Inverse design techniques in the area of fluid mechanics are normally conducted for continuous flow turbomachines rather than positive displacement devices. However, the work in this article is concerned with a class of rotary positive displacement device referred to as the limaçon-to-limaçon machine. The rotors and housings of these machines are manufactured of limaçon profiles, and are likely to suffer from interference if the rotors are not carefully profiled. Published literature indicates that solutions proposed to tackle the interference problem in these machines will adversely affect their efficiency figures. This notion motivated the work presented in this article, which first introduces the relevant mathematical models of the limaçon-to-limaçon machine and then uses these models to construct an inverse geometric design problem formulation. The proposed model has been coded in a computer program that utilises a Marquardt-Levenberg technique to converge to the required geometric parameters. Case studies are presented at the end of the article to verify the validity of the proposed inverse design model.
The limaçon of Pascal is a plane curve that possesses some characteristics which qualify it for fluid processing applications. However, the curve has not yet attracted enough attention, neither from industry nor from the mechanical engineering research community, despite the few attempts that have been conducted over the past century. The work presented here is intended to explore various mechanical linkages that can be employed to produce the limaçon and the aspects of these linkages. Mathematical analysis will show that machines based on the limaçon curve possess sinusoidal volumetric relationships and hence, simplified pressure and torque equations can be obtained in closed forms. The concepts discussed here will be demonstrated by using a compressor as a sample application