CLOSED-FORM GEOMETRY AND ANALYTICAL PERFORMANCE ESTI-MATES OF THE TWO-APEX WANKEL ROTARY COMPRESSOR

Abstract

The subject matter of the article is the working-chamber geometry and the isentropic performance of the two-apex (m = 2) Wankel rotary compressor ‒ a two-apex rotor orbiting within a single-lobe peritrochoid housing at a 2:1 eccentric-shaft gear ratio ‒ a configuration geometrically well suited to gas compression yet, unlike the classical three-apex (m = 3) engine, it is almost unstudied analytically. The goal is to derive, rigorously and from first principles, the complete set of closed-form geometric and performance relations of the m=2 machine, exact expressions for which have not previously appeared in the literature, and to render the exact profiles manufacturable. The tasks to be addressed are: to establish the kinematics and the 2:1 synchronizing-gear relation; to derive the exact housing and rotor (envelope) profiles together with the working-chamber area and volume laws; to obtain the displacement, compression ratio and apex-seal oscillation limit in closed form; to construct manufacturable profiles that decouple the running clearance from the shape factor; and to develop the asymptotic scaling laws, the closed-form isentropic performance, and a design equation relating the ideal gain with any loss model. The methods used are: the differential geometry of trochoidal envelopes, Green's theorem for the chamber area and volume, the Taylor (asymptotic) expansion of the governing shape function, and closed-form isentropic thermodynamic analysis. The following results were obtained: the working-chamber volume law
, with  and shape factor
, together with its exact crank-angle rate; exact theorems for the displacement  (about  larger than for  at identical e, R, b), the compression ratio , and the apex-seal law  giving  versus 6 for m = 3; exact normal parallel-offset profiles decoupling the running clearance  from K, removing the principal obstacle to precise CNC manufacture in earlier circular-arc treatments; four asymptotic design laws from a single expansion of — ,
 (within 1.3 % for K ≥ 4), (1 % clearance already at , no dead-volume machining), and  volumetric-compactness advantage over m = 3 at the respective kinematic limits; and the isentropic indicated work in closed form, , yielding  and a strict monotonicity theorem. Conclusions. The scientific novelty of the results obtained is as follows: 1) for the first time the complete exact closed-form geometry of the  Wankel compressor was derived from first principles, rigorously quantifying its advantages over m = 3; 2) the exact normal parallel-offset method was developed, decoupling running clearance from shape factor and rendering the exact profile directly CNC-manufacturable, overcoming the chief limitation of earlier circular-arc practice; 3) the closed-form isentropic performance and the design equation  were obtained, separating the analytically known ideal gain from any loss model and providing a verified geometric and performance basis for subsequent thermodynamic and loss analysis.