub.l=75 mm, .rho..sub.Po=75 mm. As in
the example above, the dimensions of the rotary device are fully
scalable.
[0055] FIGS. 6A to 6F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. The lobe rotor is shown rotating clockwise and the
recess rotor is shown rotating anti-clockwise. In this example
there is a single lobe interacting with a single recess. As in FIG.
5, this particular example illustrates a special case where the
lobe consists of a single circle segment. As in FIG. 4, the
snapshots shown represent the start of the cycle and the points of
penetration at the base and at the tip, intermediate positions as
well as the cycle end. In this particular case of a single lobe,
the earliest useful start of the cycle is given when the enclosed
volumes within the lobe and recess rotors first communicate with
each other, leading to a large total sweep angle and volume per
cycle. This configuration requires a large fraction of the
circumference to be enclosed by a casing. The values for parameters
chosen for this configuration are .rho..sub.Li=40 mm,
.rho..sub.Lo=100 mm, R.sub.l=74 mm, .rho..sub.Po=65 mm. As in the
examples above, the dimensions of the rotary device are fully
scalable.
[0056] FIGS. 7A to 7F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. The lobe rotor is shown rotating clockwise and the
recess rotor is shown rotating anti-clockwise. In this example
there are two lobes and three recesses. As in FIG. 4, the snapshots
shown represent the start of the cycle and the points of
penetration at the base and at the tip, intermediate positions as
well as the cycle end. The lobe is shaped such that the maximum
volume of the compression chamber is, maximized for a given total
width of the engine. The values for the parameters chosen for this
configuration are .rho..sub.Li=30.5 mm, .rho..sub.Lo=103.2 mm,
R.sub.l=70 mm, R.sub.c=25 mm, .rho..sub.Ml=40 mm, .rho..sub.Po=92.8
mm. As in the examples above, the dimensions of the rotary device
are fully scalable.
[0057] FIGS. 8A to 8F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. The lobe rotor is shown rotating clockwise and the
recess rotor is shown rotating anti-clockwise. In this example
there are two lobes and three recesses. As in FIG. 4, the snapshots
shown represent the start of the cycle and the points of
penetration at the base and at the tip, intermediate positions as
well as the cycle end. In comparison to the example shown in FIGS.
7A to 7F, this example illustrates a heavier recess rotor. The
values for the parameters chosen for this configuration are
.rho..sub.Li=33 mm, .rho..sub.Lo=100 mm, R.sub.l=60 mm, R.sub.c=50
mm, .rho..sub.Ml=42.5 mm, .rho..sub.Po=90 mm. As in the examples
above, the dimensions of the rotary device are fully scalable.
[0058] FIGS. 9A to 9F show a schematic representation of an
interacting rotor pair at various stages during a cycle of
interaction. In this example there are three lobes and four
recesses. As in FIG. 4, the snapshots shown represent the start of
the cycle and the points of penetration at the base and at the tip,
intermediate positions as well as the cycle end. Next, in FIG. 9E,
there is shown the position when the lobe and recess rotors have
rotated further as compared to the previous figure (FIG. 9D) such
that the transient chamber of variable volume is formed entirely
between the curved lobe surface and recess (as it is also in FIG.
9D) and can be seen to have significantly reduced in volume as
compared to FIG. 9D. With three lobes, the cycle length is
shortened significantly, which may be useful for applications where
it is important or desired to minimize leakage flow. The values for
the parameters chosen for this configuration are .rho..sub.Li=30
mm, .rho..sub.Lo=100 mm, R.sub.l=60 mm, R.sub.c=50 mm,
.rho..sub.Ml=46 mm, .rho..sub.Po=95 mm. As in the examples above,
the dimensions of the rotary device are fully scalable.
[0059] As explained above and also in section D in the appendix, a
general condition can be recognised for validity of a rotor
configuration. The parameters that are most favourable in order to
maximize the maximum possible volume of the transient compression
or expansion chamber of variable volume are now considered. As
explained in detail in the appendix (section E, entitled
"Maximising the Lobe Length"), a large fraction of the volume is
swept by the lobe rotor and it is thus useful to increase the
length of the outer lobe radius .rho..sub.Lo. An alternative or
additional way of achieving this, i.e. other than increasing
.rho..sub.Lo, involves reducing .rho..sub.Po followed by a
resealing of all length parameters such as to recover the same
overall size of the rotary device.
[0060] Independently, minimizing the inner lobe radius .rho..sub.Li
also contributes to an increase of the total swept volume. Thus an
independent aspect of the present invention (which may of course be
combined with other aspects or embodiments of the invention)
provides a rotary device having a lobe rotor and a recess rotor
arranged for intermeshing interaction in which the lobe rotor has
an outer radius and an inner radius and the inner radius is
minimised so as to maximise swept area or volume of the lobe.
Preferably, the rotary device comprising a first rotor rotatable
about a first axis and having at its periphery a recess bounded by
a curved surface, and a second rotor counter-rotatable to said
first rotor about a second axis, parallel to said first axis, and
having a radial lobe bounded by a curved surface, the first and
second rotors being coupled for intermeshing rotation, wherein the
first and second rotors of each section intermesh in such a manner
that on rotation thereof, a transient chamber of variable volume is
defined, the transient chamber having a progressively increasing or
decreasing volume between the recess and lobe surfaces, the
transient chamber being at least in part defined by the surfaces of
the lobe and the recess.
[0061] As explained in section E of the appendix, a criterion which
limits both these types of change is the condition on the curvature
of the main lobe segment A.sub.l, as formulated in equation (26)
and which is reformulated as equation (30). Rotor configurations
that maximize swept volume correspond to parameters such that
equation (30) is nearly satisfied as an equality, i.e. is
approximately satisfied as an equality. Thus by satisfying this
condition it is possible to maximise the swept volume in such a way
as to increase the effective working volume of the rotary device
per cycle without necessarily requiring a difference in the outer
radii of the lobe and recess rotors. Greater detail on this is
given in the appendix.
[0062] FIGS. 10A to 10F show an example of a geometry illustrating
the case of a configuration with two lobes and three pockets, where
the ratio of the rotor diameters is kept to be
.rho..sub.Lo/.rho..sub.Po=1. In this case, i.e. with parity between
the lobe and recess rotor radii, the swept volume can still be
increased by reducing the inner lobe radius. As in FIGS. 4A to 4F,
the snapshots shown include the start of the cycle (top left) and
the points of penetration at the base (top right) and at the tip
(centre right), as well as the cycle end (bottom right) and two
intermediate positions. The parameters chosen for this
configuration are .rho..sub.Li=24u, .rho..sub.Lo=96u, R.sub.l=60u,
R.sub.c=40u, .rho..sub.Ml=41.1u, .rho..sub.Po=96u.
[0063] The rotor pairs may be provided within a housing such as
that shown in and described above with reference to FIG. 1 and may
or may not be provided with a moveable containment wall so as to
enable the maximum possible volume of the transient chamber of
variable volume to be varied. In other words, in all cases the
actual volume of the chamber will vary during the cycle, from the
maximum to the minimum (zero usually) but in addition means may be
provided to vary the maximum possible volume for the chamber in any
one cycle. Indeed, it will be appreciated that rotor pairs of the
type described herein can be used in rotary devices as disclosed in
any or all of WO-A-91/06747, WO-A-98/35136 and
WO-A-2005/108745.
[0064] It will be appreciated that the above examples are
non-limiting and any suitable form may be used for the rotors. What
is important is that the radius of the lobe rotor and the recess
rotor is not the same which then enables an increased swept volume
to be achieved with the same overall size of device. In summary and
with reference to the description above of FIG. 3, it will be
appreciated that the model, for simplicity, is executed in 2
dimensions. The 3 dimensional form of the rotors is typically a
projection of the two-dimensional section (optionally helically
formed, i.e. with some rotation about the projection axis) and so
the model applies in 3 dimensions too.
[0065] As set out in the prior art referred to above, an efficient
rotational displacement device, is obtained by helically extruding
a single two-dimensional cross sectional area of the lobe and
recess rotors. By extension and reference to the prior art it is
therefore sufficient to describe the parameters defining their
two-dimensional shapes, as well as the constraints to which the
different parameters are subject.
[0066] In summary, the model operates by defining some fundamental
parameters and in dependence on these determining a shape for a
lobe rotor and the corresponding recess rotor. From the fundamental
parameters, a number of others may be derived including a number of
angles and further lengths. These two forms of parameter may be
referred to as "fundamental geometrical parameters" and "derived
geometrical parameters". The model discussed in the appendix below
uses one specific example as shown in FIG. 3. However, as explained
in section "F" entitled "Variants and Example Configurations" the
model can be used to determine a suitable shape for a lobe rotor
and recess rotor having any desired number of lobes and recesses
and to determine the shapes of lobes made up of any appropriate
number of arc segments. Thus, although in parts the appendix refers
to specific figures and examples, this has general applicability as
will be appreciated by a man skilled in the art.
[0067] Once the rotors have been designed using the method
described above the lobe rotor and the corresponding recess rotor
are made. These may be made using appropriate materials such as
steel and using any known method such as die casting, injection
moulding, extrusion of appropriate materials etc.
[0068] Embodiments of the present invention have been described
with particular reference to the examples illustrated. However, it
will be appreciated that variations and modifications may be made
to the examples described within the scope of the present
invention.
APPENDIX--MATHEMATICAL MODEL FOR USE IN DETERMINING ROTOR SHAPE
A. Fundamental Geometrical Parameters
[0069] The defining element of the rotational displacement device
(which we shall also refer to in short as the engine) is the
geometry of the lobe(s). The pocket rotor is obtained as the
involute form of the lobe geometry. The lobe rotor consists of a
n.sub.L identical lobes, offset relative to each other by an angle
2.pi./n.sub.L. Similarly, the pocket rotor features n.sub.P
identical pockets, offset by an angle 2.pi./n.sub.P. Both rotors
are linked by a pair of gears such that they rotate at a fixed
ratio of angular velocities q=n.sub.L/n.sub.P, given by the ratio
of the number of lobes n.sub.L to the number of pockets n.sub.P. As
shown in FIG. 3, the geometry of the lobe is defined by the
following elements and parameters: [0070] 1. the inner radius
defining the core of the lobe .rho..sub.Li, [0071] 2. the outer
radius at the tip of the lobe .rho..sub.Lo, [0072] 3. a circular
arc segment .sub.l of radius R.sub.l defines the bulk of the lobe
from the tip to an inflection point, [0073] 4. a second arc segment
.sub.c of radius R.sub.c defines the base of the lobe, connecting
smoothly between the segment .sub.l and the core of the lobe,
[0074] 5. to fully specify the geometry, the position of the centre
of the circular segment .sub.l has to be defined, we chose to
indicate the separation .rho..sub.m of its centre M.sub.l from the
centre of the lobe rotor.
[0075] In addition to these five parameters for the lobe, the outer
radius of the pocket rotor .rho..sub.Po completes the defining list
of defining system parameters. All further aspects of the geometry
derive from this set of six lengths as well as the ratio of number
of lobes to pockets: {.rho..sub.Li, .rho..sub.Lo, R.sub.l, R.sub.c,
.rho..sub.M.sub.l, .rho..sub.Po, q}.
B. Derived Geometrical Parameters
[0076] The length parameters given above uniquely define the
geometry. For convenience we derive from these a number of angles
and further lengths. Additional lengths which we shall refer to
below are given by the distance between the axes of the two
rotors
R=.rho..sub.Po+.rho..sub.Li, (1)
the separation of M.sub.l and M.sub.c
R.sub.lc=R.sub.l+R.sub.c, (2)
and the separation of M.sub.c and .sub.L
.rho..sub.M.sub.c=R.sub.c+.rho..sub.Li. (3)
Various angles are obtained by application of the cosine law in the
triangles present in the geometry. In particular, we define two
angles .alpha..sub.L and .alpha..sub.P, which relate to a special
state of rotation of the system. These two angles are realized in
the configuration where the tip of the lobe T first penetrates into
the interior of the pocket rotor. Considering the triangle
.DELTA.(.sub.P, .sub.L, T) at this instant, we define the two
angles .alpha..sub.L=.angle.(.sub.P, .sub.L, T), and
.alpha..sub.P=.angle.(.sub.L, .sub.P, T) (where the angle defined
is near the second of each triple of points), such that
.alpha. L = arccos [ .rho. Lo 2 + R 2 - .rho. Po 2 2 .rho. Lo R ] ,
( 4 ) ##EQU00002##
[0077] at the corner .sub.L, and
.alpha. P = arccos [ .rho. Po 2 + R 2 - .rho. Lo 2 2 .rho. Po R ] (
5 ) ##EQU00003##
[0078] at the corner .sub.P. Further angles are defined for the
lobe geometry and do not imply a particular state of rotation. All
of these angles are measured near the centre of the lobe .sub.L,
and are defined by triangles of points named in FIG. 3, in
particular .PHI..sub.M.sub.l=.phi.(M.sub.l, .sub.L, T),
.PHI..sub.M.sub.c=.angle.(M.sub.c, .sub.L, T), and
.PHI..sub.cl=.angle.(M.sub.l, .sub.L, M.sub.c).
[0079] These angles equate to
.phi. M l = arccos [ .rho. M l 2 + .rho. Lo 2 - R l 2 2 .rho. M l
.rho. Lo ] , ( 6 ) .phi. lc = arccos [ .rho. M l 2 + .rho. M c 2 -
R lc 2 2 .rho. M l .rho. M c ] , ( 7 ) .phi. M c = .phi. lc - .phi.
M l ( 8 ) ##EQU00004##
Prior patents [WO-A-91/06747, GB98/00345] have described specific
geometries of this type using the offset d of the point M.sub.l
from the radius towards the tip . This quantity can be used
interchangeably with .rho..sub.M.sub.l in the definition of the
geometry. Defining the angle .gamma..sub.T=.angle.(, T,
M.sub.l)=arccos
[(.rho..sub.Lo.sup.2+R.sub.l.sup.2-.rho..sub.M.sub.l.sup.2)/(2.rho..sub.L-
oR.sub.l)], we have d=R.sub.l sin .gamma..sub.T.
C. The Pocket Geometry
[0080] The shape of the pocket rotor follows by imprinting the
shape of the lobe under revolution of the two rotors. There are two
points of contact between the two rotors. The first point of
contact is located initially at the base of the lobe defined by the
intersection of .sub.c and and is travelling towards the tip T of
the lobe as the lobe penetrates the pocket rotor. The second point
is given by the tip of the lobe. These two points are referred to
below as the inner and outer locus. The movement of these two loci
defines the geometry of the pocket. However, some conditions need
to be verified by the lobe geometry to assure that a functional
pocket exists, which are considered in the subsequent section.
Here, we first demonstrate how to construct the shape of the
pocket.
1. Coordinate Systems
[0081] First, we need to define a convenient coordinate system in
which to express the pocket shape. We choose the system (x, y)
shown in FIG. 3, as a frame which is stationary in the rotating
frame of the pocket rotor. Its relative position to the lobe rotor
is defined by the point of first contact of the lobe tip, defined
to lie of the y-axis. In addition, we define a time t measured in
radians of rotation of the lobe rotor. The origin of the time
coordinate t=0 is associated with the state of rotation when the
base of the lobe .sub.c first penetrates the pocket rotor, i.e.,
when .sub.L, M.sub.c and .sub.P lie on a common line. Positive time
t corresponds to clockwise rotation by the angle t of the lobe
rotor. The configuration shown in FIG. 3 therefore displays time
t.sub.tip=.PHI..sub.M.sub.c-.alpha..sub.L. For typical
configurations, t.sub.tip is positive, however it can in principle
be negative.
[0082] A second useful frame of reference (.xi., .eta.) can be
defined for the lobe rotor, such that the unit vector {right arrow
over (e)}.sub..xi. continually points towards the origin of the
pocket rotor, and {right arrow over (e)}.sub..eta. is obtained by
rotating this vector by .eta./2 (counterclockwise), i.e., {right
arrow over (e)}.sub..eta.={right arrow over (e)}.sub.z.times.{right
arrow over (e)}.sub..xi., with {right arrow over (e)}.sub.z the
unit vector pointing outwards of the plane of projection of FIG. 3.
Due to rotation of the pocket system (x, y), the point .sub.L
describes the trajectory
r .fwdarw. L ( t ) = R ( cos .PHI. L sin .PHI. L ) , ( 9 )
##EQU00005##
where .phi..sub.L is measured from the x-axis
.PHI. L = .pi. 2 + .alpha. P - q ( t + .alpha. L - .phi. Mc ) . (
10 ) ##EQU00006##
Consequently, the (time-dependent) unit vectors of the system
(.xi., .eta.) are given by
e .fwdarw. = ( - cos .PHI. L - sin .PHI. L ) , and e .fwdarw. .eta.
= ( sin .PHI. L - cos .PHI. L ) . ( 11 ) ##EQU00007##
The reference system (.xi., .eta.) is not attached to the rotating
frame of the lobe. Instead, angles of points in the lobe system
decrease linearly with the time variable, t=0 corresponding to
=.rho..sub.M.sub.c{right arrow over (e)}.sub..xi..
2. Curve Segments Defining the Pocket
[0083] The motion of single points in the lobe system, such as the
lobe tip T, as well as the center points M.sub.l and M.sub.c can
now be straightforwardly expressed:
{right arrow over (.gamma.)}.sub.T(t)={right arrow over
(.gamma.)}.sub.L(t)+[cos(.PHI..sub.M.sub.c-t){right arrow over
(e)}.sub..xi.+sin(.PHI..sub.M.sub.c-t){right arrow over
(e)}.sub..eta.].rho..sub.Lo, (12)
{right arrow over (.gamma.)}.sub.M.sub.c(t)={right arrow over
(.gamma.)}.sub.L(t)+[cos(-t){right arrow over
(e)}.sub..xi.+sin(-t){right arrow over
(e)}.sub..eta.].rho..sub.M.sub.c, (13)
{right arrow over (.gamma.)}.sub.M.sub.l(t)={right arrow over
(.gamma.)}.sub.L(t)+[cos(.PHI..sub.lc-t){right arrow over
(e)}.sub..xi.+sin(.PHI..sub.lc-t){right arrow over
(e)}.sub..eta.].rho..sub.M.sub.l, (14)
The outer locus is identical with {right arrow over
(.gamma.)}.sub.T(t), while the inner locus is traced out as the
involute form of circles with centers {right arrow over
(.gamma.)}.sub.M.sub.l, and {right arrow over
(.gamma.)}.sub.M.sub.c. Its trajectory is therefore offset by the
respective radius relative to either curve, and the resulting curve
segments C.sub.l, C.sub.c can be expressed as follows
{right arrow over (.gamma.)}.sub.C.sub.c(t)={right arrow over
(.gamma.)}.sub.M.sub.c-R.sub.c{right arrow over
(e)}.sub.z.times.{right arrow over (.gamma.)}.sub.M.sub.c (15)
{right arrow over (.gamma.)}.sub.C.sub.l(t)={right arrow over
(.gamma.)}.sub.M.sub.c-R.sub.l{right arrow over
(e)}.sub.z.times.{right arrow over (.gamma.)}.sub.M.sub.l, (16)
where we have introduced the tangent vectors {right arrow over
(.tau.)}.sub.M={right arrow over ({dot over
(.gamma.)})}.sub.M/.parallel.{right arrow over ({dot over
(.gamma.)})}.sub.M.parallel. (using the notation
r .fwdarw. . .ident. d dt r .fwdarw. ##EQU00008##
for the time derivative, and .parallel.{right arrow over
(.gamma.)}.parallel. to denote the norm of a vector).
[0084] As stated above, the inner locus moves from the base of the
lobe towards its tip during the compression cycle. The curve
delineating the pocket is given as the union of three segments,
defined by {right arrow over (.gamma.)}.sub.C.sub.c, {right arrow
over (.gamma.)}.sub.C.sub.l and {right arrow over (.gamma.)}.sub.T
on the appropriate time intervals. Initially, for times t [0,
t.sub.cl] the inner locus is described by {right arrow over
(.gamma.)}.sub.C.sub.c, where the final time t.sub.cl is defined by
the intersection of the curves C.sub.c and C.sub.l, that can be
obtained by solving
{right arrow over ({dot over
(.gamma.)})}.sub.M.sub.l(t.sub.cl)[{right arrow over
(.gamma.)}.sub.M.sub.l(t.sub.cl)-{right arrow over
(.gamma.)}.sub.M.sub.c(t.sub.cl)]=0 (17)
The solution can be found analytically, and it is of the form
t cl = 2 arctan { a - a 2 + b 2 - c 2 b + c } , ( 18 )
##EQU00009##
abbreviating recurrent expressions
a = ( .rho. M c - .rho. M l cos .phi. lc ) , b = .rho. M l sin
.phi. lc , c = q + 1 q .rho. M l .rho. M c sin .phi. lc ,
##EQU00010##
For times t.sub.cl<=t<=t.sub.end, the inner locus is
described by {right arrow over (.gamma.)}.sub.C.sub.l. Finally, we
can obtain the time or angle of rotation for the end of the cycle
t.sub.end, which occurs when the inner and outer locus meet,
and
{right arrow over ({dot over
(.gamma.)})}.sub.M.sub.l(t.sub.end)[{right arrow over
(.gamma.)}.sub.M.sub.l(t.sub.end)-{right arrow over
(.gamma.)}.sub.M.sub.l(t.sub.end)]=0 (19)
It solution has a similar form as Eq. (18), but with one change in
sign:
t end = 2 arctan { d + d 2 + e 2 - f 2 e + f } , ( 20 )
##EQU00011##
and with the parameters
d = ( .rho. M l cos .phi. lc - .rho. Lo cos .phi. M c ) R , e = (
.rho. Lo sin .phi. M e - .rho. M l sin .phi. lc ) R , f = - q + 1 q
.rho. M l .rho. Lo sin ( .phi. M l ) . ##EQU00012##
D. Constraints on the System Parameters
[0085] In the previous section, we have derived mathematical
expressions for the curves defining the pocket geometry, Eqs. (12),
(15), and (16). However, not all choices of parameters
{.rho..sub.Li, .rho..sub.Lo, R.sub.l, R.sub.c, .rho..sub.M.sub.l,
.rho..sub.Po, q} yield well defined pocket geometries. We now
proceed to derive the conditions under which a functional unit is
obtained.
[0086] For a successful compressor geometry, the inner locus, as
seen in the rest-frame of the pocket rotor, performs a continuous
movement, which excludes any momentary reversals of the velocity as
well as intersections of its trajectory with itself. A valid
trajectory can be ensured by requiring a negative initial velocity
(contrary to the sense of rotation of the pocket rotor), a touching
point of the curves C.sub.c and C.sub.l at time t.sub.cl and the
absence of reversal of the velocity within curve C.sub.l. In
addition, there are some trivial geometric constraints which we
consider first.
1. Triangle Relations
[0087] On the level of basic geometry, the lengths defining the
lobe geometry have to be chosen such that the two fundamental
triangles .DELTA.(.sub.L, T, M.sub.l) and .DELTA.(.sub.L, M.sub.c,
M.sub.l) can be spanned, as described by the triangle relations
|a-b|<c<a+b [for a generic triangle .DELTA.(a, b, c)]. Six
inequalities follow, namely
R.sub.l+.rho..sub.M.sub.l>.rho..sub.Lo (21a)
.rho..sub.Lo+.rho..sub.M.sub.l>R.sub.l (21b)
R.sub.l+.rho..sub.Lo>.rho..sub.M.sub.l (21c)
for the first of the two triangles, and
.rho..sub.M.sub.l+R.sub.l>.rho..sub.Li (22a)
R.sub.l+2R.sub.c+.rho..sub.Li>.rho..sub.M.sub.l (22b)
.rho..sub.Li+.rho..sub.M.sub.l>R.sub.l (22c)
for the second.
2. Initial Velocity of the Inner Locus
[0088] In order for the initial velocity of the inner locus to be
negative (i.e., moving in the direction from the base to the tip)
it is sufficient to demand that the movement of its center has a
positive velocity at t=0. The trajectory .rho..sub.M.sub.c is then
forced to describe a loop with strong curvature, that enforces a
negative velocity {right arrow over ({dot over
(r)})}.sub.c.sub.c(0). With little algebra, this condition
translates to
R c > 1 1 + q [ q .rho. Po - .rho. Li ] ( 23 ) ##EQU00013##
3. Intersection of the Curves C.sub.c and C.sub.l
[0089] By construction, the arc segments defining the lobe .sub.c
and .sub.l share a common tangent where they join. Consequently,
the involutes of both arcs generically yield parallel curves
C.sub.c and C.sub.l at their touching point. However, C.sub.c has
an inflection point accompanied with a reversal of local velocity.
This feature must occur after the time of intersecting with
C.sub.l, in which case it does not affect the geometry. This leads
to a condition, which is equivalent to demanding a positive
argument of the root in Eq. 18. Simplifying this expression, we
arrive at the condition
- ( 1 + q ) 2 4 q 2 ( .rho. M c + .rho. M l - R c - R l ) ( R c + R
l + .rho. M c - .rho. M l ) .times. ( R c + R l - .rho. M c + .rho.
M l ) ( R c + R l + .rho. M c + .rho. M l ) + ( R c + R l ) 2 R 2
> 0. ( 24 ) ##EQU00014##
Note all the factors in parentheses for the first term are positive
by virtue of the triangle relations.
4. Bound on the curvature of {right arrow over
(.gamma.)}.sub.M.sub.l
[0090] Finally, one needs to ensure that the curve C.sub.l is well
formed. It is typically dominated by a point of inflection where
the inner locus remains nearly stationary, a and can even reverse
its direction. The latter case leads to leakage and should be
avoided. Algebraically, this can be expressed as the velocity of
the touching point {right arrow over ({dot over (r)})}.sub.c.sub.l
having a positive projection onto the velocity of the center point
{right arrow over ({dot over (r)})}.sub.M.sub.l, or in equations
{right arrow over ({dot over (r)})}.sub.C.sub.l{right arrow over
({dot over (r)})}.sub.M.sub.l>0. This translates into a
constraint on the (signed) curvature .kappa..sub.l(t) of the curve
{right arrow over (.gamma.)}.sub.M.sub.l. It is required that
.kappa. l ( t ) .ident. r M l .fwdarw. . ( e .fwdarw. z .times. r
.fwdarw. M l ) r .fwdarw. . M l 3 .ltoreq. 1 R l ( 25 )
##EQU00015##
The bound on the signed curvature .kappa..sub.l(t) can only be
satisfied if its absolute maximum max.sub.t .kappa..sub.l(t)
satisfies the bound. A pleasingly simple criterion ensues.
.rho. M l .ltoreq. 1 1 + q q 2 ( .rho. Li + .rho. Po ) 2 - 1 27 ( 1
+ 2 q ) 3 R l 2 ( 26 ) ##EQU00016##
5. Constraints from Multiple Pockets
[0091] In total, the pocket rotor has to be able to carry n.sub.P
pockets. This imposes a limitation on the maximal angle of opening
of the pocket. The total opening angle of the pocket .theta..sub.P
is given by
.theta. P = .alpha. P + q ( .phi. M c - .alpha. L ) .ltoreq. 2 .pi.
n P ( 27 ) ##EQU00017##
This criterion only tests for the size of the pockets on the
circumference of the lobe rotor. In addition, the pockets need to
be well separated in the interior of the rotor as well. This can be
checked easily by drawing a given shape of the pockets for a set of
input parameters.
6. Geometry of the Lobe
[0092] So far, we have not mentioned the shape of the trailing edge
of the lobe. As this element has no function other than ensuring
mechanical stability of the lobe, it can be designed freely except
having to avoid colliding with the pocket rotor. The maximum
allowed angle between the tip of the lobe and its trailing edge at
the base .gamma..sub.L is therefore limited to the value
.gamma. L .ltoreq. .gamma. L max = .theta. P q - .phi. M c =
.alpha. P q - .alpha. L . ( 28 ) ##EQU00018##
Typically, mechanical stability will require at least
.gamma..sub.L.sup.max>0. To extend this discussion, we consider
the constraint arising from the need that the lobe evacuates the
interior of the pocket rotor quickly enough to prevent a collision
with the trailing edge of the pocket rotor. The most protruding
feature of the trailing edge of the pocket is the point {tilde over
(T)} on the outer radius of the pocket which meets the tip of the
lobe T at time t=.phi..sub.M.sup.c-.alpha..sub.L. In a coordinate
system (x, y).sup.L defined as co-rotating with the lobe, and
oriented such that the tip of the lobe lies on its y-axis, the
trailing edge of the pocket defines the curve
r .fwdarw. T _ L = ( R - .rho. Po cos [ .beta. ( t ) ] ) ( sin (
.phi. M c - t ) cos ( .phi. . M c - t ) ) + .rho. Po sin [ .beta. (
t ) ] ( - cos ( .phi. M c - t ) sin ( .phi. M c - t ) ) , ( 29 )
##EQU00019##
introducing the abbreviation
.beta.(t)=.alpha..sub.P+q(.phi.M.sub.cv-.alpha..sub.L-t). The lobe
needs to be slim enough not to touch or cross this curve at any
point.
E. Maximizing the Lobe Length
[0093] Given the criteria for validity of a rotor configuration
discussed in section D, we may now ask which parameters are most
favorable in order to maximize the volume of the transient
compression chamber. A large fraction of the volume is swept by the
lobe rotor. It is thus useful to increase the length of the outer
lobe radius .rho..sub.Lo. Rather than thinking of increasing
.rho..sub.Lo, we may equivalently reduce .rho..sub.Po followed by a
resealing of all length parameters such as to recover the dame
overall size of the engine. Independently, minimizing the inner
lobe radius .rho..sub.Li also contributes to an increase of the
total swept volume.
[0094] The criterion which limits both these types of change is the
condition on the curvature of the main lobe segment .sub.l, Eq.
(26), which we can reformulate equivalently to read
.rho. Po + .rho. Li .ltoreq. 1 q ( 1 + q ) 2 .rho. M l 2 + 1 27 ( 1
+ 2 q ) 3 R l 2 ( 30 ) ##EQU00020##
Rotor configurations that maximize swept volume correspond to
parameters such that (30) is nearly satisfied as an equality. In
particular, previously disclosed rotor configurations in patents
WO-A-91/06747 and GB98/00345 did not approach this criterion very
closely. Even while keeping the ratio of the outer lobe radii
.rho..sub.Lo/.rho..sub.Po constant, the maximal 2D area for a
system of rotors with .rho..sub.Po=.rho..sub.Lo can be increased
substantially by reducing .rho..sub.Li. To illustrate the effect of
this modification, we modify the parameters of the engine
previously disclosed in U.S. Pat. No. 6,176,695. One can easily
achieve .rho..sub.Li/.rho..sub.Lo=1/4 as opposed to the value
.rho..sub.Li/.rho..sub.Lo=1/2 given in prior art. In FIG. 10, we
enclose a drawing of this particular configuration.
[0095] With regard to the other criteria, Eq. (23) can always be
fulfilled by choosing R.sub.c sufficiently large. However, the
remaining constraints are non-trivial. In particular, when
.rho..sub.Li is minimized, this may lead to violations of the
triangle relations (22a-c), such that .rho..sub.M.sub.l needs to be
increased while .rho..sub.Li is decreased.
F. Variants and Example Configurations
[0096] Above, we have given an explicit construction of a geometry
which implements the concept of a rotary displacement device with a
compression chamber formed by a lobe and pocket rotor that are
touching in two points of close contact. The lobe geometry
described above consists of precisely two arc segments .sub.l and
.sub.c, however, this is not the only possible way of constructing
a geometry in the spirit of patent no. WO-A-91/06747.
1. Lobe Formed of a Single Arc
[0097] As a special case of the construction presented in this
appendix, it is possible to obtain a geometry in which the lobe
consists of a single arc segment .sub.l, which touches the lobe
core tangentially. In this case, the points M.sub.l, M.sub.c and
.sub.L lie on a single line, and the arc .sub.c does then not
define any portion of the lobe and R.sub.c is not a relevant
parameter (can be formally chosen to be any positive number). In
addition, the triangle relations (22a-c) can be disregarded, and
.rho..sub.M.sub.l=R.sub.l-.rho..sub.Li. FIGS. 5 and 6 show example
configurations where the lobe consists only of a single arc segment
in this fashion.
2. Lobe Formed of More than Two Arc Segments
[0098] Following the same geometrical principles, a lobe can be
built up from multiple arc segments of different curvature.
Generalising the construction given above, the condition defining
whether a geometry can be realized is the criterion of non-reversal
of the velocity of the inner locus akin to Eq. (26). The main
difference arising in the case of multiple arc segments is to
replace this equation by a condition of the momentary curvature of
the trajectory of the relevant center point for a given segment of
the lobe. Generally, the structure of the lobe will be similar to
that given in the model of two arcs: the base of the lobe is a
convex piece either given by the inner core or a circle segment
tangential to it as in the case of the single arc structure in
section F1. The next portion of the lobe is concave, and the
portion near the lobe tip is again convex. Each of these portions
can in principle be composed of multiple arc segments of varying
curvature.
[0099] To display the versatility of the given construction with
two arc segments, a number of possible configurations are included
in the section of drawings.
* * * * *