Document Type: Research Paper
Authors
^{1} Advanced Membrane and Biotechnology Research Center, Babol Noshirvani University of Technology, Babol, Iran
^{2} Faculty of Chemical Engineering, Babol Noshirvani University of Technology, Babol, Iran
Abstract
Keywords
1. Introduction
Surface tension is a characteristic property of fluids as scientific and technological researches on many areas need data on surface tension of the materials, such as chemical engineering, materials engineering, oil recovery, environmental protection, etc [1,2]. The surface tension of pure liquids and of liquid mixtures is often required in calculations such as those involving flow through porous media or boiling heat transfer.
In addition to experimental measurements, numerous theoretical researches have been carried out on the surface tension modeling [35]. Some of the presented models have a very strong theoretical basis such as those that are based on statistical mechanical theories of liquids and density functional theory [2,6]. Also corresponding state principles have been employed to predict the surface tension of pure materials [710]. The experimental data of surface tension for nonpolar liquids have been correlated by utilizing the Riedel parameter at the critical point [7]. The critical temperature, pressure and acentric factor have been used as correlation parameters for determining the surface tension [9].
Tworeference corresponding states methods were initially proposed by Rice and Teja [11] in which critical temperature and volume were used to obtain a correlation. Later, Zuo and Stenby employed the same method for calculating surface tensions where critical pressure and temperature were considered as correlation parameters [12]. These equations have not presented satisfactory results for chemical compounds with strong hydrogenbonding forces. To overcome these hurdles, Sastri and Rao introduced a correlation based on critical pressure and temperature, normal boiling point temperature, reduced temperature and reduced boiling temperature [13]. Although these equations are usually much simpler, their dependency on experimental data for each component limits their application to the cases for which some experimental data exist. These methods usually fail to give accurate predictions for other situations. The surface tension of a liquid mixture is not a simple function of the surface tensions of the pure components. In general, there are several approaches for estimating the surface tension of mixtures (i) based on empirical or semiempirical relations suggested first for pure fluids and (ii) based on statistical mechanical grounds. It is impractical to measure the surface tension for all liquids and liquid mixtures of interest and a method for the prediction of surface tension is therefore of practical importance [11,14,15].
Because of nonlinear nature of surface tension, artificial neural network method may be considered as an alternative tool for the prediction of surface tension. Gharagheizi, et al. applied an artificial neural networkgroup contribution method to predict the surface tension of pure chemical compounds [16]. Kumar, et al. effectively used parachor, density and refractive indices as input parameters required in the neural network for the prediction of surface tension of various polar and nonpolar compounds [17]. Strechan, et al. used artificial neural network for correlations of the surface tension of molecular liquids [10]. Furthermore, artificial neural network has been applied for surface tension prediction of pure liquid metals [18]. To the authors' best knowledge, there has been no study on the application of artificial neural network for the surface tension prediction of hydrocarbon mixtures.
In this study, a feedforward artificial neural network with Levenberg–Marquardt training algorithm was applied in order to investigate its capability in prediction of the surface tension of 20 hydrocarbon mixtures. The proposed ANN model results were compared with four wellknown classical models.
2. Research Method
2.1. Classical models
Numbers of correlations based on the law of corresponding states have been developed for the prediction of surface tension (σ) which relate surface tension to the absolute temperature (T). Brock and Bird [7] proposed Eq. 1 for nonpolar liquids.
(1) 
where σ is the surface tension (dyn/cm) and is the Riedel parameter [19] at the critical point and is defined through Eq. 2.
(2) 
where T is the absolute temperature (K), P is pressure (bar) and subscripts c, r and b denote the critical, reduced and boiling values respectively.
Pitzer [20] presented a corresponding state relation for surface tension in terms of critical pressure (P_{c}), critical temperature (T_{c}) and acentric factor (ω) as shown in Eq. 3.
(3) 
Zuo and Stenby [12] used a tworeference fluid corresponding state to estimate the surface tensions as was shown in Eq. 4.
(4) 
In this method, the surface tension for the fluid of interest is related to the surface tension of two reference fluids, methane (1) and nOctane (2) by Eq. 5.
(5) 
where the surface tension of methane is calculated by Eq. 6.
(6) 
and the surface tension of noctane is calculated by Eq. 7.
(7) 
Even though the three abovementioned correspondingstates methods are satisfactory for the nonpolar liquids, they are not suitable for compounds that exhibit strong hydrogenbonding (e.g. alcohols, acids). To deal with these types of compounds, Sastri and Rao [13] modified the equations as presented in Eq. 8.
(8) 
where the values for the constants are presented in Table 1.
Table 1. Values for the constants of Eq. 8. 


K 
X 
Y 
Z 
m 
Alcohols 
2.28 
0.25 
0.175 
0 
0.8 
Acids 
0.125 
0.50 
1.5 
1.85 
1.22 
All others 
0.158 
0.50 
1.5 
1.85 
1.22 
2.2. Artificial neural network (ANN)
Neural networks are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the network function is determined largely by the connections between elements. One can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Commonly, neural networks are adjusted or trained, so that a particular input leads to a specific target output.
In an ANN, a neuron sums the weighted inputs from several connections and then outputs of neurons are produced by applying transfer function to the sum. There are many transfer functions but the most common one is sigmoid which is used in this study as presented in Eq. 9.
(9) 
where is the sum of weighted inputs to each neuron and is the output of each neuron which is calculated through Eq. 10.
(10) 
where denotes connection between node j of interlayer l to node i of interlayer l1, b_{j} is a bias term and n is the number of neuron in each layer. In any interlayer l and neuron j input values integrate and generate.
In order to minimize the difference between experimental data and the data calculated by neural network, the aforementioned process repeats for the total number of training data. After training, validation of neural network can be done by testing the data.
Numerous types of the artificial neural networks exist such as multilayer perceptron (MLP), radial basis function (RBF) networks and recurrent neural networks (RNN) where the former was used in this study. Multilayer perceptron networks are one of the most popular and successful neural network architectures which are suited to a wide variety of applications such as the prediction and process modeling [2123].
2.3. Preparation of dataset
578 datasets for surface tension of 20 hydrocarbon mixtures at different temperature and composition were collected [24] in addition to the critical temperature, critical pressure, critical volume, acentric factor, normal boiling point, molecular weight and ideal liquid density for pure components which were considered as the input for the ANN. Table 2 shows the list of hydrocarbon mixtures, the number of data point for each mixture and the temperature range. Furthermore, Table 3 shows the values of the physical properties of pure hydrocarbons. In this study, the data sets were divided into three parts: training subset (60% of all data), validation subset (10% of all data) and testing subset (30% of all data). To avoid larger number from overriding smaller number, all data is normalized between [0.10.9] using Eq. 11.
(11) 
Table 2. The list of experimental data used in this study [24]. 

Mixtures 
Number of data point 
Temperature (K) 
Tetrachloromethane  Iodomethane 
47 
288308 
Tetrachloromethane  Nitromethane 
33 
303317 
Tetrachloromethane  Methanol 
12 
308 
Tetrachloromethane  Acetonitrile 
44 
298317 
Tetrachloromethane  Iodoethane 
33 
298322 
Tetrachloromethane  Nitroethane 
33 
303317 
Tetrachloromethane  Ethanol 
18 
293345 
Tetrachloromethane  Dimethylsulfoxide 
24 
303322 
Formic acid  Acetic acid 
34 
287349 
Formic acid  Pyridine 
27 
287298 
Formic acid  2methylaniline 
8 
298 
Iodomethane  Acetic acid 
33 
293313 
Nitromethane  Acetic acid 
11 
293 
Nitromethane  Benzene 
26 
293303 
Nitromethane  Cyclohexane 
6 
293 
Methane  Propane 
46 
258338 
Methanol  Acetic acid 
11 
308 
Methanol  Ethanol 
50 
293333 
Methanol  Dimethylsulfoxide 
55 
293323 
Methanol  Benzene 
27 
273303 
Table 3. Physical properties of pure hydrocarbons used in this study. 

Hydrocarbons 
Critical Temperature (˚C) 
Critical pressure (bar) 
Critical volume (m^{3}/kgmol) 
Acentric Factor 
Normal boiling point (˚C) 
Molecular weight 
Ideal liquid density (kg/m^{3}) 
Tetrachloromethane 
283.3 
45.6 
0.2759 
0.193 
76.75 
153.8 
1601 
Nitromethane 
313.9 
63.1 
0.1732 
0.31 
101.1 
61.04 
1138 
Methanol 
239.4 
73.76 
0.1270 
0.557 
64.65 
32.04 
795.7 
Acetonitrile 
272.4 
48.2 
0.1729 
0.327 
81.65 
41.05 
782 
Iodoethane 
278.9 
47 
0.2705 
0.1563 
72.45 
156 
1961 
Nitroethane 
321.8 
51.16 
0.2295 
0.3684 
114.8 
75.07 
1052 
Ethanol 
240.8 
61.47 
0.1671 
0.6444 
78.25 
46.07 
796 
Dimethylsulfoxide 
455.9 
56.5 
0.227 
0.2806 
189.0 
78.14 
1105 
Formic acid 
296.9 
55 
0.112 
0.3525 
100.6 
46.03 
1225 
Pyridine 
346.9 
56.2 
0.254 
0.243 
115.2 
79.10 
988.8 
2 methylaniline 
420.9 
37.5 
0.35 
0.438 
200.4 
107.2 
1002 
Iodomethane 
251.9 
65.9 
0.1842 
0.1446 
42.55 
141.9 
2293 
Acetic acid 
319.6 
57.7 
0.1710 
0.447 
118.0 
60.05 
1052 
Benzene 
288.9 
49.24 
0.26 
0.215 
80.09 
78.11 
882.2 
Cyclohexane 
280.1 
40.53 
0.308 
0.2133 
80.73 
84.16 
781.8 
Methane 
82.45 
46.41 
0.099 
0.0115 
161.5 
16.04 
299.4 
Propane 
96.75 
42.57 
0.2 
0.1524 
42.1 
44.10 
506.7 
2.4. ANN modeling
Programming, validation, training and testing of the ANN model were carried out by MATLAB 7.7.0. To determine the optimized values of weights and biases, the following steps were done:
3. Results and Analysis
Three layers feeding forward neural network were used for surface tension prediction of 20 hydrocarbon mixtures and all parameters of neural network were determined by trial and error procedure. According to the experimental data and for a fair comparison with the classical models, temperature (T), composition (x), critical temperature (T_{c}), critical pressure (P_{c}), critical volume (V_{c}), acentric factor (ω), normal boiling point (T_{b}), molecular weight (M) and ideal liquid density (ρ) of mixtures were used as inputs to our proposed neural network.
The physical properties of hydrocarbon mixture were related to the physical properties of pure components, using mixing rules as were shown in Eqs. 1213 for mixture critical temperature (T_{cm}) and mixture critical pressure (P_{cm}) respectively.
(12) 

(13) 
where x_{i} and x_{j} are the composition of component i and j in the mixture. Binary critical temperature (T_{cij}) and binary critical pressure (P_{cij}) were obtained through Eqs. 1415, respectively.
(14) 

(15) 
Furthermore, mixture normal boiling point (T_{bm}), mixture critical volume (V_{cm}), mixture acentric factor (), mixture molecular weight (M_{m}) and mixture ideal liquid density () were calculated using Eqs. 1620, respectively.
(16) 

(17) 

(18) 

(19) 

(20) 
where subscripts m and i denote the mixture and pure value of each physical property, respectively.
Sigmoid function was used as transfer function in hidden layers and purelin function was used as the transfer function of output layers. Also LevenbergMarquardt back propagation learning algorithm was used for training. Usually one hidden layer is enough but the numbers of neurons in hidden layers need to be optimized for each problem. In order to optimize the number of neurons in hidden layers, average relative deviation (ARD), (calculated through Eq. 21) of testing data versus the neuron number in hidden layers is plotted, as was shown in Fig. 1. The Results showed that 951 is the best topology of the neural network (Fig. 2). The cross plot graph which shows the results of training and testing calculations are presented in Fig. 3.
(21) 
Figure 1. ARD of testing data versus neuron number in hidden layer. 
Figure 2. Topology of the proposed neural network. 
Figure 3. Cross plot graph of the results of training and testing calculations. 
Surface tension of each hydrocarbon mixtures was further calculated by four aforementioned classical models: Brock and Bird (Eq. 1), Pitzer (Eq. 3), ZuoStenby (Eq. 4) and SastriRao (Eq. 8). The mixing rules used in these four models were the same as the mixing rules used in artificial neural network method.
The accuracy of artificial neural network and four wellknown classical models were tabulated in Table 4. They indicate that ARD of artificial neural network is 3.72223% while the ARD of Brock and Bird, Pitzer, ZuoStenby and SastriRao models are 28.56102, 25.23901, 25.49967 and 13.58419 respectively.
One of the best advantages of artificial neural network is low dependency of accuracy of this method to the type of compounds. Classical methods give quite accurate results for some compounds while their answers for some other compounds may be very inaccurate. To have a quantitative measure of this quality, standard deviation parameter was calculated for each method which is presented in Table 4. The standard deviation of ARD for Brock and Bird, Pitzer, ZuoStenby and SastriRao models are 23.77569, 18.44848, 13.00388 and 9.63137 respectively while standard deviation of ARD for artificial neural network is 3.63001.
Table 4. Accuracy of artificial neural network and four wellknown classical models. 

Compound 
Brock –Bird [7] 
Pitzer [20] 
Zuo –Stenby [12] 
Sastri –Rao [13] 
ANN 
Tetrachloromethane  iodomethane 
2.7273 
2.0888 
17.3285 
3.538 
2.1173 
Tetrachloromethane  nitromethane 
10.1782 
11.6841 
19.9798 
3.4726 
2.356 
Tetrachloromethane  methanol 
41.3494 
50.4831 
50.631 
15.6244 
2.387 
Tetrachloromethane  acetonitrile 
1.8799 
4.2103 
16.5798 
4.1368 
9.9893 
Tetrachloromethane  iodoethane 
13.3881 
10.2459 
13.0284 
11.7174 
1.9742 
Tetrachloromethane  nitroethane 
7.4096 
10.5393 
13.6943 
1.5743 
1.3618 
Tetrachloromethane  ethanol 
27.1397 
34.5764 
37.6013 
7.6215 
0.9921 
Tetrachloromethane  dimethylsulfoxide 
13.8216 
17.434 
15.0959 
8.1284 
8.077 
formic acid  acetic acid 
28.1053 
30.3465 
15.0238 
12.9732 
1.2466 
formic acid  pyridine 
12.8345 
11.2285 
19.3833 
22.2164 
8.8339 
formic acid  2methylaniline 
5.4993 
3.8937 
26.2993 
16.7077 
0.0213 
Iodomethane  acetic acid 
33.716 
36.5589 
36.8824 
15.3109 
4.549 
Nitromethane  acetic acid 
32.0827 
34.4518 
19.3669 
9.3977 
1.2423 
Nitromethane  benzene 
92.2474 
12.6289 
10.7804 
20.7877 
7.4623 
Nitromethane  cyclohexane 
6.1214 
7.8804 
6.8466 
2.93 
0.8193 
Methane  propane 
37.3632 
41.1241 
25.4103 
44.6959 
2.4441 
Methanol  acetic acid 
57.8839 
65.3992 
43.6485 
19.1128 
13.4216 
Methanol  ethanol 
71.7885 
25.3453 
48.2051 
21.9246 
1.791 
Methanol  dimethylsulfoxide 
47.1162 
56.4907 
33.2154 
16.2331 
1.6447 
Methanol  benzene 
28.5681 
40.259 
40.9924 
13.5803 
1.7139 
Average 
28.56102 
25.23901 
25.49967 
13.58419 
3.72223 
Standard Deviation 
23.77569 
18.44848 
13.00388 
9.63137 
3.63001 
4. Conclusion
In this study, artificial neural network was used to predict the surface tension of hydrocarbon mixtures. The accuracy of our proposed model was compared to four wellknown empirical equations. It showed higher accuracy for the artificial neural network method. Also, results of standard deviation indicated that these empirical relations are so dependent on the type of compounds and some special parameters, while ANN is more independent. The optimized neural network parameters such as inputs and the number of neurons in hidden layers were presented for the calculation of surface tensions for the before mentioned compounds and can be used by other researchers in solving the problems which deal with the surface tension calculations.