Prediction of solubility of gases in polystyrene by Adaptive Neuro-Fuzzy Inference System and Radial Basis Function Neural Network

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Abstract

Adaptive Neuro-Fuzzy Inference System (ANFIS) and Radial Basis Function Neural Network (RBF NN) have been developed for prediction of solubility of various gases in polystyrene. Solubility of butane, isobutene, carbon dioxide, 1,1,1,2-tetrafluoroethane (HFC-134a), 1-chloro-1,1-difluoroethane (HCFC-142b), 1,1-difluoroethane (HFC-l52a) and nitrogen in polystyrene is modeled by ANFIS and RBF NN in a wide range of pressure and temperature with high accuracy. The results obtained in this work indicate that ANFIS and RBF NN are effective methods for prediction of solubility of gases in polystyrene and have better accuracy and simplicity compared with the classical methods.

Introduction

Knowledge of solubility of gases in polymers for effective process design in many polymer industrial such as polymer foaming and coating is necessary. Gas solubility data in polymers in a wide range of pressure and temperature are rare. Therefore, reliable computational methods for prediction of gas solubility in polymer are required. The statistical mechanics-based models such as the perturbed-hard chain theory (Doghieri et al., 2006, Feng et al., 2001, Lee et al., 2001), the lattice-fluid theories (Enders et al., 2005, Li et al., 2006) and cubic equation of states (EOSs) (Louli and Tassios, 2000, Zhong and Masuoka, 1998) have been extended to describe phase behavior of polymer solution.

Neural networks and fuzzy methods have been employed in recent years as an alternative to conventional methods, due to their ability of modeling complex systems from data sets to any arbitrary accuracy in different areas of research and engineering practice. The use of Artificial Neural Network (ANN) which is one of the most powerful modeling techniques reduces time and cost of required experimental measurements (Khajeh et al., 2007, Modarress et al., 2008, Safamirzaei et al., 2008). The most basic neural network, consisting of multiple layers of computational units known as neurons, is Multi-layer Perceptrons (MLPs) neural network. Another type of neural networks is Radial Basis Function Neural Network (RBF NN) that is becoming increasingly popular in many scientific areas. The RBF NN has certain advantages over other types of ANNs, including better approximation capabilities, simple network structures, fast learning algorithms and will not encounter the local minima problems (Chen et al., 2008, Panagou et al., 2007).

The fuzzy logic model is empirically-based approach instead of than attempting to model a system mathematically. Fuzzy logic can be successfully applied to perform reasonable and meaningful operations on concepts that cannot be easily coded using a classical approach. Such modification allows for a much more flexible and widespread use of reliable and consistent logic in a variety of applications (Ross, 2004, Rouvray, 1997).

Neural networks provide algorithms for learning, classification and optimization, whereas fuzzy logic deals with issues such as reasoning on a higher (semantic or linguistic) level. Consequently, the two methods complement each other. By combining neural networks with fuzzy logic, it is possible to bring the computational power and learning of neural networks into fuzzy logic systems. The synergism of integrating neural networks with fuzzy logic systems is increasing capability of fast and accurate learning, high-level thinking, generalization capacity, excellent explanation facilities in the form of semantically meaningful fuzzy rules and the ability to accommodate both data and existing expert knowledge. A specific approach in neuro-fuzzy is the Adaptive Neuro-Fuzzy Inference System (ANFIS) that is one of the first integrated hybrid neuro-fuzzy models (Jang, 1993), which has shown significant results in modeling nonlinear functions and is faster in convergence when compared to the other neuro-fuzzy models (Akcayol, 2004).

In this work, solubility of various gases in polystyrene is predicted by a RBF NN at various temperature and pressure. A hybrid grid partitioning ANFIS was established for the solubility prediction and the results of solubility prediction obtained by ANFIS method are compared with those predicted by RBF NN method. Also the results of solubility prediction by ANFIS and RBF NN are compared with those obtained by EOS approach in order to select the most accurate method.

Section snippets

Radial Basis Function Neural Network (RBF NN)

An extremely powerful neural network type is the RBF neural network, which uses a cluster algorithm in the first step (unsupervised training) and calculates the approximation in the second step (supervised training). It includes an input layer, as in many other network models, which has no calculating power and serves only to distribute the input data among the hidden neurons. A single hidden layer consists of the hidden neurons which are expressed as non-linear transfer functions and are

Adaptive Neuro-Fuzzy Inference System (ANFIS)

A fuzzy inference system is a nonlinear system that employs fuzzy if–then rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise quantitative analyses. Fuzzy logic modeling techniques can be classified into three categories, namely the linguistic (Mamdani-type) (Mamdani & Assilian, 1975), the relational equation, and the Takagi–Sugeno–Kang (TSK) (Sugeno, 1985). In linguistic models, both the antecedent and the consequence are fuzzy sets

Development of ANFIS and RBF NN models

Many ANN architectures have been proposed for process modeling (Coulibaly and Evora, 2007, Hung and Ni, 2007). One of the most commonly used models is the RBF neural network. There are a wide variety of learning strategies that have been proposed in literatures for changing the parameters of a RBF network. These strategies are of two main categories. The first category contains strategies in which centers and variances of the network are changed, including (Montazer, Sabzevari, & Khatir, 2007):

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Results and discussion

In this work a hybrid grid partitioning ANFIS by Gaussian membership function and also RBF NN were used in order to prediction of several gases solubility in polystyrene. The original data needing for establish models consist of seven gases in polystyrene that extracted from literatures of Sato et al., 2001, Sato, Takikawa et al., 2000, Sato et al., 2004, Sato, Iketani et al., 2000 in the following temperature and pressure range: for butane and isobutene (348–473 K) and pressure up to 3 MPa,

Conclusions

Predicting the solubility of gases in polymers is one of the important subjects in polymer processing. In this work, we have studied the solubility of several gases in polystyrene by using RBF NN and ANFIS. The performance of RBF NN and ANFIS was evaluated based on calculating the Average Relative Deviation (ARD%). The ARD% of testing data for all systems studied was less than 2.3 for ANFIS and less than 3.3 for RBF NN. The results show that these two methods are capable for prediction of gas

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