Elsevier

Fluid Phase Equilibria

Volume 356, 25 October 2013, Pages 11-17
Fluid Phase Equilibria

Prediction of gas solubility in polymers by back propagation artificial neural network based on self-adaptive particle swarm optimization algorithm and chaos theory

https://doi.org/10.1016/j.fluid.2013.07.017Get rights and content

Abstract

A novel prediction method based on chaos theory, self-adaptive particle swarm optimization (PSO) algorithm, and back propagation artificial neural network (BP ANN) is proposed to predict gas solubility in polymers, hereafter called CSPSO BP ANN. The premature convergence problem of CSPSO BP ANN is overcome by modifying the conventional PSO algorithm using chaos theory and self-adaptive inertia weight factor. Modified PSO algorithm is used to optimize the BP ANN connection weights. Then, the proposed CSPSO BP ANN (two input nodes consisting of temperature and pressure; one output node consisting of gas solubility in polymers) is used to investigate solubility of CO2 in polystyrene, N2 in polystyrene, and CO2 in polypropylene, respectively. Results indicate that CSPSO BP ANN is an effective prediction method for gas solubility in polymers. Moreover, compared with conventional BP ANN and PSO ANN, CSPSO BP ANN shows better performance. The values of average relative deviation (ARD), squared correlation coefficient (R2) and standard deviation (SD) are 0.1275, 0.9963, and 0.0116, respectively. Statistical data demonstrate that CSPSO BP ANN has excellent prediction capability and high accuracy, and the correlation between predicted and experimental data is good.

Introduction

The development of polymer materials over the past several decades has caused polymers to become an important part of daily life. Solubility is one of the most important physicochemical properties of polymers compounds because it determines the compatibility of components of a blending system. Researchers are focusing their attention on gas solubility in polymers [1], [2], [3], [4]. Solubility data provide useful criteria to determine requisite processing conditions that are mostly collected experimentally and by prediction [5], [6], [7]. Experimental studies mainly include phase-separation [4], [8], pressure-decay [9], gravimetric [10], volumetric [11], and chromatographic [5], [12] methods. Traditional prediction methods mainly consist of perturbed-hard chain theory, lattice-fluid theories, empirical models, and cubic equation of states [5], [13], [14], [15]. Unfortunately, solubility data of gases in polymers within a wide range of pressure and temperature are limited [16], [17], [18], [19] because some experimental studies are difficult to implement under many restricted conditions (i.e., time and material consuming). In addition, most traditional prediction methods have the shortcoming of being highly inaccurate [20], [21], [22], [23].

Gas solubility in polymers is affected by temperature, pressure, and interactions among macromolecular chains. Given the nonlinear relationship among these factors, traditional methods of predicting gas solubility in polymers are insufficient to meet application requirements. Therefore, an effective and accurate prediction method for gas solubility in polymers must be developed. Methods involving artificial neural networks (ANN) are effective tools for calculating the phase equilibrium and thermodynamic properties of polymers blending systems and have thus been used in different areas of research and engineering practice [14], [15], [24]. Considering the nonlinear nature of gas solubility in polymers, ANN method can be considered as an alternative method for solubility prediction [14], [25], [26]. Thus far, numerous ANN models for predicting physicochemical properties have been proposed. Bakhbakhi [13] compared ANN with equation of states for predicting the solubility of 2-naphthol in ternary systems. They demonstrated that the ANN method is a powerful approach with better accuracy.

Back propagation (BP) algorithm is the most frequently algorithm used for ANN training. However, ANNs trained by BP algorithm suffer from converging too slowly and being easily trapped into a local optimum [13], [27]. Researchers have recently discovered that the determination of ANN structure, parameters, and bias is crucial because the training process of ANN can be considered as a classical optimization problem [28], [29]. With the development of the interdiscipline of information science and engineering technology, many intelligent algorithms such as genetic algorithm [30], [31], [32], simulated annealing algorithm [33], [34] and particle swarm optimization (PSO) algorithm [35], [36] can be used for this determination. PSO algorithm is a global and advanced algorithm with a strong ability to search global optimum. Compared with genetic algorithm and simulated annealing, the most important advantages of PSO are the few parameters needed to adjust and the easy implementation [37]. Lazzus [38] proposed PSO for modeling the phase equilibrium of complex mixtures. Zhang [39], [40] introduced PSO algorithm for phase equilibrium calculations and for modeling vapor–liquid equilibrium data. Bonilla-Petriciolet [41] proposed a comparative study of PSO and several of its variants for solving phase stability and equilibrium problems. In addition, many ANN models based on PSO algorithm have been proposed. Lazzus [42] introduced a hybrid model based on ANN and PSO for estimation of solid vapor pressures of pure compounds at different temperatures. The studies have demonstrated that PSO is a powerful approach to ANN training [36], [43]. Although PSO ANN shows high performance, it is easily trapped into a local minimum. Obviously, traditional PSO algorithm cannot be successfully used in some higher dimensional complex optimization problems.

The aforementioned works have achieved a high level of solubility prediction accuracy in some cases. However, improving the performance of the prediction model is still the first-line goal of academic and industrial used by the aforementioned works to conduct this study, which focuses on using BP ANN based on a modified PSO algorithm and chaos theory to investigate gas solubility in polymers. To develop an effective and accurate prediction method, we propose a BP ANN trained by hybrid algorithm based on self-adaptive PSO and chaos theory, hereafter called CSPSO BP ANN. In the proposed CSPSO BP ANN, the traditional PSO is modified by chaos theory and self-adaptive inertia weight factor to overcome its premature convergence problem and accelerate the converging speed. Then, the modified PSO algorithm [also called chaotic self-adaptive PSO (CSPSO)] is used to tune and optimize the connection weights of BP ANN. Using CSPSO BP ANN, gas solubility in polymers is investigated within a wide range of temperature and pressure. A comparison among different neural networks is carried out in detail to reveal that our proposed CSPSO BP ANN outperforms BP ANN and PSO ANN.

Section snippets

Theory

In this work, BP ANN tuned by hybrid algorithm is developed to investigate its capability in predicting gas solubility in polymers. We propose a hybrid algorithm based on self-adaptive PSO and chaos theory, hereafter called CSPSO algorithm. In the proposed CSPSO algorithm, the global best fitness, average local best fitness, and acceleration coefficients generated by chaotic sequence are proposed to avoid prematurity and accelerate the converging speed.

Architecture and experimental data

In this work, CSPSO BP ANN is designed for the solubility prediction of gases in polymers. The prediction capabilities of the model is evaluated by calculating the average relative deviation (ARD), standard deviation (SD), and squared correlation coefficient (R2). ARD and SD are defined as follows:ARD=1Ni=1N|Pre(i)Exp(i)|Exp(i)SD=1Ni=1Nxixo2where N is the number of data points, Pre(i) is the predicted value of the model, Exp(i) is the experimental data, and xo is the average of N data

Results and discussion

In this study, a three-layer BP neural network trained by CSPSO algorithm (CSPSO BP ANN) is developed to predict gas solubility in polymers. The CSPSO BP ANN architecture is 2–8–1. The input layer with 2 nodes represents the temperature T and pressure P. Eight neurons in the hidden layer and one node in the output layer represent gas solubility in polymers.

CSPSO BP ANN is used to investigate solubility of CO2 in PS, N2 in PS, and CO2 in PP. Correlations between the prediction results of CSPSO

Conclusions

Gas solubility in polymers is crucial in polymer processing. In this work, we propose a novel ANN prediction model trained by hybrid algorithm combined with self-adaptive PSO algorithm and chaos theory to predict gas solubility in polymers. The aim is to replace the costly and time-consuming traditional measurement methods in laboratory. Our results show that the proposed model is reliable and useful for the modeling of gas solubility in polymers and is a useful tool for analyzing and designing

Acknowledgments

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant Numbers: 51163011, 61202313 and 51102131) and graduate student innovation fund by Nanchang University (Grant Number: cx2012011).

References (50)

  • A. Eslamimanesh et al.

    Chem. Eng. Sci.

    (2011)
  • M. Skerget et al.

    J. Supercrit. Fluids

    (2010)
  • D.W. Wei et al.

    Fluid Phase Equilibr.

    (2013)
  • N. Sunsandee et al.

    Fluid Phase Equilibr.

    (2013)
  • S.P. Nalawade et al.

    Prog. Polym. Sci.

    (2006)
  • Y. Sato et al.

    Fluid Phase Equilibr.

    (1999)
  • Y. Sato et al.

    Fluid Phase Equilibr.

    (1996)
  • M.A. Sabegh et al.

    J. Supercrit. Fluids

    (2012)
  • J. Sousa et al.

    Fluid Phase Equilibr.

    (2013)
  • Y. Bakhbakhi

    Math. Comput. Model.

    (2012)
  • M. Lashkarbolooki et al.

    Fluid Phase Equilibr.

    (2011)
  • M. Yazdizadeh et al.

    Chem. Eng. Sci.

    (2012)
  • M. Safamirzaei et al.

    Thermochim. Acta

    (2012)
  • S. Kazemi et al.

    J. Supercrit. Fluids

    (2012)
  • M. Kariznovi et al.

    Fluid Phase Equilibr.

    (2013)
  • I. Diaz et al.

    Fluid Phase Equilibr.

    (2013)
  • K. Benyelloul et al.

    Comput. Mater. Sci.

    (2013)
  • M. Lashkarbolooki et al.

    Fluid Phase Equilibr.

    (2012)
  • B. Mehdizadeh et al.

    Fluid Phase Equilibr.

    (2011)
  • R. Beigzadeh et al.

    Fluid Phase Equilibr.

    (2012)
  • A.K. Giri et al.

    Chem. Eng. J.

    (2011)
  • M.A. Ahmadi

    Fluid Phase Equilibr.

    (2012)
  • H. Karimi et al.

    Fluid Phase Equilibr.

    (2012)
  • A. Martinez-Vargas et al.

    Appl. Soft Comput.

    (2013)
  • S. Leung et al.

    Expert Syst. Appl.

    (2012)
  • Cited by (70)

    • Prediction method of solubility of carbon dioxide and methane during gas invasion in deep-water drilling

      2022, Journal of Contaminant Hydrology
      Citation Excerpt :

      Prediction technologies such as backpropagation and support vector machines have been developed. Huang et al. (2013), Yan et al. (2013), and Sedghamiz et al. (2015) successively used different machine learning algorithms to predict the solubility of gas in organic and inorganic solutions. Taherdangkoo et al. (2021) predicted the solubility of CH4 in water using four machine learning algorithms: region tree, boosted region tree, least-squares support vector machine, and Gaussian process region.

    View all citing articles on Scopus
    View full text