The mathematical modelling of the work spaces with variable volume in the Wankel rotary engine case.

Catas, Adriana ; Chioreanu, Nicolae ; Blaga, Vasile 等

1. GENERAL CONSIDERATIONS

The Wankel rotary engine is the most known rotary heat engine, patented in 1954 by Felix Wankel. The engine is built by a carcass which has an inside surface with an appropriate geometry. Inside the carcass a rotary piston is mounted over an eccentric haft. Because the rotation axes of the piston and stopper are not focused the work spaces with variable volume are created. The piston has an equilateral triangle shape with curved sides. During the rotating motion the peaks of the piston are in permanent contact with the inner surface of the carcass, forming three chambers. The volume of these chambers has a cyclical variation. For the achievement of the permanent contact between the peaks of the piston and the inner surface of the carcass, the paths of the peaks must be hypocycloids which are identical with the inner profile of the carcass. In this paper we will follow the notations used in (Chioreanu & Chioreanu, 2006). For more constructing details we can also consult (Marr, 2004 and Owen & Coley, 1990).

2. THE CARCASS PROFILE

The carcass has the bore in the form of hypocycloid (Aubin, 1976 and Libeskind, 2008), with 2 lobes. It is obtained by rolling a circle by the internal gear fixed on the rotor which is rolling on the external gearing. The engine executes three functioning cycles in four periods at one rotation of the rotor and three rotations of the engine shaft. The hypocycloid is the curve described by the point [V.sub.1], fixed relative to the mobile circle of r radius, which rolls without sliding on another fixed circle of r radius. The circles are internal. This curve closes if and only if the ratio r/R is rational. In the case of the Wankel engine the ratio of the radii is r / R = 2/3 (two lobes, the rotor rotates only once for each three rotations of the engine shaft), therefore the mobile circle runs along the fixed circle for three times. The center of the mobile circle describes a circle of e radius, e = R--r called the eccentricity, and it has its center at the point [M.sub.0,] the initial position. The center of the fixed circle is P. In order to determine the parameter equation of the hypocycloid we consider an orthogonal system of (x,y) coordinates with the origin at P. Let A be the contact point of the two circles at the initial position and let M be the center of the mobile circle after the angle rotation.

[FIGURE 1 OMITTED]

Denote c = d([M.sub.0], [V.sub.01]). The contact point between the two circles is denoted by B and by [psi] = [not less than] ([BMV.sub.1]) (figure 1). From the rolling condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we obtain r [phi] = R [psi] or [psi] = (2/3) [phi] and from the vector equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we deduce the parameter equations which generate the sectional area of the carcass

x = c sin (([phi] / 3) - e sin [phi], y = c cos ([phi] / 3) -e cos [phi]. (1)

Let us denote by M' the projection of point M on the Py axis and put [alpha] = [not less than] (M' M[V.sub.1]). One obtains [alpha] = ([pi] / 2)--([phi] / 3).

3. THE PISTON PROFILE

The peaks of the piston achieve a permanent contact with the inner surface of carcass. When the engine shaft rotates, the rotor will make a movement on a relative orbit compared to the engine shaft axis. The peaks of the piston are given by the intersection between a mobile circle, of c radius with the center at the point M and the hypocycloid of the carcass. The parameter equations of the circle are

[x.sub.c] = c sin [[gamma].sub.c] - e sin [[phi].sub.c], [y.sub.c] = c cos [[gamma].sub.c] - e cos [[phi].sub.c], (2)

where [[phi].sub.c] is the position of the center of the circle of c radius. The solution of the following system

{c sin [[gamma].sub.c] - e sin [[phi].sub.c] = c sin ([phi] / 3) - e sin [[phi].sub.c], c cos [[gamma].sub.c] - e cos [[phi].sub.c] = c cos ([phi] / 3) - e cos [phi] (3)

is [[gamma].sub.c] = ([[phi].sub.c] + 2k [pi]) / 3. For [[phi].sub.c] = [phi], the center of the circle coincides with the center of the piston P i.e. [[gamma].sub.c] = [gamma] = ([[phi].sub.c] + 2k [pi]) / 3, k = 0,1,2. We note that [DELTA] [[gamma].sub.c] = [phi] + 2 (k + 1) [pi]] / 3 - ([phi] + 2k [pi]) / 3 = 2 [pi] / 3. For [phi] = 0, the positions of the peak piston: [[phi].sub.1] = 0, [[phi].sub.2] = 2 [pi] / 3, [[phi].sub.3] = 4 [pi] / 3.

[FIGURE 2 OMITTED]

Therefore, the peaks of the piston form an equilateral triangle with the radius of the circumscribed circle equal to c and the radius of the inscribed circle equal to c / 2.

For construction reasons we have R [less than or equal to] c / 2 so [rho] : = c / e [greater than or equal to] 6. The sides of the piston are considered curve arches with the center on the line determined by the M center and the opposite peak. The rotating of the piston is achieved if and only if 2 (c - e) [greater than or equal to] [R.sub.a] - [d.sub.a], where [R.sub.a] is the radius of the curve arch and [d.sub.a] = d ([V.sub.01], [C.sub.a]) (figure 2). Let us denote by k = [d.sub.a] / e. The triangle [DELTA] [C.sub.a] [V.sub.02] [V.sub.03] is isosceles having [C.sub.a] [V.sub.02] = [C.sub.a] [V.sub.03] = [R.sub.a]; [V.sub.02] = [V.sub.03] = [square root of 3] x c. From [DELTA] [C.sub.a] [NV.sub.02] one obtains [R.sub.a] = e X [square root of [k.sup.2] + 3 x [rho] x k + 3 x [[rho].sup.2]. We denote by [k.sub.1] = j/e where j is the free running between the carcass and piston, the peak [V.sub.1] being in initial position. We obtain

2 x (c - e) - j = [R.sub.a] - [d.sub.a] and

k = [[[rho].sup.2] - 4 ([k.sub.1] +2) [rho] + [([k.sub.1] + 2).sup.2] ] / [[[rho] - 2 ([k.sub.1] + 2)]. For k = 0 [rho] = (2 + [square root of 3]) x ([k.sub.1] + 2) with [rho] > 2 x (2 + [square root of 3]) = 7.4641.

Writing the triangle [DELTA] [C.sub.a] [V.sub.02] [V.sub.03] area in two ways we get sin [beta] = ([square root of 3] /2)[[rho] (3 [rho] + 2k)] / [[k.sup.2] + 3 [rho] k + 3 [[rho].sup.2]]. For k = 0 we have [beta] = [pi] / 3. The area of circle segment [V.sub.03] [SV.sub.02] [V.sub.03] is [A.sub.s] = ([beta] - sin [beta]) [R.sup.2.sub.a] 2 = ([beta]-sin [beta])([k.sup.2] + 3 [rho] k + 3 [[rho].sup.2]) [e.sup.2] / 2. For k = 0 we get [A.sub.s] = 3 [[rho].sup.2] [([pi] / 3)-([square root of 3] / 2)] [e.sup.2] / 2.

4. THE VOLUME FUNCTION VARIATION

The three sides of the triangular piston determine with the carcass three workspace, separated one from another. The volume of this space is modified during the rotation of the piston. The volume of one chamber formed between piston and carcass has a cyclical variation as a function of the [phi] angle given by the relation [V.sub.p] (p) = [l.sub.p] x [A.sub.p] ([phi]) + [V.sub.ca], where: [l.sub.p] is the breadth of the piston, [A.sub.p] ([phi]) is the plane area of one chamber and [V.sub.ca] is the volume of the box fire. The [A.sub.p] ([phi]) is given by the Green's formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where D is the compact domain, simple with respect to the Py axis, bounded by peritrochoidal arch extended between the variable points [V.sub.1], [V.sub.2] and the segment [V.sub.1] [V.sub.2]. The curve [GAMMA] = [partial derivative]D is formed by the concatenation of the curves [[GAMMA].sub.1], [[GAMMA].sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where u = [(3 [square root of 3]) / 2] [rho] [e.sup.2] sin [(2 [phi] / 3) + [pi] / 6]. The maximal and minimal values of the function [V.sub.p] ([phi]) are

[V.sub.p max] = [V.sub.p] ([pi] / 2) = [V.sub.p] (7 [pi] / 2); [V.sub.p min] = [V.sub.p] (2 [pi]) = [V.sub.p] (5 [pi])

The cylinder capacity [V.sub.s] is [V.sub.s] = [V.sub.p max] - [V.sub.p min] = 3 [square root of 3] x [rho] x [l.sub.p] [e.sup.2] and the compression ratio is [epsilon] = [V.sub.p max] / [V.sub.p min], thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The maximum and the minimum volume of one chamber is

[V.sub.p max] = [[epsilon] / ([epsilon] - 1)] [V.sub.s]; [V.sub.p min] = [1 / ([epsilon - 1)] [V.sub.s].

In the figure 3 it is presented the variation of [V.sub.p] (p) for one chamber of an engine with the following parameters: [V.sub.s] = 250 [cm.sup.3], the compression ratio [epsilon] = 10.5 , eccentricity e = 10.5 mm, [rho] = 7.5, [l.sub.p] = 58 mm.

[FIGURE 3 OMITTED]

If it is compared a Wankel rotary engine with an internal combustion four-cycle engine, having the same cylinder capacity, which at a certain number of rotations develops the same power, we conclude: a Wankel rotary engine is equivalent to an internal combustion four-cycle engine having two cylinders.

5. REFERENCES

Aubin, T. (2001). A course in differential geometry, Providence, R.I. : American Mathematical Society, ISBN 082182709X.

Chioreanu, N. & Chioreanu, S. (2006). Heat engine for nonconventional motor vehicles, Ed. Univ. of Oradea.

Libeskind, S. (2008). Euclidian and Transformational Geometry, ISBN-13: 9780763743666, Jones and Bartlett Publishers, Inc.

Marr, A. (2004). Wankel Rotary Combustion Engines (WRCE) and Vehicles. April 7, 2000. Accessed October 18.

Owen, K. & Coley, T. (1990) Automotive Fuels Handbook. Warrendale, PA: Society of Automotive Engineers, ISBN1-56091-064-X.