**1. Introduction**

Particle swarm optimization (PSO) and differential evolution (DE) are two stochastic, population-based optimization EAs in evolutionary algorithms.

PSO was developed by Kennedy and Eberhart and was originally intended to simulate social behaviour. Every solution in PSO is a "bird" in the search space [1]. However, when the technique is implemented, it is referred to as a particle. All of the particles have fitness values that the fitness function evaluates in order to optimize them, as well as velocities that control their flight. The particles navigate through the problem space by following the current best particles. Although the PSO algorithm has attracted a lot of attention in the last decade, it unfortunately has a premature convergence issue, which is common in complicated optimization problems. To improve PSO's search performance, certain strategies for adjusting parameters such as inertia weights and acceleration coefficients have been developed.

Storn and Price [2] presented DE as a simple but effective EA for global optimization. The DE method has progressively gained popularity and has been utilized in a variety of practical applications, owing to its shown strong convergence qualities and ease of understanding [3]. DE has been successfully applied in a variety of engineering disciplines [4–8]. The selected trial vector generation technique and related parameter values have a significant impact on the performance of the traditional DE algorithm. Premature convergence can also be caused by poor methodology and parameter selection. DE scholars have proposed a number of empirical guidelines and proposals for selecting trial

**Citation:** Shen, X.; Ihenacho, D.C. Design of Gas Cyclone Using Hybrid Particle Swarm Optimization Algorithm. *Appl. Sci.* **2021**, *11*, 9772. https://doi.org/10.3390/app11209772

Academic Editors: Peng-Yeng Yin, Ray-I Chang, Youcef Gheraibia, Ming-Chin Chuang, Hua-Yi Lin and Jen-Chun Lee

Received: 4 September 2021 Accepted: 12 October 2021 Published: 19 October 2021

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**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/).

vector generation techniques and their associated control parameter settings during the last decade [9–12].

Despite the fact that PSO has been successfully applied to a wide range of challenges, including test and real-world scenarios, it has faults that might cause the algorithm performance to suffer. The major problem is a lack of diversity, which leads to a poor solution or a slow convergence rate [13]. The hybridization of algorithms, in which two algorithms are combined to generate a new algorithm, is one of the groups of modified algorithms used to improve the performance. DE is applied to each particle for a certain number of iterations in order to choose the best particle, which is then added to the population [14,15]. The barebones DE [16] is a proposed hybrid version of PSO and DE. The evolving candidate solution is created using DE or PSO and is based on a specified probability distribution [17]. In a hybrid metaheuristic [13], the strengths of both techniques are retained.

Cyclone separators are a low-cost and low-maintenance way of separating particulates from air streams. A cyclone is made up of two parts: an upper cylindrical element called the barrel and a lower conical part called the cone, as seen in Figure 1. The air stream enters the barrel tangentially and goes downhill into the cone, generating an outer vortex. The particles are separated from the air stream by a centrifugal force caused by the increased air velocity in the outer vortex. When the air reaches the bottom of the cone, an inner vortex forms, reversing the direction of the air and exiting out the top as clean air, while particulates fall into the dust collection chamber attached to the bottom of the cyclone.

**Figure 1.** Schematic flow diagram of a cyclone [18].

The simulation of an optimum gas cyclone with low cost is the primary focus of this research with the use of PSO, DE, and hybrid DEPSO algorithms using an objective function which is of a minimization type. The efficient global optimization is a major advantage of the DE algorithm. Furthermore, the diversity of the entire population is easily maintained throughout the process, preventing individuals from falling into a local optimum. PSO, on the other hand, has the advantage of a quick convergence speed. The best individual particle across the entire iteration is saved to obtain the lowest fitness values. Combining the benefits of DE and PSO, the use of a hybrid DEPSO strategy is proposed for this research with the goal of fast convergence and efficient global optimization. The proposed DEPSO method first reduces the search space using the DE algorithm, and then the obtained populations are used as the initial population by the PSO to achieve a fast convergence rate to a final global optimum. Moreover, DEPSO utilizes the crossover operator feature of DE to improve the distribution of information between candidates, and based on the fitness

function, the hybrid algorithm can determine the global minimum cost value much better than the use of PSO alone due to its drawbacks, such as high computational complexity, slow convergence, sensitivity to parameters, and so forth.

The overview of this paper is as follows:

