Prediction of the specific volume of polymeric systems using the artificial neural network-group contribution method

Fluid Phase Equilibria 356 (2013) 176–184

Contents lists available at ScienceDirect

Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid

Prediction of the specific volume of polymeric systems using the artificial neural network-group contribution method Majid Moosavi ∗ , Nima Soltani Department of Chemistry, University of Isfahan, Isfahan 81746-73441, Iran

a r t i c l e

i n f o

Article history: Received 30 April 2013 Received in revised form 28 June 2013 Accepted 1 July 2013 Available online 25 July 2013 Keywords: Polymeric system Specific volume Artificial neural network Group contribution method

a b s t r a c t In this work, the specific volumes of some polymeric systems have been estimated using a combined method that includes an artificial neural network (ANN) and a simple group contribution method (GCM). A total of 2865 data points of specific volume at several temperatures and pressures, corresponding to 25 different polymeric systems have been used to train, validate and test the model. This study shows that the ANN–GCM model represent an excellent alternative for the estimation of the specific volume of different polymeric systems with a good accuracy. The average relative deviations for train, validation, and test sets are 0.0403, 0.0439, and 0.0482, respectively. A wide comparison between our results and those of obtained from some previous methods show that this work can provide a simple procedure for prediction the specific volume of different polymeric systems in a better accord with experimental data up to high temperature, high pressure (HTHP) conditions © 2013 Elsevier B.V. All rights reserved.

1. Introduction Polymeric systems almost affect on all aspects of our life. These systems are widely used for industrial and academic purposes. For example, Poly ethylene glycols (PEGs), are frequently used in the pharmaceutical and cosmetic fields as solvents, carriers, humectants, lubricants, binders, bases, and coupling agents and also for extraction, separation, and purification of biological materials [1]. In recent years, great interest has been focused on the measurement, correlation, and prediction of thermodynamic properties of polymers. Thermodynamic properties of polymeric systems play an important role in the polymer industry and are often a key factor in polymer production, processing, and material development, especially for the design of advanced polymeric materials [2–8]. During the past decades, some attempts have been made to predict the specific volume and other thermodynamic properties of polymeric systems [2–14]. Different authors used different equations and methods to predict and reproduce the thermodynamic properties of polymer systems. Most of these attempts are restricted to the limited systems and little systematic work has been devoted to test the ability of these equations to predict the thermodynamic properties of these systems.

∗ Corresponding author. Tel.: +98 311 7932722; fax: +98 311 668 9732. E-mail addresses: [email protected], majid [email protected] (M. Moosavi). 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.07.004

An artificial neural network (ANN) can be a suitable alternative to model the different thermodynamic properties. Neural networks generally consist of a number of interconnected processing elements or neurons. How the inter-neuron connections are arranged and the nature of the connections determines the structure of a network. How the strengths of the connections are adjusted or trained to achieve a desired overall behavior of the network is governed by its learning algorithm [15]. In other words, ANN is an especially efficient algorithm to approximate any function with a finite number of discontinuities by learning the relationships between the input and output vectors [16]. Thus, an ANN is an appropriate technique to model the nonlinear behavior of chemical properties. Recently, neural networks have been used to estimate the different thermodynamic properties such as density, melting point, vapor pressure, etc. for different classes of materials [17–27]. In the last years, some limited attempts have been made to develop ANN models to predict the specific volume of polymeric systems [13,14,28]. Yousefi and Karimi [13,14] developed the ANN models for prediction the specific volume of limited polymeric systems. They used the genetic algorithm to train the neural networks in an unsupervised manner. Zhang and Friedrich [28] reviewed the application of artificial neural networks to polymer composites. This study was undertaken to investigate the specific volumes of 25 different polymeric systems at different temperatures and pressures using a combined method that includes an artificial neural network (ANN) and a simple group contribution method (GCM) up to the extremely HTHP conditions.

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

177

Scheme 1. The ANN–GCM model used in this work.

2. Methodology and modeling Artificial neural networks (ANNs) are designed by simulation of human brain procedures and have been extensively used in various scientific and engineering areas such as estimations of physical and chemical properties [29]. These powerful tools can be usually applied to study the complicated systems especially at extreme conditions. The theoretical explanations about neural networks can be found in many references such as reference [30]. The database is divided into three subdata sets including the training, validation, and test sets. The training set is used to generate the ANN structure in which a neural network modifies the weights and biases in answer to initial information. The validation (optimization) set is applied for optimization of the model, and the test (prediction) set is used to investigate the prediction capability and validity of the obtained model. Testing stage has no effect on training and so provides an independent measure of network performance. The process of division of database into three sub data sets is performed randomly. In this work, the total number of experimental data used to design the network is 2865, of which about 75%, 10%, and 15% of the main data set are randomly selected for training (about 2150 data points), validation (around 285 data points),

and test (about 430 data points), respectively. The effect of the percent allocation of the three subdata sets from the database on the accuracy of the ANN model has been studied elsewhere [31]. Gharagheizi showed that the percent of test set allocated from the main dataset should be between 5 and 35%. If this percent is lower than 5% the accuracy of the model over the training set is greater than the test set. Also, if the percent is greater than 40% the obtained model cannot predict the test set as well as the training set. On the other hand, his work showed that the optimum percent of the test set is dependent to the nature of the problem. The optimum percent is the percent on which the accuracy of the model over the test set approaches the training set. The experimental data points of the specific volumes at several temperatures and pressures, corresponding to 25 different polymeric systems have been used to train, validate and test the ANN–GCM model using the MATLAB software. Temperature (T), pressure (P), molecular mass (Mw ), and the structural groups that form the molecules were given as input variables. The output parameter is specific volume. All data used in this work have been experimentally determined and the data obtained from theoretical models and also those data for which their accuracy is not guaranteed were not considered.

178

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

Table 1 Names, temperature and pressure ranges, and the number of experimental data of the polymeric systems used in this work. T (K)

Polymer Polycarbonate tetramethyl bisphenol-A (PCTB) Poly vinylidene fluoride (PVDF) Poly ␣-methylstyrene (PMS) Ethylene/1-octene copolymer (EOC) Ethylene 1-butene copolymer (EBC) Polypropylene (PP) Polycarbonate bisphenol-A (PC) Poly vinyl methyl ether (PVME) Poly butylene succinate (PBS) Polystyrene (PS) Polypropylene (PP) Poly phenyl methyl siloxane (PPMS) Poly 1-octene (PO1) Poly ␧-caprolactone (PCL) Poly butyl acrylate (PBA) Poly dimethyl siloxane (PDMS) Ethylene/propylene copolymer (EPC) Poly 1-butene (PB1) Polyethylene (PE) Poly vinyl chloride (PVC) Ethylene/acrylic acid copolymer (EAAC) Poly ether sulfone (PES) Poly butylene terephthalate (PBT) Poly ethylene glycol methyl ether (PEGME-350) Poly ethylene glycol (PEG-200)

P (MPa)

491.55–562.75 453.15–493.15 481.65–522.3 423.1–503.31 424.48–504.43 455.25–535.5 443.7–603.4 311.5–415.5 413.9–493.4 391.45–557.25 353.15–393.15 339.6–397.3 455.9–536.75 373.75–421.35 303.15–483.15 291.25–423.05 422.35–502.28 454.4–534.6 452.1–532.85 373.35–423.25 393.2–473.2 497.05–569.35 514.55–590.85 298.15–338.15 298.15–338.15

0.1–160 0.1–120 0.1–160 0.1–200 0.1–200 0.1–200 0.1–200 0.1–120 0.1–200 0.1–200 0.1–100 10–100 0.1–200 0.1–200 10–200 0.1–202.5 0.1–200 0.1–200 0.1–200 0.1–170 3.4–200.1 0.1–100 0.1–200 0.1–30 0.1–30

Total

=

fjh

Ref.

99 45 63 180 180 189 54 84 35 189 55 95 189 126 138 152 189 189 189 81 63 99 126 28 28

[33] [34] [35] [36] [36] [36] [37] [38] [39] [40] [2] [38] [36] [2] [41] [42] [36] [36] [36] [2] [43] [44] [45] [46] [46]

2865

In terms of their structures, neural networks can be divided into two types: feed forward networks and recurrent networks. In a feed forward network, the neurons are generally grouped into layers. MLPs (Multi-Layer Perceptrons) are perhaps the best known type of feed forward networks. MLP has three layers: an input layer, an output layer and an intermediate or hidden layer. Neural networks are trained by two main types of learning algorithms: supervised and unsupervised learning algorithms. A supervised learning algorithm adjusts the strengths or weights of the inter-neuron connections according to the difference between the target and actual network outputs corresponding to a given input. Example of supervised learning algorithms is back propagation algorithm. In this work, a feed-forward backpropagation neural network was used, which is one that is very effective in representing nonlinear relationships among variables. The specific volumes of polymeric systems have been estimated using a combined method that includes an artificial neural network and a simple group contribution method (ANN–GCM). A neural network consists of a number of simple processing elements, called neurons. Since there is no a specific approach to determine the number of neurons of the hidden layer, the optimum number of neurons was determined by adding neurons in a systematic form during the training process, i.e. in a trial and error method. The output of a neuron is computed from the following equation: Njh

NP

 n 

adjustment of the network’s weights and biases. At the beginning, all weights and biases are initialized randomly. Then, the network is trained (i.e. its weights are adjusted) by an optimization algorithm until it correctly emulates the input/output mapping. The weights and biases should be obtained by minimization of an objective function. The objective function used in this study is the mean of squares of errors between the outputs of the neural network and the target values. This minimization was performed by Levenberg–Marquardt algorithm. This algorithm is rapid and accurate in the process of training neural networks [29–32]. Different types of transfer functions have been proposed for artificial neural networks such as linear (purelin) function, logarithmic sigmoid (logsig), and hyperbolic tangent sigmoid (tansig) [31]. In the present study, different combinations of these mentioned transfer functions have been tested to choose the best transfer Table 2 Structural groups considered in the ANN–GCM model.

COOH

CH3

COO

>CH2

OCOO

>CH

O

>C<

SOO

>Si<



wijh pi

+ bhj

(1)

i=1

where p corresponds to the vector of the inputs of the training, j is the hidden neuron, wij is the weight of the connection among the input neurons with the hidden neuron, the term bj corresponds to the bias of the neuron j of the hidden layer, and f h is the transfer function of the neuron. Similar calculations are carried out to obtain the results of each neuron of the following layer until the output layer. During the training algorithm, input data are fed to the input layer of the network and the difference between the results from the output layer and the desired outputs is used as a criterion for

Ph

Ph

(meta)

CF2

CHCl

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

179

Table 3 Optimum weights and biases for the used ANN–GCM model. Input weights

O C O

HO C O

−2.19812 0.708577 −0.8361 0.183169 −0.01174 −1.21148 −3.7916 −0.32198 −0.14685 2.146535 1.489485 1.952423 0.141648 −0.17453 −0.32665 0.279845 1.211221 0.4313

Neuron # 1 Neuron # 2 Neuron # 3 Neuron # 4 Neuron # 5 Neuron # 6 Neuron # 7 Neuron # 8 Neuron # 9 Neuron # 10 Neuron # 11 Neuron # 12 Neuron # 13 Neuron # 14 Neuron # 15 Neuron # 16 Neuron # 17 Neuron # 18 Input weights

CH2

Neuron # 1 Neuron # 2 Neuron # 3 Neuron # 4 Neuron # 5 Neuron # 6 Neuron # 7 Neuron # 8 Neuron # 9 Neuron # 10 Neuron # 11 Neuron # 12 Neuron # 13 Neuron # 14 Neuron # 15 Neuron # 16 Neuron # 17 Neuron # 18

CH3

1.143122 0.11567 −0.38924 0.077539 −0.64003 −2.13676 0.752031 −0.20213 −0.41991 0.297674 −4.20832 −0.69265 −0.58903 0.42398 4.138437 −0.74757 −0.0815 −0.72339

Input Weights

0.695898 −0.19402 −0.12972 −0.32086 −0.04728 0.5493 −0.24464 0.974772 −0.23309 −1.07977 −0.44178 0.574284 −1.14139 0.358856 −0.96054 −0.12659 −0.3837 0.127397

OCOO

Neuron # 1 Neuron # 2 Neuron # 3 Neuron # 4 Neuron # 5 Neuron # 6 Neuron # 7 Neuron # 8 Neuron # 9 Neuron # 10 Neuron # 11 Neuron # 12 Neuron # 13 Neuron # 14 Neuron # 15 Neuron # 16 Neuron # 17 Neuron # 18

O

−2.03057 0.011357 0.239526 0.120886 0.312837 −0.18151 −0.80609 0.154047 −0.03926 2.614637 1.699909 −0.32901 1.522334 0.841973 −1.66925 0.902533 −0.33406 −0.11646

CHCl −1.54696 −0.54356 0.107006 −1.1697 −0.29408 0.197329 −0.65441 −0.42641 0.390554 1.851228 1.364204 −0.60824 1.182377 −0.18065 −2.76316 −0.5105 0.114235 0.365766

0.768617 −1.08806 −2.01152 0.402157 0.755026 0.417692 0.656151 −0.57916 0.184647 −1.12753 0.271121 0.375573 −1.39444 −0.25043 −1.10811 −0.54607 −0.09164 −0.17729

S

0.175545 0.049631 0.445312 0.540178 0.75052 −0.54418 0.194614 −0.69163 0.577508 3.159433 −2.86001 0.463472 0.002102 0.003579 1.978288 −0.51738 −1.00835 2.391142

−0.27297 0.009722 −0.41681 −0.95766 0.85479 0.577089 −0.80163 0.491475 0.849672 −0.51625 2.505027 −1.04693 −0.95007 0.082422 −2.48905 0.855343 −0.00736 0.859902

>C<

>Si<

−0.0569 0.497791 −1.16053 −1.47693 0.250958 −0.19148 0.681629 0.249682 0.060397 −1.12081 −0.59196 −0.09666 −1.29284 −0.42365 −1.49177 0.545546 0.334453 −0.21149

0.837944 0.106723 −0.81163 0.830959 0.154519 0.93251 −0.28731 0.568046 0.951526 −0.71528 1.41719 1.419684 −0.93067 −0.35858 −0.47249 0.727875 0.526178 1.156839

CF2

>CH

−0.86717 0.176678 −0.22827 −1.47218 −0.28634 0.383186 −1.99816 0.084905 0.609113 −0.1397 −0.21929 −1.25805 1.383011 0.274193 −1.92815 −0.32396 0.181381 −0.23873

1.23637 −0.03806 −0.06955 0.029527 −0.21718 −2.63955 1.303597 −0.17884 0.421242 −0.97308 −0.10806 1.158334 −0.83981 0.172767 −0.77356 −0.41497 0.116242 −2.97388

Ph

Ph

−0.07172 0.260412 0.34399 0.46662 −0.32887 0.4102 −0.80734 −1.18672 0.25613 0.165259 2.794958 −0.70564 0.743864 −1.41402 −3.51666 1.027887 0.109916 0.339575

−3.85704 0.323272 −0.27038 −0.45233 0.453596 −0.02589 −4.40428 −0.43485 −0.00161 −0.46056 −0.59168 1.050367 1.969598 0.016944 −1.47494 −1.53881 0.153257 0.381193

T (K)

P (Mpa)

Mw (kg/mol)

Bias

5.431737 −0.45627 −0.52906 −0.54098 −0.17927 −0.95315 6.03094 0.069713 −0.04956 −0.43877 −0.0021 1.657117 −0.71747 −1.06154 −0.00294 1.122864 −0.05917 0.458915

−2.52914 0.976193 1.123676 1.785141 0.966446 0.041938 −0.79387 −0.07086 −0.45205 0.027603 −0.01103 −0.14984 0.021364 0.380954 0.010959 −0.01453 0.026158 −1.11212

−0.51028 −0.09435 4.292862 −0.01654 0.328558 0.77156 0.045573 −0.03086 0.136031 2.890054 −7.85229 1.567028 0.515398 0.193275 22.14363 0.80722 0.004668 −1.57732

2.218229 2.702669 2.156235 −1.47556 −0.90163 0.669187 1.013526 0.65411 −0.28253 0.742793 −1.48783 1.233884 0.487986 −0.97409 3.40103 −1.50438 1.860401 −1.53975

Neuron

1

2

3

4

5

6

7

8

9

10

11

Layer Weights

−1.05978

−1.66145

−1.43324

0.05805

0.161098

1.92152

4.226347

3.674358

−0.74973

4.845394

16.84358

Neuron

12

13

14

15

16

17

18

Bias

Layer Weights

−0.07948

−2.68407

−0.10249

16.45263

−1.3873

−6.36732

−1.23155

3.106436

function for input and output layer. The purelin, logsig, and tansig transfer functions are defined in Eqs. (2)–(4), respectively: purelin(n) = n log sig(n) =

1 1 + exp(−n)

tansig(n) =

2 −1 1 + exp(−2n)

(4)

(2) (3)

Scheme 1 presents a diagram for the ANN–GCM model used in this work.

180

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

Table 4 Overal minimum, maximum, and the average deviations for the calculated specific volumes of the polymeric systems using the ANN–GCM model. ANN model Experimental data No. of Data points Deviations Devmin (%) Devmax (%) Bias AAD RMS No. of Dev < 0.5% No. of Dev > 1%

Training set

Testing set

Validation set

Total set

2148

430

287

2865

0.00001 0.8431 0.0007 0.0403 0.0765 2140 0

0.00001 0.9917 0.0104 0.0482 0.0973 425 0

0.00006 0.4556 −0.0021 0.0439 0.0741 287 0

0.00001 0.9917 0.0019 0.0419 0.0797 2852 0

3. Results and discussion A list of polymeric systems used in the ANN–GCM model, temperature and pressure ranges, and the number of experimental data [33–40,2,41–46] has been given in Table 1. Table 2 shows the structural groups used as entrance variables. The value associated with each structural group in monomeric unit of a polymer was defined as 0 when the group does not appear in the substance and n, when the group appears n times in the substance. Several network architectures were tested to select the most accurate scheme. Since no additional information about the recommended number of neurons has been found for the calculation of properties for any type of substances, the optimal number of neurons was determined by trial and error.

The accuracy of the model to reproduce and predict the specific volumes of polymeric systems at different temperatures and pressures may be evaluated using the statistical parameters [47–50], namely, the absolute average deviation (AAD), the average percentual deviation (bias), the mean square error (MSE) and the root mean square error (RMS) which are defined as follows:



 (5)





 exp,i − cal,i  1  100 ×   N exp,i

(6)

1 (exp,i − cal,i )2 N

(7)

N

AAD =

exp,i − cal,i exp,i

i=1

i=1 N

MSE =

i=1

(a) 0.8 Training Validation Testing

 N 1 RMS =

(exp,i − cal,i )2 N

0.6

AAD %

1 100 × N N

bias =

The AAD characterizes the fact that the calculated values are more or less close to experimental points. The bias characterizes the quality of the distribution of the calculated data on either side of the experimental points. Among the different transfer functions, the tansig–purelin transfer function with 18 neurons makes the least error. Therefore, this architecture was used to design ANN–GCM model. Fig. 1 shows the AAD and the RMS found in correlating the specific volume of different polymeric systems as a function of the number of neurons in

0.4

0.2

0.0 5

10

15

(8)

i=1

20

Neurones in hidden layer 10 0

(b) 0.020 Training Validation Testing

Mean Squared Error (mse)

RMS

0.015

0.010

0.005

Training Validation Testing Best

10 -1 10 -2 10 -3 10 -4 10 -5 10 -6

0.000 0

5

10

15

20

Neurones in hidden layer

10 -7

0

200

400

600

800

1000

1200

1400

Number of Epochs Fig. 1. (a) Average absolute deviation (AAD in %) and (b) root mean square error (RMS) found in correlating the specific volume of different polymeric systems as a function of the number of neurons in the hidden layer for training, validation, and testing sets of data.

Fig. 2. The profiles of the MSE for the training, validation and testing with respect to the number of epochs for the best network topology. The best validation performance is 5.26 × 10−7 at epoch 1418.

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

181

Fig. 4. The percent deviations between the experimental specific volume data of polymeric systems and those of calculated using the ANN–GCM model for training, validation, and test sets of data.

network topology. The best validation performance is 5.26 × 10−7 at epoch 1418. Fig. 3 shows a comparison between the experimental and calculated values of specific volume of polymeric systems for training, validation, and prediction sets. The correlation coefficient, R2 , for each plot has been given. As these plots show, there is very good agreement between experimental data and the results obtained from the ANN–GCM model predictions in training, validation, and prediction sets of data. The percent deviations between the experimental specific volumes of polymeric systems and those calculated using the ANN–GCM model for training, validation, and test sets of data have been shown in Fig. 4. Fig. 5 shows the percent deviation of the model results as a function of the data number (in a bar chart) for each of 2865 experimental data for training, validation, and test sets. Table 4 shows the overall minimum, maximum, and the average deviations for the calculated specific volume of the polymeric systems using the proposed network. The average relative deviations for train, validation, and test sets are 0.0403, 0.0439, and 0.0482, respectively. The results show that the ANN–GCM model can be accurately trained and the chosen topology can estimate the specific volume of polymeric systems up to high temperature and pressure conditions with good accuracy.

Fig. 3. Comparison between experimental and calculated values of the specific volume of polymeric systems for (a) training, (b) validation, and (c) test data.

the hidden layer for training, validation, and prediction sets of data. As observed in the figure, the optimal number of neurons in the hidden layer is 18. Thus, the best network topology was obtained as (17-18-1). Optimum weights and biases for designing the stated artificial neural network have been presented in Table 3. The profiles of the MSE for the training, validation and testing with respect to the number of epochs have been shown in Fig. 2 for the best

Fig. 5. Deviation of the model results as a function of the data number for each of 2865 experimental specific volume data for training, validation, and test sets.

182

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

Table 5 Comparison between this work and some other equations for the prediction of specific volume for selected polymeric systems. Compound

Polypropylene (PP) Poly vinyl methyl ether (PVME) Poly vinyl chloride (PVC) Poly 1-butene (PB1) Poly ␧-caprolactone (PCL) Polyethylene (PE) Poly ethylene glycol (PEG-200) Polystyrene (PS) polycarbonate bisphenol-A (PC) Poly vinylidene fluoride (PVDF) polypropylene glycol (PPG) Overall

AAD ISM EoS [6]

TM EoS [7]

Tait correlation [8]

ANN–GCM (this work)

0.38 0.33 0.43 0.98 0.79 0.92 0.25 0.75 1.07 0.33 0.50 0.61

0.29 0.28 0.59 0.84 0.34 0.62 0.02 – – – 0.21 0.40

0.11 2.07 0.17 0.70 0.12 0.27 – – – – – 0.57

0.02 0.04 0.06 0.03 0.02 0.02 0.06 0.12 0.03 0.06 – 0.05

To assess the performance of the this model in prediction of specific volumes of different polymeric systems, the AAD values between experimental and calculated specific volumes from ANN–GCM model and those predicted by some other methods, namely, modified Ihm-Song-Mason equation of state (ISM EoS) [14], Tao-Mason EoS (TM EoS) [13], and also Tait equation [13] have

been compared in Table 5 for selected polymeric systems. Fig. 6 shows a comparison between the calculated specific volumes of the ANN–GCM model (present work) and the results obtained by Sabzi and Boushehri EoS [8] for polypropylene (PP), polyethylene

Fig. 7. Deviation plot of the calculated specific volumes from the experimental data versus temperature for the ANN–GCM model (this work) and those of previous ANN model [13] for seven polymers including PEG, PP, PVC, PB1, PCL, PE and PVME.

Fig. 6. Comparison between the deviation plot for the calculated specific volume at different temperatures and pressures compared with the experiment [2,46] for (a) PP, (b) PEG, and (c) PVC. The open markers represent deviation of the calculated specific volumes for Sabzi and Boushehri EoS [8] and the filled markers represent deviations of the calculated specific volumes for the present work at different temperatures.

Fig. 8. Deviation plot of the calculated specific volumes from the experimental data versus temperature for the ANN–GCM model (this work) and those of previous ANN model [14] for ten selected polymers including PS, PC, PVDF, PEG, PP, PVC, PB1, PCL, PE and PVME.

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

glycol (PEG), and polyvinyl chloride (PVC). The deviation plot of the calculated specific volumes from the experimental data versus temperature for the ANN–GCM model (this work) and those of previous ANN model [13] for seven polymers including polyethylene glycol (PEG), polypropylene (PP), polyvinyl chloride (PVC), poly 1-butene (PB1), poly ␧-caprolactone (PCL), polyethylene (PE) and polyvinyl methyl ether (PVME) has been given in Fig. 7. The average absolute deviation for simple ANN model presented in references [13] and the ANN–GCM model presented in this work is 0.28 and 0.03, respectively for these seven selected polymers. Fig. 8 shows the deviation plot of the calculated specific volumes from the experimental data versus temperature for the ANN–GCM model (this work) and those of previous simple ANN model [14] for ten selected polymers including polystyrene (PS), polycarbonate bisphenol-A (PC), polyvinylidene fluoride (PVDF), polyethylene glycol (PEG), polypropylene (PP), poly vinylchloride’s properties (PVC), poly 1-butene (PB1), poly ␧-caprolactone (PCL), polyethylene (PE) and polyvinyl methyl ether (PVME). The average absolute deviation for simple ANN model presented in references [14] and the ANN–GCM model presented in this work is 0.20 and 0.05, respectively for these ten selected polymers. Much better results obtained from the ANN–GCM model provide good evidence for the ability of this model in predicting the specific volume of different polymeric systems up to extreme conditions. Artificial neural networks belong to a category of models called systems theoretic or blackbox models. In spite of ANN, other conceptual models and equations of state have several adjustable parameters for each compound. So, it can be claimed that the ANN is more general compared to other methods. The advantages of ANN compared to conceptual models are its high speed, simplicity and large capacity which reduce engineering attempts.

4. Conclusions We have developed an ANN–GCM method to predict the specific volume of the polymeric systems up to the extremely high temperature, high pressure (HTHP) conditions and found that it can be applied for these compounds successfully. An ANN–GCM model with (17-18-1) structure was designed to predict the specific volumes of different kinds of polymeric systems. Total number of experimental data used to design the stated networks is 2865, of which 75% were randomly chosen to train the networks, 10% for validation, and 15% to test it. Our results are in a good agreement with the experimental and the previous literature data up to high temperatures and pressures. In general, the ANN–GCM model can provide a simple procedure for prediction of the specific volume of the polymeric systems. This is a significant benefit in practical applications. The average relative deviations for train, validation, and test sets are 0.0403, 0.0439, and 0.0482, respectively.

Acknowledgements We gratefully acknowledge financial support from the Iran National Science Foundation (INSF) (research project 91002519) and Research Council of the University of Isfahan.

Appendix A. Instructions for running the program: The application of the model is very easy. Drag and drop the mat file (GC ANN Poly.mat file which is available as Supporting Information) into the current folder of the MATLAB software (any version). Then, load it to the workspace (using this command:  load (‘GC ANN Poly.mat’)). To get a response from the model, you can follow the following example step by step:

183

Assume that you want to predict the density of polybutyl acrylate at 303.15 K and 10 MPa. The group contribution parameters should be defined from the chemical structure of polybutyl acrylate (refer to the Supporting Information). The number of occurrences of the different groups in the monomeric structure of this polymer has been given in the parentheses as: O C O (1), HO C O (0), O (0), S (0), CF2 (0), >CH (1), CH2 (4), CH3 (1), >C< (0), >Si< (0), Ph (0), Ph (0), OCOO (0), CHCl (0). So, the following commands should be entered in the MATLAB command window:  input = [1 0 0 0 0 1 4 1 0 0 0 0 0 0 303.15 10 101]’;  SpecificVolume = sim(net,input) The result is 0.9136, where its experimental value is equal to 0.9137 (the percent deviation is −0.0077%). Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.fluid.2013.07.004. References [1] P.A. Albertsson, Partition of Cell Particles and Macromolecules, 3rd ed., John Wiley and Sons, New York, 1986. [2] P.A. Rodgers, Journal of Applied Polymer Science 48 (1993) 1061–1080. [3] G.R. Brannock, I.C. Sanchez, Macromolecules 26 (1993) 4970–4972. [4] G. Bogdanic, A. Fredenslund, Industrial and Engineering Chemistry Research 33 (1994) 1331–1340. [5] R.B. Gupta, J.M. Prausnitz, Fluid Phase Equilibria 117 (1996) 77–83. [6] A.C. Colin, R.G. Rubio, A. Compostizo, Polymer 41 (2000) 7407–7414. [7] K. Akbarzadeh, M. Moshfeghian, Fluid Phase Equilibria 187/188 (2001) 347–361. [8] F. Sabzi, A. Boushehri, European Polymer Journal 40 (2004) 1105–1110. [9] A.R. Berenji, E.K. Goharshadi, Polymer 47 (2006) 4726–4733. [10] E.K. Goharshadi, M. Imani, R. Rahimi-Zarei, F. Razghandi, M. Abareshi, A.R. Berenji, European Polymer Journal 46 (2010) 587–591. [11] M.M. Papari, M. Kiani, R. Behjatmanesh-Ardakani, J. Moghadasi, A. Campo, Journal of Molecular Liquids 161 (2011) 148–152. [12] M.M. Papari, R. Behjatmanesh-Ardakani, M. Kiani, J. Moghadasi, A. Campo, Colloid and Polymer Science 289 (2011) 1081–1087. [13] F. Yousefi, H. Karimi, European Polymer Journal 48 (2012) 1135–1143. [14] F. Yousefi, H. Karimi, Journal of Industrial and Engineering Chemistry 19 (2013) 498–507. [15] A. Mohebbi, M. Taheri, A. Soltani, International Journal of Refrigeration 31 (2008) 1317–1327. [16] K. Liu, Y. Wu, M.A. McHugh, H. Baled, R.M. Enick, B.D. Morreale, Journal of Supercritical Fluids 55 (2010) 701–711. [17] J.A. Lazzus, Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 213–232. [18] F. Gharagheizi, Journal of Hazardous Materials 170 (2009) 595–604. [19] F. Gharagheizi, A. Eslamimanesh, A.H. Mohammadi, D. Richon, Industrial and Engineering Chemistry Research 50 (2011) 5815–5823. [20] F. Gharagheizi, O. Babaie, S. Mazdeyasna, Industrial and Engineering Chemistry Research 50 (2011) 6503–6507. [21] A.A. Rohani, G. Pazuki, H.A. Najafabadi, S. Seyfi, M. Vossoughi, Expert Systems with Applications 38 (2011) 1738–1747. [22] F. Gharagheizi, A. Eslamimanesh, A.H. Mohammadi, D. Richon, Journal of Chemical & Engineering Data 56 (2011) 2460–2476. [23] C.L. Aguirre, L.A. Cisternas, J.O. Valderrama, International Journal of Thermophysics 33 (2012) 34–46. [24] F. Gharagheizi, P. Ilani-Kashkouli, A.H. Mohammadi, Fluid Phase Equilibria 329 (2012) 1–7. [25] J.A. Lazzus, Fluid Phase Equilibria 313 (2012) 1–6. [26] A.Z. Hezave, M. Lashkarbolooki, S. Raeissi, Fluid Phase Equilibria 314 (2012) 128–133. [27] M. Moosavi, N. Soltani, Thermochimica Acta 556 (2013) 89–96. [28] Z. Zhang, K. Friedrich, Composites Science and Technology 63 (2003) 2029–2044. [29] J. Taskinen, J. Yliruusi, Advanced Drug Delivery Reviews 55 (2003) 1163–1183. [30] M. Hagan, M.H.B. Demuth, M.H. Beale, Neural Network Design, International Thomson Publishing, Boston, 2002. [31] F. Gharagheizi, Computation Materials Science 40 (2007) 159–167. [32] Project 801, Evaluated Process Design Data, Public Release Documentation, Design Institute for Physical Properties (DIPPR), American Institute of Chemical Engineers (AIChE), 2006. [33] C.K. Kim, D.R. Paul, Polymer 33 (1992) 1630–1639. [34] N. Mekhilef, Journal of Applied Polymer Science 80 (2001) 230–241.

184

M. Moosavi, N. Soltani / Fluid Phase Equilibria 356 (2013) 176–184

[35] R.D. Maier, J. Kressler, B. Rudolf, P. Reichert, F. Koopman, H. Frey, R. Mühlhaupt, Macromolecules 29 (1996) 1490–1497. [36] R.D. Maier, Thermodynamik und Kristallisation von Metallocen-Polyolefinen (experimental data by R.-D. Maier), Albert-Ludwigs-Universität, Freiburg i. Br., 1999 (Ph.D. Thesis). [37] Y. Sato, Y. Yamasaki, S. Takishima, H. Masuoka, Journal of Applied Polymer Science 66 (1997) 141–150. [38] S. Janssen, D. Schwahn, K. Mortensen, T. Springer, Macromolecules 26 (1993) 5587–5591. [39] Y. Sato, K. Inohara, S. Takishima, H. Masuoka, M. Imaizumi, H. Yamamoto, M. Takasugi, Polymer Engineering and Science 40 (2000) 2602–2609. [40] T. Ougizawa, G.T. Dee, D.J. Walsh, Polymer 30 (1989) 1675–1679. [41] J.H. Chu, D.R. Paul, Polymer 40 (1999) 2687–2698. [42] V.K. Sachdev, U. Yashi, R.K. Jain, Journal of Polymer Science Part B: Polymer Physics 36 (1998) 841–850.

[43] R.W. Wind, Untersuchungen zum Phasenverhalten von Mischungen aus Ethylen, Acrylsäure und Ethylen-Acrylsäure-Copolymeren unter hohem Druck, TH Darmstadt, 1992 (Ph. D. Thesis). [44] J.H. Kim, M.S. Whang, C.K. Kim, Macromolecules 37 (2004) 2287–2293. [45] Y.A. Fakhreddine, P. Zoller, Journal of Polymer Science Part B: Polymer Physics 29 (1991) 1141–1146. [46] M.J. Lee, C.K. Lo, H. Lin, Journal of Chemical and Engineering Data 43 (1998) 1076–1081. [47] R. Haghbakhsh, H. Adib, P. Keshavarz, M. Koolivand, S. Keshtkari, Thermochimica Acta 551 (2013) 124–130. [48] J. Li, R. Tillner-Roth, H. Sato, K. Watanabe, International Journal of Thermophysics 20 (1999) 1639–1651. [49] G. Giuliani, S. Kumar, P. Zazzini, F. Polonara, Journal of Chemical and Engineering Data 40 (1996) 903–908. [50] F. Amato, J.L. Gonzalez-Hernandez, J. Havel, Talanta 93 (2012) 72–78.