# Particle Swarm Optimization and Application to Liquid-Liquid Equilibrium

## Introduction

Low volatile phosphonium ILs have proved to be better solvents as compared to volatile organic solvents from our liquid-liquid equilibrium (LLE) experiments in Chapter 2. However, the experiments were carried out on the laboratory scale. The separation has not been implemented on an industrial scale to date. For transforming laboratory data for an industrial application, a process optimization study is necessary. In the past, many popular stochastic algorithms such as Genetic Algorithm (GA) (Goldberg, 1989), Simulated Annealing (SA) (Kirkpatrick, Gelatt, & Vecchi, 1983), particle swarm optimization (PSO) (Eberhart & Kennedy, 1995a, 1995b), Ant colony optimization (ACO) (Colorni, Dorigo, & Maniezzo, 1991; Dorigo, 1992), Differential Evolution (DE) (Storn & Price, 1997) and Self-Organising Migrating Algorithm (SOMA) (Zelinka, 2004) have been investigated for optimization in science and engineering.

PSO is an evolutionary algorithm based on social behavior of birds in swarm. The initial position and velocity of each particle are initiated randomly. During simulation, each particle in swarm (population) updates its position and velocity based on its experience as well as neighbors' experience within the search space. PSO is robust as it evaluates fewer function values during simulation than GA (Hassan, Cohanim, & de Weck, 2005; Sivanandam & Deepa, 2009). Ethni, Zahawi, Giaouris and Acarnley (2009) showed that PSO shows more success rate to reach the target optimum value than SA. PSO is also more preferable than ACO due to high success rate and solution quality (Elbeltagi, Hegazy, & Grierson, 2005). Keeping the shortcomings of other methodologies in mind, we have chosen the PSO technique for optimizing the flow rate and number of stages in a multi-stage extractor.

A multi-stage extractor containing more than two components requires detailed design like temperature, pressure, flow rate and composition in each stage. These are achieved by solving material balance equations (M), phase equilibrium relation (E), mole fraction summation for each stage (S) and energy balance equations (H), better known as MESH equations.

In this work, the traditional Isothermal Sum Rate (ISR) method (Tsuboka & Katayama, 1976) has been considered for the stage-wise calculation. We have tried to obtain the optimum number of stages and solvent flow rate by minimizing the multi-stage extractor cost for extraction of butanol and ethanol from aqueous solution using ILs: [TDTHP][DCA] and [TDTHP][Phosph].