machines Article On the Design of a Class of Rotary Compressors Using Bayesian Optimization Kui Lu , Truong H. Phung * and Ibrahim A. Sultan   Citation: Lu, K.; Phung, T.H.; Sultan, I.A. On the Design of a Class of Rotary Compressors Using Bayesian Optimization. Machines 2021, 9, 219. https://doi.org/10.3390/ machines9100219 Academic Editors: Kim Tiow Ooi and Kuan Thai Aw Received: 16 August 2021 Accepted: 27 September 2021 Published: 29 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). School of Engineering, IT and Physical Sciences, Federation University Australia, Mount Helen, Ballarat, VIC 3353, Australia; k.lu@federation.edu.au (K.L.); i.sultan@federation.edu.au (I.A.S.) * Correspondence: t.phung@federation.edu.au Abstract: The optimization process of compressors is usually regarded as a ‘black-box’ problem, in which the mathematical form underlying the relationship between design parameters and the design objective is impractical and costly to be obtained. To solve the ‘black-box’ problem, Bayesian optimization has been proven as an accurate and efficient method. However, the application of such a method in the design of compressors is rarely discussed, particularly no work has been reported in terms of the positive displacement type compressor. Therefore, this paper aims to introduce the Bayesian optimization to the design of positive displacement compressors through the optimization process of the novel limaçon compressor. In this paper, a two-stage optimization process is presented, in which the first stage optimizes the geometric parameters as per design requirements and the second stage focuses on revealing an optimum setting of port geometries that improves machine performance. A numerical illustration is offered to prove the validity of the presented approach. Keywords: positive displacement; limaçon of Pascal; rotary compressor; optimization; Bayesian opti- mization 1. Introduction In compressor design, information solely obtained from the simulation of the mathe- matical model is usually insufficient to reflect the thorough relationship between the design parameters and design objective in terms of performance. As such, designers often need to rely on optimization strategies to reveal the optimum design scenario before reaching the final decision on the prototype. The published literature shows that various optimization techniques have been ap- plied to the design optimization of the positive displacement machine and compressor in particular. Ooi [1] applied the direct-search method to seek a set of six machine dimensions and seven design constraints, which can minimize the mechanical losses of the rolling piston compressor. The author reported that a predicted 50% reduction in mechanical loss, which increases 14% of the coefficient of performance, can be achieved with a proper combination of design dimensions. Liu et al. [2] employed the gradient search method to determine optimum dimensions of bearing components that can reduce the frictional loss occurring in the scroll compressor. Based on the optimization result, the author found that the frictional loss can be reduced in the range of 14.1% to 18.1%. Sultan and Kalim [3] adopted the simultaneous perturbation stochastic approximation (SPSA) method to find the best piston trajectory of the reciprocating compressor, and the authors also employed the gradient-based optimization to determine the machine dimensions which can real- ize such a trajectory. Recently, the SPSA approach has also been utilized in the work of Phung and Sultan [4] to design a new embodiment of the limaçon machine referred to as the limaçon-to-circular machine. Like other limaçon machine embodiments, this new design can be used as expanders, compressors, and potentially pumps. Cavazzini et al. [5] adopted topology optimization, which combines the particle swarm method with the computational fluid dynamics, to determine the geometric parameters that can maximize Machines 2021, 9, 219. https://doi.org/10.3390/machines9100219 https://www.mdpi.com/journal/machines