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Geometrical analysis of a quadrilateral rotary piston engine
Mechanism and Machine Theory 93 (2015) 112–126
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Geometrical analysis of a quadrilateral rotary piston engine Osama M. Al-Hawaj Mechanical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
a r t i c l e
i n f o
Article history: Received 6 April 2015 Received in revised form 19 June 2015 Accepted 22 June 2015 Available online xxxx Keywords: Internal combustion engine Trochoid Geometrical analysis Rotary engine Quadrilateral
a b s t r a c t This paper presents geometrical analysis of a novel rotary device suitable for operation as an engine, an expander and a pump. The device comprises a stator having an elongated oval-shaped chamber encompassing a quadrilateral deformable rotor. The analysis includes derivation of the closed-form equations describing the inner stator profiles, compatible rotor-segment profiles, volume variation, compression ratio, and displacement volume. The study shows the pivotal importance of the chamber eccentricity ratio and inner offset parameters on the performance of the device. Verification of the analytical results was made by comparing them with those obtained by numerical integration at selected points. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The evolution of mechanisms and machines covering the aspects of theory and design has been addressed in [1–3]. This paper presents a theoretical study for an innovative rotary device referred to as rhomboidal rotary machine (RRM) or quadrilateral rotary machine (QRM). The device is a positive displacement one and configurable to operate as a power-producing device such as an expander or an engine equivalent to four-stroke internal combustion engine, as depicted schematically in Fig. 1, or as a power-consuming device such as a pump or a compressor. One class of rotary devices, which bears some relevance to the present device, is termed planetary rotation machines or trochoidaltype machines [4]. These machines comprise two components, a rotor and a stator (i.e. a chamber), whereby the rotor in the stator executes a general plane motion (i.e. a combination of rotation and curvilinear translation along a circular orbit) whereas the rotor is continuously in contact with the inner stator at a number of points, and hence defines a multiplicity of varying volume subchambers. In such machines, one of the two components (i.e. rotor or stator) comprises a profile mathematically defined by a trochoid, and the other respective component (i.e. rotor or stator) is referred to as an envelope. The envelope comprises a family of curves that include limiting cases referred to as conjugate envelopes. Examples of such devices include trochoidal-type rotary pump and Wankel machine [5]. The present device under consideration, QRM, primarily differs from the trochoidal-type machines, firstly, in the adoption of a deformable rotor as opposed to rigid rotor and, secondly, in the execution of a pure rotational motion of the rotor assembly as opposed to a combination of rotation and curvilinear translation motion (Table 1). According to the above table, the pure rotational motion and gearless transmission represent the important advantages of QRM over a Wankel machine. Historically, the earliest reporting of the concept of QRD was made in a US patent by E.H. Werner, 1902 [6], and this followed by a number of patents [7–11]. Literature search on the subject of QRM yields no theoretical work on the subject, and thereby, the present study, according to author’s knowledge, is the first theoretical investigation on the subject.
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.mechmachtheory.2015.06.014 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
O.M. Al-Hawaj / Mechanism and Machine Theory 93 (2015) 112–126
(a)
113
(b)
Fig. 1. Quadrilateral rotary device (a) rhombus orientation (b) square orientation.
This study focuses on the geometrical aspects of QRM, as presented on the following sections. Section 2 derives the mathematical equation for the inner stator profile. Section 3 identifies potential rotor segment profiles compatible with inner stator profiles. Section 4 derives closed-form equations of volume variation for different classes of rotor-stator profiles. Sections 5 and 6 derive closed-form expressions for compression ratio and displacement volumes. Section 7 addresses the effect of apex seal protrusion on performance. Section 8 discusses the results and provides numerical verification of the results. Lastly, Section 9 summarizes important findings and suggests topics for future work. 2. Stator geometry Although the general oval-shaped profile is well known in the literature for such devices, no exact mathematical definitions were reported upon reviewing the literature. Thus, the objective of this section is to derive the exact mathematical formula for such profiles. The selected profile must permit the inclusion of deformable quadrilateral rotor that rotates without seizure while making constant contact at four points. For this purpose, consider a symmetrical oval-shaped profile, shown schematically in Fig. 2, oriented with major axis along the horizontal x-axis and having a characteristic length, R, and eccentricity, e, such that major and minor radii are equal to R + e and R − e, respectively. Using dimensionless polar coordinates, (r, θ), where r = r′/R and θ measured from the horizontal positive in counterclockwise direction, and the eccentricity parameter, k, defined as k = e/R . Now, the quadrangular rotor with four vertices as represented schematically by a rhombus shape with four equal sides and two perpendicular diagonals is equivalent to a set of four mirror-imaged right angle triangles. For the purpose of derivation, it is sufficient to consider a single right angle triangle as depicted in the figure. Thus, considering a rotating right-angled triangle having a fixed dimensionless hypotenuse of length, L (i. e. L = L′/R), rotating in a counterclockwise direction about its right angle vertex, 0, coinciding with the center of the oval-shaped profile with both leading vertex pl and trailing vertex pt tracing the oval-shaped profile. Letting the angular positional vector of the triangle coincide with a line bisecting the 90° vertex such that when θ = 0, the right angle hypotenuse aligns vertically, and when θ = π/2 the hypotenuse, it aligns horizontally. Hence, at any angular position, the triangle is mathematically described by the Pythagorean relation as π 2 π 2 2 r l θ þ ; k þ r t θ− ; k ¼ L ; 4 4
ð1Þ
where rl and rt represents the leading and trailing sides of the rotating triangle. Among the different available oval-shaped profiles, the one satisfying the above equation is to be derived in the following paragraph. Considering a general oval-shaped profile described in polar coordinates by the general equation 0
r ðθÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 þ C 2 cosðθÞ2 ;
Table 1 Comparison between QRM and Wankel machine.
Chamber shape Rotor shape Rotor flexibility Rotor speed Rotor motion Power transmission
QRM
Wankel
Epitrochoid Quadrangle Deformable Equal shaft speed Pure rotation Gearless
Epitrochoid Triangular Fixed One-third shaft speed A combination of rotation plus curvilinear translation Internal gear
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Fig. 2. A schematic of a rotating right angle triangle and chamber profile.
where C1 and C2 are constants to be determined from the conditions 0
r ð0Þ ¼ R þ e: 0 r ðπ=2Þ ¼ R−e; This gives 2
C 1 ¼ ðR−eÞ C 2 ¼ 4Re
Normalizing by R and employing the definitions r = r′/R, and k = e/R, gives 2
2
2
r ¼ ð1−kÞ þ 4k cos θ;
ð2Þ
It can be shown that the above equation is identically satisfied by the following parametric equations: x ¼ k cos3 θ þ cos θ
ð3aÞ
y ¼ k sin3 θ þ sin θ
ð3bÞ
Substituting Eq. (2) in Eq. (1) and simplifying it gives the following relation: L¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k2 þ 1
ð4Þ
The above result being independent of angular position, θ, is proof of the claim that Eqs. (2), (3a), and (3b) describe the locus of points traced by the two vertices of a rotating right angle triangle about its right angle vertex coinciding with oval-shape center with fixed hypotenuse. Thus, the above equations describe the only valid oval-shaped profile defining the inner stator wall within which a concentrically disposed deformable quadrilateral rotor rotates without seizure while making contact at the four vertices. Also, Eq. (4) gives a relation between the dimensionless rotor segment length, L (i.e. hypotenuse length), and the stator eccentricity ratio, k. Moreover, the above equation provides a means for sizing the rotor given the eccentricity ratio and the characteristic length of the stator, or alternatively for sizing the stator given the dimension of segment length and characteristic length. It is notable that the dimensionless length segment is equal to square root of sum of squared normalized major radius, k + 1, and minor radius, k − 1. Incidentally, Eq. (2) and its parametric form, Eq. (3a,3b), are recognizable as equations for an epitrochoid defined as the locus of a point fixed to circle rolling without slipping on a bigger fixed circle having twice the radius. Furthermore, according to the doublegeneration theorem, the above equation is also the equation for a peritrochoid defined as the locus of a tip point of an arm fixed on a revolving larger circle of radius R as it rolls on a smaller base circle without slipping. In certain applications, the leading and trailing vertices are disposed at an inner offset with respect to the inner stator profile due to a variation in design or to account for the presence of an apex seal protrusion, and thereby defining a clearance, δ (i. e. δ = δ′/R). Accordingly, a parallel epitrochoid curve modification of Eq. (2), takes the following form: 2 2 2 2 r ¼ ð1−δÞ þ k þ 2kð1−δÞ 2 cos θ−1
ð5Þ
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And the dimensionless length of rotor segment, L (i.e. hypotenuse length) becomes
L¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 k2 þ ð1−δÞ2
ð6Þ
Fig. 3 illustrates the effect of eccentricity parameter, k, on the shape of the stator profile. In general, larger values of eccentricity result in elongation of the oval shape along major radii and contraction at minor radii with the appearance of necking after the critical value of the eccentricity parameter. On the other hand, the opposite is true for decreasing values of eccentricity such that the oval shape assumes, expectedly, a circular shape as the eccentricity approaches zero. 3. Rotor geometry Having determined the exact mathematical form of the inner stator profile, the second objective of this study was to determine feasible rotor-segment profiles compatible with the epitrochoidal stator profile. The quadrilateral rotor comprises four hinged segments defining a rhombus shape with equal sides that change orientation from a square to a rhombus shape as it rotates without seizure within a predetermined stator profile, as schematically depicted in Fig. 4. The rhombus and square shapes are parallelogram shapes distinguished by having all equal sides, whereas the rhombus shape is analogically described as a square shape deformed symmetrically along the diagonal resulting in equal opposite angles. Practically, the quadrilateral rotor segment has a finite thickness comprising an outer profile ρ(θ) and an inner profile. The inner segment profile, not of interest in this study, is arbitrary but must be configured with consideration to strength, weight, and to accommodate a power transmission mechanism which may include a medial pivot joint connected to a crank fixedly coupled to a power shaft concentric with the rotor and stator geometric center. The outer profile ρ(θ) is to be selected from a number of possible profiles ranging from flat to curved profiles, whereas the curved profiles can have either a medially concave or convex geometry. The important consideration in the selection of outer rotor profiles primarily includes compatibility with the stator profile, performance and cost of manufacturing. Mathematically, the outer rotor profile comprises four identical segment curve sequentially connected to form a quadrangle, which can be described in polar coordinates by a piecewise continuous function that changes shape from square to rhombus four times as it completes one revolution about its geometric center. Considering the square orientation for convenience, the quadrilateral rotor can be described mathematically by the following equation: 8 ρðθÞ > > > π > > < ρ θ− 2 ρðθÞ ¼ ρðθ−πÞ > > > > > ρ θ− 3π : 2
−π=4 ≤θbπ=4 π=4≤θb3π=4 3π=4≤θb5π=4 5π=4≤θb7π=4
Fig. 3. Effect of eccentricity on stator inner profile shape.
ð7Þ
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Fig. 4. Schematic of quadrilateral rotor orientations.
For proper operation as a positive displacement device, the QRM segment profile must satisfy the conditions of symmetry with respect to a radial line bisecting the segment, sealing contact at the ends of segment and inclusiveness at all intermediate points and in particular the midpoint. Hence, considering a rotor in square orientation for convenience, a segment profile, ρ(θ), and an inner stator profile, r(θ, k), the above constraints are mathematically expressed by the following equations: • Symmetry ρðt Þ ¼ ρð−t Þ 0 ≤t ≤
π 4
ð8aÞ
• Sealing
ρ
ð2n−1Þπ ð2n−1Þπ ¼r ; k n ¼ 1; 2; 3; 4 4 4
ð8bÞ
• Inclusiveness nπ nπ ρ ; k n ¼ 1; 2; 3; 4 ≤r 2 2
ð8cÞ
Taking into account the above consideration, two major families of segment rotor profiles are identified for which the analytic mathematical forms are described below: 3.1. Flat rotor segment This represents the simplest family of profiles and it can be used as an approximation for curved segment profile that is predominantly flat except for end roundedness at vertices. The equation for the upper horizontal flat segment is expressed in polar coordinates as ρðt Þ ¼
d cosðt Þ
0≤t ≤
π 4
ð9aÞ
Recognizing that t ¼ π2 −θ and d equal half the segment length, the above equation can be rewritten as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1 2 ρðθÞ ¼ π cos −θ 2
π π ≤θ ≤ 4 2
ð9bÞ
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To satisfy the inclusiveness condition, Eq. (8c), the inner stator profile must satisfy the following limiting condition: ρðπ=2Þ ¼ r
π ; k0 2
This condition leads to the following quadratic equation in k0, 2
k0 −4k0 þ 1 ¼ 0 The solution of the above quadratic equations gives the two roots k0 ¼ 2
pffiffiffi 3
Excluding the non-physical root leading to eccentricity greater than unity, the acceptable solution becomes pffiffiffi k0 ¼ 2− 3 ¼ 0:286 Hence, Eq. (9b) can be rewritten to accommodate the limitation imposed by Eq. (13) such that the upper flat segment profile in square orientation can be rewritten as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1 2 ρðθÞ ¼ π cos −θ 2
π π ≤θ ≤ and 0≤k ≤k0 4 2
ð9cÞ
Examples of this family of flat rotor profiles are illustrated in Fig. 5. The limiting case of maximum stator eccentricity k0 results in a slightly necked stator profile with minimum volume splits into two equal smaller sub-volumes as depicted in Fig. 5a, and it is expected that the limiting case results in the smallest and maximum sub-volumes in square orientation. 3.2. Curved rotor segment As shown previously, the usage of a flat segment profile limits the maximum stator eccentricity parameter and thereby limits the expected performance. To circumvent such a limitation, it is desirable to consider a rotor with curved segment profile. Here, a curved profile having the same functional form as the inner stator profile and defined within a 90° interval symmetrically with respect to the minimum radii is considered. Hence, a general epitrochoidal curve function is chosen to describe the outer profile of the rotor segment. π π 2 2 2 2 π −θ −1 ≤θ ≤ ρ ¼ ð1−δr Þ þ kr þ 2kr ð1−δr Þ 2 cos 2 4 2
ð10Þ
Here, the two parameters, kr and δr , are the rotor eccentricity and rotor inner offset parameters, respectively. Obviously, the above equation satisfies the symmetry condition, Eq. (8a), the two parameters are to be chosen to satisfy the other two constraints. The following two subsections identify and describe two potential curved rotor segment families classified equal and unequal rotor-stator eccentricity. 3.2.1. Equal stator-rotor eccentricity (i.e. k = kr) Considering equal eccentricity (i.e. k = kr), Eq. (10) takes the following form: π π 2 2 2 2 π −θ −1 ≤θ ≤ ρ ¼ ð1−δr Þ þ k þ 2kð1−δr Þ 2 cos 2 4 2
ð11Þ
The profile described by Eq. (11) satisfies the symmetry and inclusiveness conditions while violating the sealing condition at the vertices due to the presence of clearance. This dilemma is circumvented by considering a rotor with apex seals such that the protruded portion of the apex seal has a dimensionless radial length equal to δr to fill the clearance gap at the vertices. Examples of this family of curved rotor profiles are illustrated in Fig. 6. As illustrated in the figure, due to equal rotor-stator eccentricity, the rotor segment convexity is equal stator convexity at minimum radius, and the rotor’s shape is highly dependent on the stator’s shape. 3.2.2. Unequal stator-rotor eccentricity (i.e. kr ≥ k) In certain applications, specific performance level with a specific stator eccentricity value are not attainable using flat or curved rotor of equal rotor-stator eccentricity designs without further rotor segment profile modification such as the inclusion of recess volume or multi-humps. Alternatively, specific performance level with a specific stator eccentricity can be attained using curved rotor segment profile having an eccentricity greater than stator eccentricity.
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Fig. 5. Schematic of flat segment QRM in square orientation (effect of eccentricity): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
For this purpose, consider a stator profile as described by Eq. (2) with eccentricity k and zero offset, and a rotor segment profile described by Eq. (11) with eccentricity kr greater than stator eccentricity k and inner offset parameter δr. The rotor eccentricity kr being greater than stator eccentricity together with the inner offset parameter, δr, assures the satisfaction of the inclusiveness
Fig. 6. Schematic of curved segment QRM in square orientation (effect of eccentricity) (k = kr, δ N 0): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
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condition, Eq. (8c), at least proximal to the minor radii, which is of importance. The sealing condition constraint, Eq. (8b), requires the following relation be satisfied at both ends of the segment interval, and because of the symmetry, only one end condition needs to be satisfied. Substituting Eq. (11) and Eq. (2) into Eq. (8b) yields a quadratic equation in kr with an acceptable root satisfying the condition of positive eccentricity as given by the relation kr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ δr ð2−δr Þ
ð12Þ
By substitution in Eq. (11), the segment profile takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π π 2 2 2 π ρ ¼ 1 þ k þ 2ð1−δr Þ k2 þ δr ð2−δr Þ 2 cos −θ −1 ≤θ≤ 2 4 2
ð13Þ
Figs. 7 and 8 are examples of unequal rotor-stator configuration that illustrate the effect stator eccentricity, k, and rotor inner offset parameter, δr on relative rotor shape with respect to stator profile, respectively. In general, the curved rotor segment profile depends on the rotor eccentricity, and that depends on stator eccentricity and inner offset parameter. The effect of stator eccentricity is to limit rotor eccentricity, not to exceed stator eccentricity, and the effect of inner offset parameter is to increase rotor convexity beyond this limiting value. Depending on the combination of the two parameters, the shape of the rotor ranges from a four-leaf shape to an approximately square shape with rounded corners. Moreover, for small values of the two parameters, the rotor shape approaches a circular shape and the case reduces to the special case of equal rotor-stator eccentricity. 4. Volume variation The volume of the working chamber of the quadrilateral device is defined as the product of the side area enclosed by rotor segment contour and the inner surface of the epitrochoid and the width of the rotor casing. Considering a unit width, the normalized chamber 0 volume is equal to the normalized side chamber area, Ac, ði:e: Ac ¼ Ac =R2 Þ. Here, the reference area taken as R2 is one-fourth of a square area with side length 2R encompassing a circle of radius, R, corresponding to an epitrochoid with zero eccentricity. In the following section, the analytical equations describing the volume variation for flat and curved rotor segments are derived.
Fig. 7. Schematic of curved segment QRM in square orientation (effect of eccentricity) (kr N k, δr = δ0): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
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Fig. 8. Schematic of curved segment QRM in square orientation (effect of offset δr) [kr N k, k = k0: (a) δ = δ0 = 0.05; (b) δ = 0.75δ0; (c) δ = 0.5δ0; (d) δ = 0.25 δ0].
4.1. Flat rotor segment In the flat rotor case, the lateral rotor segment area is flat and the rotor segment profile is a chord straight line joining the two vertices of the right angle triangle. Hence, the normalized side chamber area, Ac, is the area resulting from an intersection of the chord straight line with the inner stator profile, which can be calculated as the difference between the 90° epitrochoidal sector area, Asec, and the overlapping triangular area, Atrg, with an inner offset, δ, Thus, Ac ¼ Asec −Atrg; Z θþπ=4 2 Asec ¼ 0:5 r ðt; kÞdt
ð14Þ
θ−π=4
¼
π 2 2 1 þ k þ k 2 cos ðθÞ−1 4
π π Atrg ¼ 0:5 r θ þ ; k; δ r θ− ; k; δ 4 4
ð15Þ
ð16Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π π r θ− ; k; δ ¼ ð1−δÞ2 þ k2 þ 2kð1−δÞ 2 cos2 θ− −1 4 4
ð17aÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi π π 2 2 2 r θ þ ; k; δ ¼ ð1−δÞ þ k þ 2kð1−δÞ 2 cos θ þ −1 4 4
ð17bÞ
4.2. Curved rotor segment The side chamber area for a curved rotor segment is obtainable by modifying the flat chamber area by adding or subtracting an additional area, Ad , to account for the deviation from flatness or curved profile modification. Hence, the following relation expresses the chamber volume as Ac ¼ Asec −Atrg −Ad ;
ð18Þ
According to the above equation, positive values of Ad represent a curved segment profile, equivalent to a flat profile with an additional added hump. On the other hand, negative values of Ad represent a curved segment equivalent to a flat profile with a recess.
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Since Ad is a constant quantity and is independent of rotor orientation, it can be calculated at any arbitrary angular position. Conveniently, the square orientation is used such that Ad ¼ Asecr ðπ=2Þ−Atrg ðπ=2Þ;
Asecr ðπ=2Þ ¼
ð19Þ
ð1−δÞ2 þ k2 þ δr ð2−δr Þ π 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −ð1−δÞ k2 þ δr ð2−δr Þ
h i 2 2 Atrg ðπ=2Þ ¼ 0:5 ð1−δÞ þ k
Ad ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πδ ð2−δ Þ π 1 2 2 r − ð1−δÞ þ k þ r −ð1−δÞ k2 þ δr ð2−δr Þ 4 2 4
ð20Þ
ð21Þ
ð22Þ
4.2.1. Equal rotor-stator eccentricity This case is obtained by setting inner rotor offset δr to zero (i.e. δr = 0), which results in k = kr and accordingly Eq. (20) through Eq. (22) reduces to
Asecr ðπ=2Þ ¼
ð1−δÞ2 þ k2 π 4
−kð1−δÞ
h i 2 2 Atrg ðπ=2Þ ¼ 0:5 ð1−δÞ þ k
Ad ¼
i π 1 h 2 2 − ð1−δÞ þ k −kð1−δÞ 4 2
ð23aÞ
ð23bÞ
ð23cÞ
4.2.2. Unequal rotor-stator eccentricity This is the general case with the added allowance for a clearance offset at the vertices to account for apex seal protrusion, δ, and the curved area correction are described in Eq. (20)–Eq. (22). 5. Compression ratio In this section, expressions for compression ratio for the two families of rotor profile segments are be determined. 5.1. Flat rotor segment The maximum and minimum chamber volumes occurs when rotor assumes a square orientation, such that for the assumed chamber orientation and the chosen coordinates, this occur at rotor angular positions corresponding to θ = 0 and θ = π/2, respectively. π 2 1þk −1=2 þ δð1−0:5δÞ þ k 4 π 2 ¼ 1þk −1=2 þ δð1−0:5δÞ−k 4
Acfmx ¼
ð24Þ
Acfmn
ð25Þ
Hence, the compression ratio becomes. π 2 1þk −1=2 þ δð1−0:5δÞ þ k 4 CR f ¼ π 1 þ k2 −1=2 þ δð1−0:5δÞ−k 4
ð26Þ
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5.2. Curved rotor segment The maximum and minimum chamber volumes occur at angular positions 0 and π/2, respectively Acmx ¼ Acfmx −Ad ðk; δ; δr Þ
Acmx ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i πh 2 2 2ðδ−δr Þ þ δr −δ þ ð1−δÞ k2 þ δr ð2−δr Þ þ k 4
Acmn ¼ Acfmn −Ad ðk; δ; δr Þ
Acmn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i πh 2 2 2ðδ−δr Þ þ δr −δ þ ð1−δÞ k2 þ δr ð2−δr Þ –k 4
ð27aÞ
ð27bÞ
ð28aÞ
ð28bÞ
Thus, the compression ratio becomes CRc ¼
Acmx Acmn
ð29Þ
6. Displacement volume The dimensionless displacement volume, D, defined as the difference between maximum and minimum chamber volumes, is expressed by the equation, D ¼ Acmx −Acmn
ð30Þ
For both flat and curved rotor segment cases, the above equation reduces to D ¼ 2k
ð31Þ
This result is analogical to reciprocating machines in which the dimensionless displacement volume is equal to twice the dimensionless crank length. 7. Segment end recess The preceding analysis assumes that the vertex comprises a point in contact with the inner stator wall, which represents an approximation for one type of hinge design. In another design, as depicted in Fig. 1, the hinge is provided by a cylindrical hub of radius,
Fig. 9. Effect of stator eccentricity k on chamber volume variation for flat segment QRM [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
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Fig. 10. Effect of stator eccentricity k on chamber volume variation for curved rotor segment (k = kr) [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
rb , with the center defining the vertex and disposed at radial distance, rv, from the inner stator wall such that rv N rb and a seal with radial dimension, δs, is provided to fill the radial gap so that the following relation holds: r v ¼ r b þ δs
ð32Þ
In this design, a truncation of the outer profile is needed to accommodate a seal strip plus an additional gap recess to allow for flexing at the hinges as the rotor changes orientation from a rhombus to a square. This additional recess adds to the chamber volume and estimation of this gap is required for accurate estimation of chamber volumes and compression ratio. Fortunately, this additional end recess volume, Aer, is constant and can be readily calculated from the equations h i 2 2 Aer ¼ 0:5 ðr v −δÞ −r b Δβmx
ð33Þ
Δβmx ¼ βmx −βmn
ð34Þ
−1
β mx ¼ tan
1þk 1−k
ð35Þ
Fig. 11. Effect of stator eccentricity k on chamber volume variation for curved rotor segment QRM (k ≤ kr, δr = δ0), [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
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Fig. 12. Effect of rotor offset parameter δr on chamber volume variation for curved rotor segment (k ≤ kr, k = k0, δ0 = 0.05) [(a) δr = 3δ0; (b) δr = 2δ0; (c) δr = δ0; (d) δr = 0].
βmn ¼
π −βmx 2
ð36Þ
For example, given typical dimensionless values of rv = 0.1, δ = 0.01, rb = 0.05, and k = kr = k0, we get using the above equations, the following values for δs = 0.05, βmx = π/3, Δβmx = π/6 and Aer = 0.003. Considering a typical quadrilateral device with a flat segment, a maximum dimensionless area of 0.6 and a minimum dimensionless area of the order 0.05, the end recess area amounts to less than 0.5% of the maximum chamber volume and 6% of the minimum volume and its effect on the compression ratio is very small. Hence, the assumption of neglecting the apex seal recess is justifiable. 8. Discussion The closed-form analytic equations for volume variation and compression ratios for the three cases of flat and two curved rotors are displayed graphically in Figs. 9 through 15.
Fig. 13. Effect of offset parameter δ on compression ratio variation for flat segment QRM [(a) δ = δ0 = 0.05; (b) δ = 0.75 δ0; (c) δ = 0.50 δ0; (d) δ = 0.25 δ0; (e) δ = 0].
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125
Fig. 14. Effect of rotor inner offset δ on compression ratio for curved segment QRM, (k = kr) [(a) δ = δ0 = 0.05; (b) δ = 0.75δ0; (c) δ = 0.5δ0; (d) δ = 0.25 δ0].
Figs. 9 through 11 illustrate the three cases of volume variation over a half-cycle with stator eccentricity as a parameter. As expected, all three cases exhibit periodic character and larger volume variation as stator eccentricity increases, and all were reduced to the special case of constant volume as eccentricity becomes zero. For the case of equal rotor-stator eccentricity, the minimum chamber volume is insensitive to changes in stator eccentricity. This is expected since the minimum volume is approximately equal to the length of the curve segment times the clearance gap δ and because the latter being practically smaller by at least an order of magnitude, this variation is negligible. Moreover, the effect of inner rotor offset parameter, δr , on chamber volume variation for curved rotor segment as shown in Figs. 11 and 12 results in a shift in volume variation and is analogically similar in the effect clearance volume variation in reciprocating devices. Figs. 13 through 15 depict the compression ratio variation as a function of stator eccentricity with inner offset δ as a parameter corresponding to the three cases of flat and two curved segments. In general, all three cases display an increase in compression ratio as stator eccentricity increases and as inner offset decreases. This is expected since the effect of larger eccentricity results in elongation along the major axis and contraction at the minor axis, which results in larger maximum volume compared to minimum, and the strong influence of inner offset on magnitude of the minimum volume or equivalent clearance volume. Moreover, the compression ratio for the flat rotor segment as shown in Fig. 13, is limited by the maximum allowable eccentricity, k = 0.286, amounting to around 15. Also, for the case of curved rotor segment with equal-rotor eccentricity, as shown in Fig. 14, compression ratio versus stator
Fig. 15. Effect of rotor inner offset δ on compression ratio for curved segment QRM, (kr ≥ k) [(a) δr = δ0 = 0.05; (b) δr = 0.75 δ0; (c) δr = 0.5 δ0; (d) δr = 0.25 δ0].
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O.M. Al-Hawaj / Mechanism and Machine Theory 93 (2015) 112–126
Table 2 Comparison between analytic and numerical results of chamber areas. Case
θ
1 2 3 4
0 0 0 0
k
δ
δr
0.20 0.25 0.25 0.25
0 0 0.05 0.05
0 0 0 0.01
(radians)
Ac
Ac
Analytic
Numeric
0.4 0.5 0.56407 0.58364
0.4 0.5 0.56407 0.58364
Error%
0 0 0 0
eccentricity exhibits almost linear variation due to the negligible effect of eccentricity on minimum volume, and the minimum volume (i.e. equivalent clearance volume) has a scaling effect on compression ratio. In this paper, the closed-form equation for stator, rotor, volume, compression and displacement volume obtained represent the exact solutions. To verify the correctness of the results, it was sufficient to verify the analytic closed-form for volume variation at specific points using another method starting from the basic integral form. Acmx ¼ Asec ðk; 0Þ−Asecr ðk; δ; δr ; 0Þ; Z ¼ 0:5
π=4 −π=4
h
2
ð37aÞ
2
ρðk; δ; δr ; t Þ −r ðk; t Þ
i
dt
ð37bÞ
Table 2 presents a comparison between the analytical results obtained from Eqs. (19) to (22), and the results obtained from numerical computation of chamber areas using Eqs. (37a) and (37b). The numerical computation shows almost identical results, thus providing proof of the correctness of the analytic closed-form expressions for chamber volume equations. 9. Conclusions In this paper, the geometry of a quadrilateral device was theoretically investigated for the first time according to the author. The study includes the following findings: 1. Derivation of the closed-form equation for the oval-shaped inner stator profile of the QRM is made and it is shown to be mathematically described by an epitrochoid. 2. Derivation of the closed-form equations for segment rotor profile were compatible with the epitrochoidal stator profile. 3. The identification of three classes of rotor segment profiles: (a) Flat profiles (b) Curved epitrochoidal profile with eccentricity parameter equal to stator eccentricity (c) Curved epitrochoidal profile with eccentricity parameter greater than stator eccentricity 4. Derivation of closed-form equations for volume variation. 5. Derivation of closed-form equations for compression volumes. 6. Compression ratios range from low to limited for the flat rotor segment class, moderate to high for the unequal rotor-stator eccentricity class, and high for the equal rotor-stator eccentricity class. 7. The displacement volume is directly proportional to twice the stator eccentricity for all three cases of rotor segment profiles. 8. The effect of end segment truncation due to finite seal, apex seal and pivot is negligibly small. It is hoped that this study will shed some light on the characteristics of QRM and its potential applicability as an engine, expander, compressor and pumps. Also, it is hoped to provide a tool for engineers in designing and optimizing quadrilateral devices for the various stated applications. Future research will address the kinematics and dynamics of the seal under steady and transient conditions. Moreover, the vibratory characteristics of the QRM operating under varying loads and speeds is a subject worthy of future investigations. The author hopes that this study encourages further research on the quadrilateral device in different applications including internal combustion engines. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
A.D. Dimarogonas, Machine Design A CAD Approach, John Wiley and Sons, Inc., N.Y., 2001 T.G. Chondros, Archimedes life works and machines, J. Mech. Mach. Theory 45 (2010) 1766–1775. Marco Ceccarelli, Distinguished Figures in Mechanism and Machine Science: Their Contributions and Their Legacies, part 1, Springer, 2014. J.B. Shung, G.R. Pennock, Geometry for trochoidal-type machines with conjugate envelopes, Mech. Theory 29 (1) (1994) 25–42. F. Wankel, R.F. Ansdale, Rotary piston machines, Classification of Design Principles for Engines, Pumps and Compressors, Iliffe Books, London, 1965. Werner, E.H., Rotary engine, 1902, United state patents, US 716,970 Novak, A. 1965, Rotary engine, United States patents, US 3,196,854 Jordan, A. 1968, Rotary internal combustion engine, United States patents, US 3,369,529 Ishida, C. 1975, Rotary internal combustion engine, United States patents, US 3,918,415 Saint-Hilaire et al. 2000, Quasi-turbine zero vibration-continuous rotary engine compressor or pump. US patent no. 6,164,263. Al-Hawaj, O., 2014. Rotary mechanism with articulating rotor. United States patents, US 8,904,991.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Geometrical analysis of a quadrilateral rotary piston engine Osama M. Al-Hawaj Mechanical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
a r t i c l e
i n f o
Article history: Received 6 April 2015 Received in revised form 19 June 2015 Accepted 22 June 2015 Available online xxxx Keywords: Internal combustion engine Trochoid Geometrical analysis Rotary engine Quadrilateral
a b s t r a c t This paper presents geometrical analysis of a novel rotary device suitable for operation as an engine, an expander and a pump. The device comprises a stator having an elongated oval-shaped chamber encompassing a quadrilateral deformable rotor. The analysis includes derivation of the closed-form equations describing the inner stator profiles, compatible rotor-segment profiles, volume variation, compression ratio, and displacement volume. The study shows the pivotal importance of the chamber eccentricity ratio and inner offset parameters on the performance of the device. Verification of the analytical results was made by comparing them with those obtained by numerical integration at selected points. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The evolution of mechanisms and machines covering the aspects of theory and design has been addressed in [1–3]. This paper presents a theoretical study for an innovative rotary device referred to as rhomboidal rotary machine (RRM) or quadrilateral rotary machine (QRM). The device is a positive displacement one and configurable to operate as a power-producing device such as an expander or an engine equivalent to four-stroke internal combustion engine, as depicted schematically in Fig. 1, or as a power-consuming device such as a pump or a compressor. One class of rotary devices, which bears some relevance to the present device, is termed planetary rotation machines or trochoidaltype machines [4]. These machines comprise two components, a rotor and a stator (i.e. a chamber), whereby the rotor in the stator executes a general plane motion (i.e. a combination of rotation and curvilinear translation along a circular orbit) whereas the rotor is continuously in contact with the inner stator at a number of points, and hence defines a multiplicity of varying volume subchambers. In such machines, one of the two components (i.e. rotor or stator) comprises a profile mathematically defined by a trochoid, and the other respective component (i.e. rotor or stator) is referred to as an envelope. The envelope comprises a family of curves that include limiting cases referred to as conjugate envelopes. Examples of such devices include trochoidal-type rotary pump and Wankel machine [5]. The present device under consideration, QRM, primarily differs from the trochoidal-type machines, firstly, in the adoption of a deformable rotor as opposed to rigid rotor and, secondly, in the execution of a pure rotational motion of the rotor assembly as opposed to a combination of rotation and curvilinear translation motion (Table 1). According to the above table, the pure rotational motion and gearless transmission represent the important advantages of QRM over a Wankel machine. Historically, the earliest reporting of the concept of QRD was made in a US patent by E.H. Werner, 1902 [6], and this followed by a number of patents [7–11]. Literature search on the subject of QRM yields no theoretical work on the subject, and thereby, the present study, according to author’s knowledge, is the first theoretical investigation on the subject.
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.mechmachtheory.2015.06.014 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
O.M. Al-Hawaj / Mechanism and Machine Theory 93 (2015) 112–126
(a)
113
(b)
Fig. 1. Quadrilateral rotary device (a) rhombus orientation (b) square orientation.
This study focuses on the geometrical aspects of QRM, as presented on the following sections. Section 2 derives the mathematical equation for the inner stator profile. Section 3 identifies potential rotor segment profiles compatible with inner stator profiles. Section 4 derives closed-form equations of volume variation for different classes of rotor-stator profiles. Sections 5 and 6 derive closed-form expressions for compression ratio and displacement volumes. Section 7 addresses the effect of apex seal protrusion on performance. Section 8 discusses the results and provides numerical verification of the results. Lastly, Section 9 summarizes important findings and suggests topics for future work. 2. Stator geometry Although the general oval-shaped profile is well known in the literature for such devices, no exact mathematical definitions were reported upon reviewing the literature. Thus, the objective of this section is to derive the exact mathematical formula for such profiles. The selected profile must permit the inclusion of deformable quadrilateral rotor that rotates without seizure while making constant contact at four points. For this purpose, consider a symmetrical oval-shaped profile, shown schematically in Fig. 2, oriented with major axis along the horizontal x-axis and having a characteristic length, R, and eccentricity, e, such that major and minor radii are equal to R + e and R − e, respectively. Using dimensionless polar coordinates, (r, θ), where r = r′/R and θ measured from the horizontal positive in counterclockwise direction, and the eccentricity parameter, k, defined as k = e/R . Now, the quadrangular rotor with four vertices as represented schematically by a rhombus shape with four equal sides and two perpendicular diagonals is equivalent to a set of four mirror-imaged right angle triangles. For the purpose of derivation, it is sufficient to consider a single right angle triangle as depicted in the figure. Thus, considering a rotating right-angled triangle having a fixed dimensionless hypotenuse of length, L (i. e. L = L′/R), rotating in a counterclockwise direction about its right angle vertex, 0, coinciding with the center of the oval-shaped profile with both leading vertex pl and trailing vertex pt tracing the oval-shaped profile. Letting the angular positional vector of the triangle coincide with a line bisecting the 90° vertex such that when θ = 0, the right angle hypotenuse aligns vertically, and when θ = π/2 the hypotenuse, it aligns horizontally. Hence, at any angular position, the triangle is mathematically described by the Pythagorean relation as π 2 π 2 2 r l θ þ ; k þ r t θ− ; k ¼ L ; 4 4
ð1Þ
where rl and rt represents the leading and trailing sides of the rotating triangle. Among the different available oval-shaped profiles, the one satisfying the above equation is to be derived in the following paragraph. Considering a general oval-shaped profile described in polar coordinates by the general equation 0
r ðθÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 þ C 2 cosðθÞ2 ;
Table 1 Comparison between QRM and Wankel machine.
Chamber shape Rotor shape Rotor flexibility Rotor speed Rotor motion Power transmission
QRM
Wankel
Epitrochoid Quadrangle Deformable Equal shaft speed Pure rotation Gearless
Epitrochoid Triangular Fixed One-third shaft speed A combination of rotation plus curvilinear translation Internal gear
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Fig. 2. A schematic of a rotating right angle triangle and chamber profile.
where C1 and C2 are constants to be determined from the conditions 0
r ð0Þ ¼ R þ e: 0 r ðπ=2Þ ¼ R−e; This gives 2
C 1 ¼ ðR−eÞ C 2 ¼ 4Re
Normalizing by R and employing the definitions r = r′/R, and k = e/R, gives 2
2
2
r ¼ ð1−kÞ þ 4k cos θ;
ð2Þ
It can be shown that the above equation is identically satisfied by the following parametric equations: x ¼ k cos3 θ þ cos θ
ð3aÞ
y ¼ k sin3 θ þ sin θ
ð3bÞ
Substituting Eq. (2) in Eq. (1) and simplifying it gives the following relation: L¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k2 þ 1
ð4Þ
The above result being independent of angular position, θ, is proof of the claim that Eqs. (2), (3a), and (3b) describe the locus of points traced by the two vertices of a rotating right angle triangle about its right angle vertex coinciding with oval-shape center with fixed hypotenuse. Thus, the above equations describe the only valid oval-shaped profile defining the inner stator wall within which a concentrically disposed deformable quadrilateral rotor rotates without seizure while making contact at the four vertices. Also, Eq. (4) gives a relation between the dimensionless rotor segment length, L (i.e. hypotenuse length), and the stator eccentricity ratio, k. Moreover, the above equation provides a means for sizing the rotor given the eccentricity ratio and the characteristic length of the stator, or alternatively for sizing the stator given the dimension of segment length and characteristic length. It is notable that the dimensionless length segment is equal to square root of sum of squared normalized major radius, k + 1, and minor radius, k − 1. Incidentally, Eq. (2) and its parametric form, Eq. (3a,3b), are recognizable as equations for an epitrochoid defined as the locus of a point fixed to circle rolling without slipping on a bigger fixed circle having twice the radius. Furthermore, according to the doublegeneration theorem, the above equation is also the equation for a peritrochoid defined as the locus of a tip point of an arm fixed on a revolving larger circle of radius R as it rolls on a smaller base circle without slipping. In certain applications, the leading and trailing vertices are disposed at an inner offset with respect to the inner stator profile due to a variation in design or to account for the presence of an apex seal protrusion, and thereby defining a clearance, δ (i. e. δ = δ′/R). Accordingly, a parallel epitrochoid curve modification of Eq. (2), takes the following form: 2 2 2 2 r ¼ ð1−δÞ þ k þ 2kð1−δÞ 2 cos θ−1
ð5Þ
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And the dimensionless length of rotor segment, L (i.e. hypotenuse length) becomes
L¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 k2 þ ð1−δÞ2
ð6Þ
Fig. 3 illustrates the effect of eccentricity parameter, k, on the shape of the stator profile. In general, larger values of eccentricity result in elongation of the oval shape along major radii and contraction at minor radii with the appearance of necking after the critical value of the eccentricity parameter. On the other hand, the opposite is true for decreasing values of eccentricity such that the oval shape assumes, expectedly, a circular shape as the eccentricity approaches zero. 3. Rotor geometry Having determined the exact mathematical form of the inner stator profile, the second objective of this study was to determine feasible rotor-segment profiles compatible with the epitrochoidal stator profile. The quadrilateral rotor comprises four hinged segments defining a rhombus shape with equal sides that change orientation from a square to a rhombus shape as it rotates without seizure within a predetermined stator profile, as schematically depicted in Fig. 4. The rhombus and square shapes are parallelogram shapes distinguished by having all equal sides, whereas the rhombus shape is analogically described as a square shape deformed symmetrically along the diagonal resulting in equal opposite angles. Practically, the quadrilateral rotor segment has a finite thickness comprising an outer profile ρ(θ) and an inner profile. The inner segment profile, not of interest in this study, is arbitrary but must be configured with consideration to strength, weight, and to accommodate a power transmission mechanism which may include a medial pivot joint connected to a crank fixedly coupled to a power shaft concentric with the rotor and stator geometric center. The outer profile ρ(θ) is to be selected from a number of possible profiles ranging from flat to curved profiles, whereas the curved profiles can have either a medially concave or convex geometry. The important consideration in the selection of outer rotor profiles primarily includes compatibility with the stator profile, performance and cost of manufacturing. Mathematically, the outer rotor profile comprises four identical segment curve sequentially connected to form a quadrangle, which can be described in polar coordinates by a piecewise continuous function that changes shape from square to rhombus four times as it completes one revolution about its geometric center. Considering the square orientation for convenience, the quadrilateral rotor can be described mathematically by the following equation: 8 ρðθÞ > > > π > > < ρ θ− 2 ρðθÞ ¼ ρðθ−πÞ > > > > > ρ θ− 3π : 2
−π=4 ≤θbπ=4 π=4≤θb3π=4 3π=4≤θb5π=4 5π=4≤θb7π=4
Fig. 3. Effect of eccentricity on stator inner profile shape.
ð7Þ
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Fig. 4. Schematic of quadrilateral rotor orientations.
For proper operation as a positive displacement device, the QRM segment profile must satisfy the conditions of symmetry with respect to a radial line bisecting the segment, sealing contact at the ends of segment and inclusiveness at all intermediate points and in particular the midpoint. Hence, considering a rotor in square orientation for convenience, a segment profile, ρ(θ), and an inner stator profile, r(θ, k), the above constraints are mathematically expressed by the following equations: • Symmetry ρðt Þ ¼ ρð−t Þ 0 ≤t ≤
π 4
ð8aÞ
• Sealing
ρ
ð2n−1Þπ ð2n−1Þπ ¼r ; k n ¼ 1; 2; 3; 4 4 4
ð8bÞ
• Inclusiveness nπ nπ ρ ; k n ¼ 1; 2; 3; 4 ≤r 2 2
ð8cÞ
Taking into account the above consideration, two major families of segment rotor profiles are identified for which the analytic mathematical forms are described below: 3.1. Flat rotor segment This represents the simplest family of profiles and it can be used as an approximation for curved segment profile that is predominantly flat except for end roundedness at vertices. The equation for the upper horizontal flat segment is expressed in polar coordinates as ρðt Þ ¼
d cosðt Þ
0≤t ≤
π 4
ð9aÞ
Recognizing that t ¼ π2 −θ and d equal half the segment length, the above equation can be rewritten as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1 2 ρðθÞ ¼ π cos −θ 2
π π ≤θ ≤ 4 2
ð9bÞ
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To satisfy the inclusiveness condition, Eq. (8c), the inner stator profile must satisfy the following limiting condition: ρðπ=2Þ ¼ r
π ; k0 2
This condition leads to the following quadratic equation in k0, 2
k0 −4k0 þ 1 ¼ 0 The solution of the above quadratic equations gives the two roots k0 ¼ 2
pffiffiffi 3
Excluding the non-physical root leading to eccentricity greater than unity, the acceptable solution becomes pffiffiffi k0 ¼ 2− 3 ¼ 0:286 Hence, Eq. (9b) can be rewritten to accommodate the limitation imposed by Eq. (13) such that the upper flat segment profile in square orientation can be rewritten as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1 2 ρðθÞ ¼ π cos −θ 2
π π ≤θ ≤ and 0≤k ≤k0 4 2
ð9cÞ
Examples of this family of flat rotor profiles are illustrated in Fig. 5. The limiting case of maximum stator eccentricity k0 results in a slightly necked stator profile with minimum volume splits into two equal smaller sub-volumes as depicted in Fig. 5a, and it is expected that the limiting case results in the smallest and maximum sub-volumes in square orientation. 3.2. Curved rotor segment As shown previously, the usage of a flat segment profile limits the maximum stator eccentricity parameter and thereby limits the expected performance. To circumvent such a limitation, it is desirable to consider a rotor with curved segment profile. Here, a curved profile having the same functional form as the inner stator profile and defined within a 90° interval symmetrically with respect to the minimum radii is considered. Hence, a general epitrochoidal curve function is chosen to describe the outer profile of the rotor segment. π π 2 2 2 2 π −θ −1 ≤θ ≤ ρ ¼ ð1−δr Þ þ kr þ 2kr ð1−δr Þ 2 cos 2 4 2
ð10Þ
Here, the two parameters, kr and δr , are the rotor eccentricity and rotor inner offset parameters, respectively. Obviously, the above equation satisfies the symmetry condition, Eq. (8a), the two parameters are to be chosen to satisfy the other two constraints. The following two subsections identify and describe two potential curved rotor segment families classified equal and unequal rotor-stator eccentricity. 3.2.1. Equal stator-rotor eccentricity (i.e. k = kr) Considering equal eccentricity (i.e. k = kr), Eq. (10) takes the following form: π π 2 2 2 2 π −θ −1 ≤θ ≤ ρ ¼ ð1−δr Þ þ k þ 2kð1−δr Þ 2 cos 2 4 2
ð11Þ
The profile described by Eq. (11) satisfies the symmetry and inclusiveness conditions while violating the sealing condition at the vertices due to the presence of clearance. This dilemma is circumvented by considering a rotor with apex seals such that the protruded portion of the apex seal has a dimensionless radial length equal to δr to fill the clearance gap at the vertices. Examples of this family of curved rotor profiles are illustrated in Fig. 6. As illustrated in the figure, due to equal rotor-stator eccentricity, the rotor segment convexity is equal stator convexity at minimum radius, and the rotor’s shape is highly dependent on the stator’s shape. 3.2.2. Unequal stator-rotor eccentricity (i.e. kr ≥ k) In certain applications, specific performance level with a specific stator eccentricity value are not attainable using flat or curved rotor of equal rotor-stator eccentricity designs without further rotor segment profile modification such as the inclusion of recess volume or multi-humps. Alternatively, specific performance level with a specific stator eccentricity can be attained using curved rotor segment profile having an eccentricity greater than stator eccentricity.
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Fig. 5. Schematic of flat segment QRM in square orientation (effect of eccentricity): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
For this purpose, consider a stator profile as described by Eq. (2) with eccentricity k and zero offset, and a rotor segment profile described by Eq. (11) with eccentricity kr greater than stator eccentricity k and inner offset parameter δr. The rotor eccentricity kr being greater than stator eccentricity together with the inner offset parameter, δr, assures the satisfaction of the inclusiveness
Fig. 6. Schematic of curved segment QRM in square orientation (effect of eccentricity) (k = kr, δ N 0): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
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condition, Eq. (8c), at least proximal to the minor radii, which is of importance. The sealing condition constraint, Eq. (8b), requires the following relation be satisfied at both ends of the segment interval, and because of the symmetry, only one end condition needs to be satisfied. Substituting Eq. (11) and Eq. (2) into Eq. (8b) yields a quadratic equation in kr with an acceptable root satisfying the condition of positive eccentricity as given by the relation kr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ δr ð2−δr Þ
ð12Þ
By substitution in Eq. (11), the segment profile takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π π 2 2 2 π ρ ¼ 1 þ k þ 2ð1−δr Þ k2 þ δr ð2−δr Þ 2 cos −θ −1 ≤θ≤ 2 4 2
ð13Þ
Figs. 7 and 8 are examples of unequal rotor-stator configuration that illustrate the effect stator eccentricity, k, and rotor inner offset parameter, δr on relative rotor shape with respect to stator profile, respectively. In general, the curved rotor segment profile depends on the rotor eccentricity, and that depends on stator eccentricity and inner offset parameter. The effect of stator eccentricity is to limit rotor eccentricity, not to exceed stator eccentricity, and the effect of inner offset parameter is to increase rotor convexity beyond this limiting value. Depending on the combination of the two parameters, the shape of the rotor ranges from a four-leaf shape to an approximately square shape with rounded corners. Moreover, for small values of the two parameters, the rotor shape approaches a circular shape and the case reduces to the special case of equal rotor-stator eccentricity. 4. Volume variation The volume of the working chamber of the quadrilateral device is defined as the product of the side area enclosed by rotor segment contour and the inner surface of the epitrochoid and the width of the rotor casing. Considering a unit width, the normalized chamber 0 volume is equal to the normalized side chamber area, Ac, ði:e: Ac ¼ Ac =R2 Þ. Here, the reference area taken as R2 is one-fourth of a square area with side length 2R encompassing a circle of radius, R, corresponding to an epitrochoid with zero eccentricity. In the following section, the analytical equations describing the volume variation for flat and curved rotor segments are derived.
Fig. 7. Schematic of curved segment QRM in square orientation (effect of eccentricity) (kr N k, δr = δ0): (a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.5 k0; (d) k = 0.25 k0.
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Fig. 8. Schematic of curved segment QRM in square orientation (effect of offset δr) [kr N k, k = k0: (a) δ = δ0 = 0.05; (b) δ = 0.75δ0; (c) δ = 0.5δ0; (d) δ = 0.25 δ0].
4.1. Flat rotor segment In the flat rotor case, the lateral rotor segment area is flat and the rotor segment profile is a chord straight line joining the two vertices of the right angle triangle. Hence, the normalized side chamber area, Ac, is the area resulting from an intersection of the chord straight line with the inner stator profile, which can be calculated as the difference between the 90° epitrochoidal sector area, Asec, and the overlapping triangular area, Atrg, with an inner offset, δ, Thus, Ac ¼ Asec −Atrg; Z θþπ=4 2 Asec ¼ 0:5 r ðt; kÞdt
ð14Þ
θ−π=4
¼
π 2 2 1 þ k þ k 2 cos ðθÞ−1 4
π π Atrg ¼ 0:5 r θ þ ; k; δ r θ− ; k; δ 4 4
ð15Þ
ð16Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π π r θ− ; k; δ ¼ ð1−δÞ2 þ k2 þ 2kð1−δÞ 2 cos2 θ− −1 4 4
ð17aÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi π π 2 2 2 r θ þ ; k; δ ¼ ð1−δÞ þ k þ 2kð1−δÞ 2 cos θ þ −1 4 4
ð17bÞ
4.2. Curved rotor segment The side chamber area for a curved rotor segment is obtainable by modifying the flat chamber area by adding or subtracting an additional area, Ad , to account for the deviation from flatness or curved profile modification. Hence, the following relation expresses the chamber volume as Ac ¼ Asec −Atrg −Ad ;
ð18Þ
According to the above equation, positive values of Ad represent a curved segment profile, equivalent to a flat profile with an additional added hump. On the other hand, negative values of Ad represent a curved segment equivalent to a flat profile with a recess.
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Since Ad is a constant quantity and is independent of rotor orientation, it can be calculated at any arbitrary angular position. Conveniently, the square orientation is used such that Ad ¼ Asecr ðπ=2Þ−Atrg ðπ=2Þ;
Asecr ðπ=2Þ ¼
ð19Þ
ð1−δÞ2 þ k2 þ δr ð2−δr Þ π 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −ð1−δÞ k2 þ δr ð2−δr Þ
h i 2 2 Atrg ðπ=2Þ ¼ 0:5 ð1−δÞ þ k
Ad ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πδ ð2−δ Þ π 1 2 2 r − ð1−δÞ þ k þ r −ð1−δÞ k2 þ δr ð2−δr Þ 4 2 4
ð20Þ
ð21Þ
ð22Þ
4.2.1. Equal rotor-stator eccentricity This case is obtained by setting inner rotor offset δr to zero (i.e. δr = 0), which results in k = kr and accordingly Eq. (20) through Eq. (22) reduces to
Asecr ðπ=2Þ ¼
ð1−δÞ2 þ k2 π 4
−kð1−δÞ
h i 2 2 Atrg ðπ=2Þ ¼ 0:5 ð1−δÞ þ k
Ad ¼
i π 1 h 2 2 − ð1−δÞ þ k −kð1−δÞ 4 2
ð23aÞ
ð23bÞ
ð23cÞ
4.2.2. Unequal rotor-stator eccentricity This is the general case with the added allowance for a clearance offset at the vertices to account for apex seal protrusion, δ, and the curved area correction are described in Eq. (20)–Eq. (22). 5. Compression ratio In this section, expressions for compression ratio for the two families of rotor profile segments are be determined. 5.1. Flat rotor segment The maximum and minimum chamber volumes occurs when rotor assumes a square orientation, such that for the assumed chamber orientation and the chosen coordinates, this occur at rotor angular positions corresponding to θ = 0 and θ = π/2, respectively. π 2 1þk −1=2 þ δð1−0:5δÞ þ k 4 π 2 ¼ 1þk −1=2 þ δð1−0:5δÞ−k 4
Acfmx ¼
ð24Þ
Acfmn
ð25Þ
Hence, the compression ratio becomes. π 2 1þk −1=2 þ δð1−0:5δÞ þ k 4 CR f ¼ π 1 þ k2 −1=2 þ δð1−0:5δÞ−k 4
ð26Þ
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5.2. Curved rotor segment The maximum and minimum chamber volumes occur at angular positions 0 and π/2, respectively Acmx ¼ Acfmx −Ad ðk; δ; δr Þ
Acmx ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i πh 2 2 2ðδ−δr Þ þ δr −δ þ ð1−δÞ k2 þ δr ð2−δr Þ þ k 4
Acmn ¼ Acfmn −Ad ðk; δ; δr Þ
Acmn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i πh 2 2 2ðδ−δr Þ þ δr −δ þ ð1−δÞ k2 þ δr ð2−δr Þ –k 4
ð27aÞ
ð27bÞ
ð28aÞ
ð28bÞ
Thus, the compression ratio becomes CRc ¼
Acmx Acmn
ð29Þ
6. Displacement volume The dimensionless displacement volume, D, defined as the difference between maximum and minimum chamber volumes, is expressed by the equation, D ¼ Acmx −Acmn
ð30Þ
For both flat and curved rotor segment cases, the above equation reduces to D ¼ 2k
ð31Þ
This result is analogical to reciprocating machines in which the dimensionless displacement volume is equal to twice the dimensionless crank length. 7. Segment end recess The preceding analysis assumes that the vertex comprises a point in contact with the inner stator wall, which represents an approximation for one type of hinge design. In another design, as depicted in Fig. 1, the hinge is provided by a cylindrical hub of radius,
Fig. 9. Effect of stator eccentricity k on chamber volume variation for flat segment QRM [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
O.M. Al-Hawaj / Mechanism and Machine Theory 93 (2015) 112–126
123
Fig. 10. Effect of stator eccentricity k on chamber volume variation for curved rotor segment (k = kr) [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
rb , with the center defining the vertex and disposed at radial distance, rv, from the inner stator wall such that rv N rb and a seal with radial dimension, δs, is provided to fill the radial gap so that the following relation holds: r v ¼ r b þ δs
ð32Þ
In this design, a truncation of the outer profile is needed to accommodate a seal strip plus an additional gap recess to allow for flexing at the hinges as the rotor changes orientation from a rhombus to a square. This additional recess adds to the chamber volume and estimation of this gap is required for accurate estimation of chamber volumes and compression ratio. Fortunately, this additional end recess volume, Aer, is constant and can be readily calculated from the equations h i 2 2 Aer ¼ 0:5 ðr v −δÞ −r b Δβmx
ð33Þ
Δβmx ¼ βmx −βmn
ð34Þ
−1
β mx ¼ tan
1þk 1−k
ð35Þ
Fig. 11. Effect of stator eccentricity k on chamber volume variation for curved rotor segment QRM (k ≤ kr, δr = δ0), [(a) k = k0 = 0.286; (b) k = 0.75 k0; (c) k = 0.50 k0; (d) k = 0.25 k0; (e) k = 0].
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Fig. 12. Effect of rotor offset parameter δr on chamber volume variation for curved rotor segment (k ≤ kr, k = k0, δ0 = 0.05) [(a) δr = 3δ0; (b) δr = 2δ0; (c) δr = δ0; (d) δr = 0].
βmn ¼
π −βmx 2
ð36Þ
For example, given typical dimensionless values of rv = 0.1, δ = 0.01, rb = 0.05, and k = kr = k0, we get using the above equations, the following values for δs = 0.05, βmx = π/3, Δβmx = π/6 and Aer = 0.003. Considering a typical quadrilateral device with a flat segment, a maximum dimensionless area of 0.6 and a minimum dimensionless area of the order 0.05, the end recess area amounts to less than 0.5% of the maximum chamber volume and 6% of the minimum volume and its effect on the compression ratio is very small. Hence, the assumption of neglecting the apex seal recess is justifiable. 8. Discussion The closed-form analytic equations for volume variation and compression ratios for the three cases of flat and two curved rotors are displayed graphically in Figs. 9 through 15.
Fig. 13. Effect of offset parameter δ on compression ratio variation for flat segment QRM [(a) δ = δ0 = 0.05; (b) δ = 0.75 δ0; (c) δ = 0.50 δ0; (d) δ = 0.25 δ0; (e) δ = 0].
O.M. Al-Hawaj / Mechanism and Machine Theory 93 (2015) 112–126
125
Fig. 14. Effect of rotor inner offset δ on compression ratio for curved segment QRM, (k = kr) [(a) δ = δ0 = 0.05; (b) δ = 0.75δ0; (c) δ = 0.5δ0; (d) δ = 0.25 δ0].
Figs. 9 through 11 illustrate the three cases of volume variation over a half-cycle with stator eccentricity as a parameter. As expected, all three cases exhibit periodic character and larger volume variation as stator eccentricity increases, and all were reduced to the special case of constant volume as eccentricity becomes zero. For the case of equal rotor-stator eccentricity, the minimum chamber volume is insensitive to changes in stator eccentricity. This is expected since the minimum volume is approximately equal to the length of the curve segment times the clearance gap δ and because the latter being practically smaller by at least an order of magnitude, this variation is negligible. Moreover, the effect of inner rotor offset parameter, δr , on chamber volume variation for curved rotor segment as shown in Figs. 11 and 12 results in a shift in volume variation and is analogically similar in the effect clearance volume variation in reciprocating devices. Figs. 13 through 15 depict the compression ratio variation as a function of stator eccentricity with inner offset δ as a parameter corresponding to the three cases of flat and two curved segments. In general, all three cases display an increase in compression ratio as stator eccentricity increases and as inner offset decreases. This is expected since the effect of larger eccentricity results in elongation along the major axis and contraction at the minor axis, which results in larger maximum volume compared to minimum, and the strong influence of inner offset on magnitude of the minimum volume or equivalent clearance volume. Moreover, the compression ratio for the flat rotor segment as shown in Fig. 13, is limited by the maximum allowable eccentricity, k = 0.286, amounting to around 15. Also, for the case of curved rotor segment with equal-rotor eccentricity, as shown in Fig. 14, compression ratio versus stator
Fig. 15. Effect of rotor inner offset δ on compression ratio for curved segment QRM, (kr ≥ k) [(a) δr = δ0 = 0.05; (b) δr = 0.75 δ0; (c) δr = 0.5 δ0; (d) δr = 0.25 δ0].
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Table 2 Comparison between analytic and numerical results of chamber areas. Case
θ
1 2 3 4
0 0 0 0
k
δ
δr
0.20 0.25 0.25 0.25
0 0 0.05 0.05
0 0 0 0.01
(radians)
Ac
Ac
Analytic
Numeric
0.4 0.5 0.56407 0.58364
0.4 0.5 0.56407 0.58364
Error%
0 0 0 0
eccentricity exhibits almost linear variation due to the negligible effect of eccentricity on minimum volume, and the minimum volume (i.e. equivalent clearance volume) has a scaling effect on compression ratio. In this paper, the closed-form equation for stator, rotor, volume, compression and displacement volume obtained represent the exact solutions. To verify the correctness of the results, it was sufficient to verify the analytic closed-form for volume variation at specific points using another method starting from the basic integral form. Acmx ¼ Asec ðk; 0Þ−Asecr ðk; δ; δr ; 0Þ; Z ¼ 0:5
π=4 −π=4
h
2
ð37aÞ
2
ρðk; δ; δr ; t Þ −r ðk; t Þ
i
dt
ð37bÞ
Table 2 presents a comparison between the analytical results obtained from Eqs. (19) to (22), and the results obtained from numerical computation of chamber areas using Eqs. (37a) and (37b). The numerical computation shows almost identical results, thus providing proof of the correctness of the analytic closed-form expressions for chamber volume equations. 9. Conclusions In this paper, the geometry of a quadrilateral device was theoretically investigated for the first time according to the author. The study includes the following findings: 1. Derivation of the closed-form equation for the oval-shaped inner stator profile of the QRM is made and it is shown to be mathematically described by an epitrochoid. 2. Derivation of the closed-form equations for segment rotor profile were compatible with the epitrochoidal stator profile. 3. The identification of three classes of rotor segment profiles: (a) Flat profiles (b) Curved epitrochoidal profile with eccentricity parameter equal to stator eccentricity (c) Curved epitrochoidal profile with eccentricity parameter greater than stator eccentricity 4. Derivation of closed-form equations for volume variation. 5. Derivation of closed-form equations for compression volumes. 6. Compression ratios range from low to limited for the flat rotor segment class, moderate to high for the unequal rotor-stator eccentricity class, and high for the equal rotor-stator eccentricity class. 7. The displacement volume is directly proportional to twice the stator eccentricity for all three cases of rotor segment profiles. 8. The effect of end segment truncation due to finite seal, apex seal and pivot is negligibly small. It is hoped that this study will shed some light on the characteristics of QRM and its potential applicability as an engine, expander, compressor and pumps. Also, it is hoped to provide a tool for engineers in designing and optimizing quadrilateral devices for the various stated applications. Future research will address the kinematics and dynamics of the seal under steady and transient conditions. Moreover, the vibratory characteristics of the QRM operating under varying loads and speeds is a subject worthy of future investigations. The author hopes that this study encourages further research on the quadrilateral device in different applications including internal combustion engines. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
A.D. Dimarogonas, Machine Design A CAD Approach, John Wiley and Sons, Inc., N.Y., 2001 T.G. Chondros, Archimedes life works and machines, J. Mech. Mach. Theory 45 (2010) 1766–1775. Marco Ceccarelli, Distinguished Figures in Mechanism and Machine Science: Their Contributions and Their Legacies, part 1, Springer, 2014. J.B. Shung, G.R. Pennock, Geometry for trochoidal-type machines with conjugate envelopes, Mech. Theory 29 (1) (1994) 25–42. F. Wankel, R.F. Ansdale, Rotary piston machines, Classification of Design Principles for Engines, Pumps and Compressors, Iliffe Books, London, 1965. Werner, E.H., Rotary engine, 1902, United state patents, US 716,970 Novak, A. 1965, Rotary engine, United States patents, US 3,196,854 Jordan, A. 1968, Rotary internal combustion engine, United States patents, US 3,369,529 Ishida, C. 1975, Rotary internal combustion engine, United States patents, US 3,918,415 Saint-Hilaire et al. 2000, Quasi-turbine zero vibration-continuous rotary engine compressor or pump. US patent no. 6,164,263. Al-Hawaj, O., 2014. Rotary mechanism with articulating rotor. United States patents, US 8,904,991.