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ionic liquids mixtures using intelligent models
Accepted Manuscript The prediction of liquid-liquid equilibria for benzene/alkane/ionic liquids mixtures using intelligent models
Fariborz Shaahmadi, Mohammad Amin Anbaz PII: DOI: Reference:
S0167-7322(16)32771-4 doi: 10.1016/j.molliq.2017.02.108 MOLLIQ 7021
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 September 2016 6 February 2017 25 February 2017
Please cite this article as: Fariborz Shaahmadi, Mohammad Amin Anbaz , The prediction of liquid-liquid equilibria for benzene/alkane/ionic liquids mixtures using intelligent models. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.02.108
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ACCEPTED MANUSCRIPT
The Prediction of Liquid-Liquid Equilibria for
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Benzene/Alkane/Ionic Liquids Mixtures Using Intelligent
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Models
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Fariborz Shaahmadi*, Mohammad Amin Anbaz Gas Engineering Department, Petroleum University of Technology (PUT), Post Box: 63431,
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Ahwaz, Iran
*
Corresponding author. E-mail addresses: [email protected].
ACCEPTED MANUSCRIPT Abstract Liquid-liquid extraction of aromatics from aliphatic hydrocarbons is a main process in petrochemical industry. Therefore, accurate predicting the liquid-liquid equilibria (LLE) for ternary systems of aromatic/alkane/ionic liquid can result in better liquid-liquid extraction. In this
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study, three intelligence methods including artificial neural network (ANN), support vector
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machine (SVM) and least square support vector machine (LSSVM) have been applied to predict the thermodynamic phase behavior of LLE for benzene/alkane/ionic liquid ternary systems.
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Optimization techniques such as particle swarm optimization (PSO), genetic algorithm (GA) and shuffled complex evolution (SCE) have been used to obtain adjustable parameters of SVM and
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LSSVM models. The results of prediction operation demonstrate that there was good agreement
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between the estimation of intelligent models and the experimental data of LLE.
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Key words: Liquid-liquid extraction, Ternary mixture, Artificial neural network, Support vector
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machine
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1- Introduction The separation of mixtures of aromatic and aliphatic hydrocarbons is a main problem in
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petrochemical industry, because these components have near boiling points and their
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combinations form azeotropic mixtures [1]. Liquid-liquid extraction is widely used as an
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industrial separation process for azeotropic mixtures unfit for distillation. There are traditional organic solvents employed industrially for liquid-liquid extraction such as sulfolane, N-methyl
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pyrrolidone (NMP), N-formyl morpholine, ethylene glycols and propylene carbonate [2-7].
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However, these solvents have some problems such as volatility, flammability and toxicity. Ionic liquids can be environmentally considered as a proper alternative for the traditional solvents in
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liquid-liquid extraction because of their properties such as negligible vapor pressure (non-
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volatility), non-flammable, high chemical and thermal stability.
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In recent years, the use of ionic liquids (IL) has been gradually increased as solvents for the liquid-liquid extraction of aromatic hydrocarbons from their mixtures with aliphatic hydrocarbons
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[8-15]. Consequently, there are a considerable number of publications conducted by researchers
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around the world reporting experimental liquidβliquid equilibrium (LLE) data for ternary systems of aromatic/aliphatic hydrocarbon/ionic liquid to develop the liquid-liquid extraction process. Most of these studies used the thermodynamic models such as NRTL and UNIQUAC to correlate the experimental LLE data of ternary systems [16-19]. These models usually required several tuning parameters adjusted according to actual experimental data of each ternary system and cannot be applied as a predictive tool for other ternary systems. It should be noted that these limited models are not accurate and reliable without adjusting their parameters for each system.
ACCEPTED MANUSCRIPT Bonilla-Petriciolet et al. [20] utilized the optimization tools such as particle swarm optimization (PSO) to adjust the fitting parameters of thermodynamic models for phase stability and equilibrium calculations through global minimizing of Gibbs energy. It becomes a necessity to have a simple and robust model to accurately predict the thermodynamic
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phase behavior of LLE for aromatic/alkane/ionic liquid systems. Therefore, intelligence methods
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can be used as appropriate alternatives for the conceptual models, physically based models, and conventional statistical models. Such intelligence methods include artificial neural network
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(ANN), support vector machine (SVM), least square support vector machine (LSSVM), etc. ANN model is a computational system based on the operation of biological neural networks. ANN
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method has become popular as feasible tool in a variety of fields of study in recent years [21-28].
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Torrecilla et al. [29] used ANN method for the prediction of the LLE data of only four toluene/heptane/ionic liquid systems at 313.2 and 348.2 K. They concluded that ANN was
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successfully used as a predictive tool for ternary LLE data. Another intelligence model is SVM,
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proposed by Vapnik [30] increasingly gained popularity to be employed in various engineering and science disciplines. According to SVM models, the results are obtained by quadratic
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programming (QP). Although several studies successfully used the SVM method in various
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applications [31-34], this method has not been applied to evaluate the LLE data of these ternary systems. Finally, the LSSVM method, introduced by Suykens and Vandewalle [35] as a modified version of conventional SVM, has the idea of the equality constraints which is the major advantage of LSSVM over the original SVM . In this study, the thermodynamic phase behavior of LLE of benzene/alkane/ionic liquid ternary systems has been predicted by intelligence methods including ANN, SVM and LSSVM models. These methods have been constructed by wide range of experimental LLE data gathered from
ACCEPTED MANUSCRIPT literature (664 data points) for this ternary system with 28 different ionic liquids. The solute distribution coefficient (π½) of benzene/alkane/ionic liquid systems has been estimated as a main parameter to evaluate the reliability of the ionic liquids as solvents for liquid-liquid extraction.
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2- Model development
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2-1. Data acquisition
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The reliable experimental data are required to have an accurate and precise model. Therefore, the
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experimental LLE data of benzene + Alkane + ionic liquid system at atmospheric pressure are gathered from published studies in the literature over the years. In this study, a data set consisting
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of 664 data points for 28 different ionic liquids and n-alkanes range in carbon number from C6 to C16 have been collected as shown in Table 1 [36-63]. The IUPAC names and chemical formulas of
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studied ionic liquids have been presented in Table 2. Table 3 shows the parameter ranges of input
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and output data in which the input data are feed compositions (π1 , π2 and π3 ), temperature,
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molecular weight of alkane and properties of ionic liquid (ππ , ππ , ππ and π), and the output parameter is the solute distribution coefficient of benzene/alkane/ionic liquid system. The
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[64, 65].
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properties of ionic liquids were obtained by group contribution method presented by Valderama
2-2. Artificial neural network ANNs, considered as mathematical tools, can learn from an experience, improve its performance and adopt to the environment changes. ANNs are known for their capability of performing nonlinear statistical modeling, relating input and output parameters and generalizing the results. In this study, multilayer perceptron (MLP) neural network as one of the most powerful types of
ACCEPTED MANUSCRIPT ANNs has been used to estimate the thermodynamic phase behavior of LLE for benzene/alkane/ionic liquid ternary systems. MLP neural networks originally are made of three different types of layers including input layer, hidden layer(s), and output layer. Figure 1 shows the structure of the MLP neural network
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considered in this study. As can be seen from Figure 1, each layer is composed of some fully
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interconnected processing units called neurons. It should be mentioned that the number of neurons in the input and output layers are corresponded to number of input and output data, respectively.
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The number of neurons in hidden layers was obtained by a trial and error method. Input layer received all input signals, and then these signals are transmitted to other neurons for processing in
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hidden layers. Then, the output layer received the information from hidden layers. Synaptic
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weights and biases are the parameters of this type of network. These parameters are adjusted by using network error as a benchmark through receiving the input data within the training algorithm.
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MLP neural network was trained by an iterative optimization procedure starting with random
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guesses on synaptic weights and biases. This training process continues to reach a minimum value
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in squared weights and errors over several iterations. 2-3. Support vector machine (SVM)
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SVM model is an effective strategy and analytical tool developed from the machine-learning community for classification and regression purposes. According to this method, consider an assumed train sample of N data samples in which x is input vector with a dimension of N Γ n (n is the number of input parameters) and π¦π is the relevant target sample. The aim of the SVM method is to find an regressed function π(x) as follows:
ACCEPTED MANUSCRIPT π(x) = π€ π π(x) + π
(1)
Where π€ π is weight vector, π(x) represents the kernel function, and b is a bias term. In order to calculate π€ π and b, Vapnik [30] proposed minimization of the following cost function: π
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1 πΆππ π‘ ππ’πππ‘πππ = π€ π + π β(ππ β ππβ ) 2
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π=1
(3)
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π = 1, 2, β¦ , π π = 1, 2, β¦ , π π = 1, 2, β¦ , π
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To satisfy constraints: π¦π β π€ π π(xk ) β π β€ π + ππ {π€ π π(xk ) + π β π¦π β€ π + ππ ππ . ππβ β₯ 0
(2)
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Where xk and π¦π represent the input vector at the train sample k and the relevant target sample, respectively. The Ξ΅ determines the fixed precision of the function approximation and ππ (or ππβ ) is
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slack variable. The variable c is the deviation from the desired Ξ΅ in Eq. (3) considered as the
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tuning parameter of the SVM. Lagrangian is utilized to minimize the cost function as follows: π
π
π
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1 πΏ(π. πβ ) = β β (ππ β ππβ )(ππΌ β ππΌβ )πΎ(xk , xI ) β π β(ππ β ππβ ) + β π¦π (ππ β ππβ ) 2 π
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π.πΌ=1
β(ππ β ππβ ) = 0 π=1
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ππ . ππβ β [0, π]
πΎ(xk . xI ) = π(xk )π π(xI )
π = 1, 2, β¦ , π
π=1
(4)
π=1
(5)
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Where ππ and ππβ are Lagrangian multipliers. Finally, the resulting form of SVM method is obtained as follows:
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π(x) = β (ππ β ππβ ) πΎ(x, xk ) + π
(7)
π.πΌ=1
The unknowns (ππ , ππβ , and b) can be obtained by solving a quadratic programming problem.
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Therefore, the tuning parameters of SVM model are Ξ΅, c and a parameter of kernel function.
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2-4. Least-squares support vector machine (LSSVM)
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The least square modification of SVM (LSSVM) is developed by Suykens and Vandewalle [35]
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through introducing a system of equations instead of a nonlinear quadratic programming in original SVM. The method applied in the LSSVM model is briefly introduced as following. A
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new optimization issue is applied in LSSVM as follows:
(8)
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k=1
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1 1 πΆππ π‘ ππ’πππ‘πππ = π€ π π€ + πΎ β ππ2 2 2
Where Ξ³ is the factor of regularization considered as a tuning parameter in LSSVM method and
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πk represents the error of LSSVM in training phase. An equality constraint is established to
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minimize the equation (8): π¦k = π€ π π(xk ) + π + πk
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(9)
The Lagrangian is applied to locate the solutions of the optimization in equation (8) along with its constraint defined in Eq. (9) as follows: π
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1 1 πΏ(π€. π. π. π) = π€ π π€ + πΎ β ππ2 β β ππ (π€ π π(xk ) + π + ππ β π¦π ) 2 2 π=1
π=1
(10)
ACCEPTED MANUSCRIPT Where ππ are support values or Lagrange multipliers. In order to acquire the solution, the derivatives of Eq. (10) should be equated to zero. Therefore, the following equations are obtained: π
ππΏ = 0 β π€ = β ππ π(xk ) ππ€ ππΏ = 0 β β ππ = 0 ππ π=1
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ππΏ = 0 β ππ = πΎππ π = 1,2, β¦ , π πππ ππΏ = 0 β π€ π π(xk ) + π + ππ β π¦π = 0 π = 1,2, β¦ , π {πππ
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π=1
π
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Eq. (11) shows that there are 2N + 2 equations and 2N + 2 unknown parameters (ππ , ππ , π€, and
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b). Thus, the parameters of LSSVM can be obtained by solving the system of equations defined in
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π 1ππ 0 ] [ ]=[ ] β1 π¦ Ξ©+πΎ πΌ πΌ
(12)
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[
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Eq. (11). The system of linear equations can be rewritten in matrix form as:
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Where π¦ = [π¦1 β¦ π¦π ]π , 1π = [1 β¦ 1]π , πΌ = [πΌ1 β¦ πΌπ ]π , I is an identity matrix, and Ξ© = π(xπ )π in which π(xπ ) = πΎ(xπ , xl ) β π, π = 1,2, β¦ , π. πΎ(xπ . xl ) is the kernel function taking into
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account the Mercer limitation [66]. The radial basis function (RBF) Kernel has been used in this study as follows:
πΎ(x. xk ) = exp(
β|| xk β x||2 ) π2
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ACCEPTED MANUSCRIPT where π 2 is squared bandwidth which is optimized through an external optimization technique during the training process as shown in Figure 2. The resulting formulation of LSSVM model for function estimation is then written as follows: π
π¦(x) = β πΌπ πΎ(x, xk ) + π
(14)
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π=1
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Finally, it should be noted that LSSVM model is exactly influenced by two adaptable parameters
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π 2 and regularization parameter (Ξ³). 2-5. Data normalization
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Data must be well processed and scaled prior to input to the intelligent models. Therefore, all the
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inputs and output were normalized between 0 and 1, because higher valued input variables may be likely to suppress the impact of the smaller ones. The following equation has been utilized for
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normalization as below:
(15)
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x xπ = ( ) Γ 0.8 + 0.1 1.5 Γ xπππ₯
Where xπ represents normalized data, xπππ₯ is the maximum value of the data and x is the real
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data. It should be noted that the normalization of data has no effect on the model results. At the end, these values were returned to their original values. 2-6. Computational procedure The database was randomly divided into two subsets involving train and test sets to construct the studied models. In this study, 80% of all data points were randomly selected for train set to generate the model structure. The remaining 20% of data points have been applied for test set to
ACCEPTED MANUSCRIPT investigate modelsβ prediction validity and capability. As previously mentioned, nine input parameters have been applied including temperature, composition of three components, molecular weight of alkane, critical temperature of IL, critical pressure of IL, molecular weight of IL and the acentric factor of IL. These input data have been used to develop the intelligent models. In this
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study, the structure of ANN model was determined through evaluating different networks. The
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tuning parameters of SVM (c, Ξ΅, and π 2 ) and LSSVM (Ξ³ and π 2 ) have been adjusted with the
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experimental train data using non-traditional optimization techniques such as PSO, genetic
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algorithm (GA) and shuffled complex evolution (SCE) [67]. SVM and LSSVM models have been constructed after finding the optimum parameters. The prediction data were obtained by replacing
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the test data sample into these models. The validity of models can be evaluated through these
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predictions.
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2-7. Accuracy of intelligent models
Several statistical and graphical error analyses have been utilized to check the precision and
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reliability of studied intelligent models such as coefficient of determination (π 2 ), root mean
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square error (RMSE), and standard deviation error (STD). The formulation of these statistical parameters are as follows:
π πππΈ = (
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2 βπ π=1(π½ππππ (π) β π½πΈπ₯π (π)) π =1β π Μ (π))2 βπ=1(π½ππππ (π) β π½πΈπ₯π 2
βπ π=1 (π½ππππ (π) β π½πΈπ₯π (π)) π
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2
)0.5
(17)
ACCEPTED MANUSCRIPT π
Μ (π))2 (π½ππππ (π) β π½πΈπ₯π πππ· = β( )0.5 π
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π=1
Where π½πΈπ₯π and π½ππππ are the experimental and predicted solute distribution coefficient, Μ respectively. And π½πΈπ₯π is the mean of the experimental data values of solute distribution
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coefficient.
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In this study, crossplot and error distribution curves have been applied to visualize the validity of studied intelligent models. In crossplot curve, the predicted values of intelligent models are
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sketched versus the experimental values. The consistency of the model can be obtained by
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concentrating the data points around the 45 Β° line which indicates the perfect model line.
3- Results and discussion
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distribution around the zero error line.
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According to error distribution curve, the error trend can be observed by showing error
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Three intelligent models have been applied to evaluate the phase behavior of ternary LLE
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systems. The feed compositions, temperature, molecular weight of alkane and properties of ionic liquid have been considered as correlating parameters of the solute distribution coefficient for
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benzene/alkane/ionic liquid system. The optimum structure of ANN model was obtained by try and error method through evaluating different networks. The results showed that the optimum structure had two hidden layers with 8 and 6 neurons, respectively. The structure of studied ANN model has been shown in Figure 1. Tables 4 and 5 also show the parameters of synaptic weights and bias for three types of layers in this structure of AAN model. Three optimization techniques (PSO, GA and SCE) are applied to adjust the parameters of SVM and LSSVM models. The results of modelβs predictions have been presented in Table 6. According to Table 6, SCE
ACCEPTED MANUSCRIPT technique had better results in comparison with PSO and GA methods for parameter optimization. It can be observed from Table 6 that the π 2 values for train sets were 0.9996, 0.9994 and 0.9997 for ANN, SCE-SVM and SCE-LSSVM models, respectively. Moreover, the π 2 values of test sets were 0.9925, 0.9867 and 0.9964 for ANN, SCE-SVM and SCE-LSSVM models, respectively. It
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can be inferred from the values of these statistical parameters in Table 6 that the solute
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distribution coefficient has been successfully estimated with studied intelligent models. The
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optimized values of c, π and π 2 for SVM model obtained by SCE were 25.8262, 0.0014 and 0.1235, respectively. The optimized LSSVM parameters also obtained by SCE were 1.8267 and
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912.6581 for π 2 and πΎ, respectively. The schematic representations of SCE-SVM and SCE-
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LSSVM have been shown in Figure 2. Figures 3-6 have been presented to visually evaluate the accuracy of each model in various π½ ranges based on the best parameters obtained by SCE
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technique for SVM and LSSVM models. Figure 3, a crossplot curve, shows π½ππ₯π versus π½πππ for
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all three intelligent models. The concentration of the data points was almost around the 45β¦ line
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for all three models. The ANN and SCE-SVM outputs show a small deviation from 45Β° line for the values of π½ in the range of 0-1.5 and 1.5-2, respectively. However, a small deviation from
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experimental values around π½ =4 has been observed for SCE-LSSVM model. The error
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distribution curves (Figures 4 and 5) were applied to evaluate the trend of error for various π½ ranges in details. As it can be seen from Figure 4, the error deviation ranges were β0.27 π‘π 0.2, β0.4 π‘π 0.2 and β0.2 π‘π 0.1 for ANN, SCE-SVM and SCE-LSSVM models, respectively. The deviation respect to the real (experimental) value can be seen in Figure 5. The relative deviation for ANN model was in π½ range of 0.05 π‘π 0.1 and 0.1 π‘π 0.2 which all of them belonged to π½ < 2 and the model was in good agreement with experimental data for π½ > 2. As can be seen from Figures 3-5, the most of modelsβ deviations were occurred in the range of π½ < 2 for test set.
ACCEPTED MANUSCRIPT However, the results of SCE-LSSVM model showed a major deviation around π½ = 4 for both test and train sets. The one reason for these deviations may be the limited experimental data around π½ = 4. The final results of intelligent models for all literature data have been shown in Figure 6. According to Figure 6, the phase behavior of LLE for benzene/alkane/ionic liquid ternary systems
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have been successfully predicted by intelligent models. Wide literature data, best selected input
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parameters and proper parameters for defining ionic liquids can be the main reasons for the good
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accuracy of these models. Finally, Figure 7 shows the LLE prediction of intelligent models for
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benzene/nonane/[Bmim][NTf2] ternary system presented by Dominguez et al. [38]. The composition of both IL-rich and alkane-rich phases were obtained by assuming that the
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composition of ionic liquid was zero in alkane-rich phase.
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4- Conclusion
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The implement of computer-based algorithms including ANN, SVM and LSSVM was used to estimate the phase behavior of LLE for benzene/alkane/ionic liquid ternary systems. The wide
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range of experimental data from the literature was utilized to construct the models for prediction
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of the LLE for various alkanes and ionic liquids. The optimized parameters of SVM and LSSVM models have been adjusted by three optimization tools (PSO, GA and SCE). The results in form of
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both graphical and statistical error parameters showed good consistency between experimental data and predicted values. The SCE-LSSVM model presented better consistency with experimental data according to higher correlation coefficient of π 2 = 0.9997 and lower values of π πππΈ = 0.010 and πππ· = 0.5811 in comparison with ANN and SCE-SVM models. Finally, it can be concluded that all constructed intelligent methods can be sued as invaluable tools for liquid-liquid extraction of aromatic hydrocarbons from their mixtures with aliphatic hydrocarbons.
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atmospheric pressure, Journal of Chemical & Engineering Data, 51 (2006) 988-991.
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AN
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Thermodynamics, 42 (2010) 1234-1239.
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atmospheric pressure: Effect of the size of the aliphatic hydrocarbons, The Journal of Chemical Thermodynamics, 42 (2010) 104-109.
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IP
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ACCEPTED MANUSCRIPT Nomenclature Abbreviations Artificial neural network
GA
Genetic algorithm
IL
Ionic liquid
LLE
Liquid-liquid equilibria
LSSVM
Least square support vector machine
MLP
Multilayer perceptron
PSO
Particle swarm optimization
QP
Quadratic programming
RBF
Radial basis function
RMSE
Root mean square error
SCE
Shuffled complex evolution
SVM
Variables
Standard deviation error Support vector machine
CR US
AN
M
ED
PT
CE
AC
STD
IP
T
ANN
ACCEPTED MANUSCRIPT Lagrangian multiplier
C
Adjustable parameter of SVM model
πk
Error of LSSVM in training phase
ππ
Critical pressure of ionic liquid
ππ
Critical temperature of ionic liquid
π€π
Weight vector
xk
Input vector at the train sample k
xπ
Normalized data
π¦π
Target vector at the train sample k
π1
Feed composition of benzene
π2
Feed composition of alkane
π3
Feed composition of ionic liquid
CR US
AN
M
ED
PT
CE
AC
Greek symbols π½
IP
T
ππ (ππβ )
Solute distribution coefficient
Ξ³
Regularization parameter
Ξ΅
Adjustable parameter of SVM model
ππ (ππβ )
Slack variable
ACCEPTED MANUSCRIPT π2
Squared bandwidth
π(x)
Kernel function
π
Acentric factor of ionic liquid
Max
Maximum value
Pred
Predicted
CR
Experimental
AC
CE
PT
ED
M
AN
US
Exp
IP
T
Subscripts
ACCEPTED MANUSCRIPT Figure captions Figure 1. Structure of studied MLP neural network. Figure 2. Schematic presentation of SCE-SVM and SCE-LSSVM models.
T
Figure 3. Crossplot of predicted solute distribution coefficient data points versus experimental
CR
IP
data points for LSSVM, SVM and ANN models at both train and test sets.
Figure 4. The error distribution between experimental and predicted solute distribution coefficient
US
data point for LSSVM, SVM and ANN models.
AN
Figure 5. The relative error distribution between experimental and predicted solute distribution coefficient data point for LSSVM, SVM and ANN models.
M
Figure 6. Comparison between the experimental data and the results of intelligent models for all
ED
data points.
PT
Figure 7. Tie-lines of the ternary system of Benzene/Nonane/[Bmim][NTf2] at T=298.15 K and
CE
atmospheric pressure. Dashed lines and empty squares indicate the experimental tie-lines and solid lines and fill points indicate the tie-lines calculated by intelligent model: a) LSSVM model;
AC
b) SVM model and c) ANN model
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 1-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 2-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 3-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 4-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 5-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 6-
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 7- a)
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 7- b)
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
Figure 7- c)
ACCEPTED MANUSCRIPT Table 1- List of experimental data points used in this study. Molecular weight of ionic liquid min max
Reference
Number of data points
min
max
[36]
20
298.15
298.15
250.32
[37]
20
298.15
298.15
[38]
31
298.15
[39]
3
[40]
18
[41]
max
4
8 ,9
0.539
0.673
419.36
419.36
2
7 ,8
0.861
1.721
298.15
405.34
419.36
2, 23
8 ,9
0.818
1.619
298
298.15
247.31
764
28
6
0.564
5.773
313
313
384.32
388.31
16, 26
6
0.474
1.238
7
298.15
298.15
288.29
288.29
6
7
1.069
6.277
[42]
63
283.15
303.15
247.31
247.31
15
6 ,7
0.59
1
[43]
24
298.2
328.2
236.29
236.29
13
6
0.432
0.694
[44]
8
313.2
313.2
407.31
407.31
1
7
0.691
1.455
[45]
10
298.15
298.15
284.18
284.18
3
8 ,11
0.174
0.88
[46]
8
298.15
298.15
340.29
340.29
25
8 ,11
0.326
0.895
[47]
17
298.2
298.2
320.45
394.53
14, 20
7 ,16
0.102
1.199
[48]
36
298.15
298.15
419.36
531.58
6
0.816
2.211
[49]
36
283.15
298.15
233.28
233.28
17
6 ,7
0.385
0.602
[50]
35
283.15
298.15
247.31
247.31
15
8 ,9
0.474
0.843
[51]
34
298.15
298.15
236.29
AN
2, 10, 11, 24
236.29
13
6 ,7 ,8 ,9
0.449
0.608
[52]
52
298.2
298.2
254.08
312.24
18, 19
7 ,12 ,16
0.266
0.922
[53]
58
298.15
298.15
201.22
257.33
5, 22
6 ,7 ,8
0.418
1.159
[54]
13
298.2
298.2
230.78
230.78
21
7 ,12 ,16
0.456
0.867
[55]
14
313.2
333.2
237.05
237.05
9
6
0.635
1.364
[56]
24
298.2
328.2
250.32
250.32
4
6
0.619
0.899
[57]
28
303.2
328.2
265.12
265.12
7
6
0.561
0.83
[58]
18
303.2
328.2
240.3
240.3
8
6
0.801
1.811
[59]
9
298.15
298.15
284.18
284.18
3
6
3.387
4.556
[60]
16
298.15
313.15
391.31
391.31
12
6
0.757
1.371
[61]
35
283.15
298.15
233.28
233.28
17
8 ,9
0.376
0.468
[62]
19
303.15
328.15
216.28
216.28
27
6
0.655
1.455
[63]
8
298.15
298.15
247.31
247.31
15
7
0.564
5.773
IP
CR
US
M
ED
T
min
250.32
AC
Carbon number
PT
π·
Ionic liquid index
CE
Temperature, K
ACCEPTED MANUSCRIPT Table 2- Names and chemical formulas for ionic liquids of ternary systems used in this study. Index number
IUPAC name
Chemical formula
1
C8H11F6N3O5S2
1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
2
C10H15F6N3O4S2
1-butyl-3-methylimidazolium hexafluorophosphate
3
C8H15F6N2P
1-butyl-3-methylimidazolium methylsulfate
4
C9H18N2O4S
T
1-(2-hydroxyethyl)-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
5
IP
1-butyl-3-methylimidazolium nitrate
C8H15N3O3
6
C9H15F3N2O3S
7
C14H16BN5
8
C14H16N4
9
C10H16BF4N
10
C16H27F6N3O4S2
1-dodecyl-3-methyl-1H-imidazolium bis(trifluoromethylsulfonyl)amide
11
C18H31F6N3O4S2
1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
12
C8H11F6N3O4S2
1-ethyl-3-methylimidazolium ethyl sulfate
13
C8H16N2O4S
14
C14H28N2O4S
15
C10H17NO4S
1-ethylpyridinium bis[(trifluoromethyl)sulfonyl]imide
16
C9H10F6N2O4S2
1-ethylpyridinium ethylsulfate
17
C9H15NO4S
1-hexyl-3-methylimidazolium hexafluorophosphate
18
C10H19F6N2P
1-hexyl-3-methylimidazolium tetrafluoroborate
19
C10H19BF4N2
1-methyl-3-octylimidazolium 2-(2-methoxyethoxy)ethyl sulfate
20
C17H34N2O6S
1-methyl-3-octylimidazolium chloride
21
C12H23ClN2
1-methyl-3-octylimidazolium nitrate
22
C12H23N3O3
1-methyl-3-propylimidazolium bis[(trifluoromethyl)sulfonyl]imide
23
C9H13F6N3O4S2
1-octyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
24
C14H23F6N3O4S2
1-octyl-3-methylimidazolium hexafluorophosphate
25
C12H23F6N2P
2-hydroxyethyl-N,N,N-trimethylammonium bis(trifluoromethylsulfonyl)imide
26
C7H14F6N2O5S2
N-butyl-3-methylpyridinium dicyanamide
27
C12H16N4
trihexyl(tetradecyl)phosphonium bis[(trifluoromethyl)sulfonyl]imide
28
C34H68F6NO4PS2
CR
1-butyl-3-methylimidazolium trifluoromethanesulfonate 1-butyl-3-methylpyridinium tetracyanoborate
1-butyl-4-methylpyridinium tetrafluoroborate
M
AN
1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)amide
AC
CE
PT
ED
1-ethyl-3-methylimidazolium octyl sulfate 1-ethyl-3-methylpyridinium ethylsulfate
US
1-butyl-3-methylpyridinium tricyanomethanide
ACCEPTED MANUSCRIPT
Table 3. Type and range of studied data points. Maximum
Temperature, K
283.15
333.2
Z1
0.0155
0.977
Z2
0
0.937
Z3 Molecular weight of ionic liquid Tc ,K
0.0215
764
625.8 8.5
40.6
0.1671
1.3276
86.18
226.44
Ξ²
0.1015
6.2771
ED
M
US
AN
Output
PT
201.22
Molecular weight of alkane
Ο
CE
0.4881
1586.7
Pc, bar
AC
T
Minimum
IP
Inputs
Property
CR
Type of data
ACCEPTED MANUSCRIPT Table 4- Synaptic weights and bias values of input layer in ANN model.
π»π
ππ
π
Molecular weight of alkane
Bias
28.19173
18.68931
-0.83718
-23.4028
8.695186
-4.25175
0.885245
-10.7624
7.836228
9.942562
-0.54197
-13.7927
19.13606
7.517467
0.14734
15.88987
-17.9302
-1.1312
62.64396
57.59585
28.29098
-2.09597
15.3728
-10.2341
-3.25386
-1.03335
-43.7159
-3.02793
12.84845
5.703048
10.45933
3.727307
-20.7846
33.92001
6.999725
6.968783
-22.8848
0.445422
1.351546
-0.83988
2.541374
0.061344
0.040065
-0.41822
-0.13339
-0.2638
-1.14714
6
-1.86944
23.63197
25.1401
10.05138
3.134528
-6.10607
-7.16134
-2.30675
-0.87074
-12.7862
7
0.284779
-8.86003
-4.66846
-8.49389
2.603556
-7.01541
5.521464
2.470344
0.975593
6.713943
8
3.08028
23.47227
20.22212
11.08549
0.713912
-1.89859
0.939572
0.560179
-0.42201
-17.582
1
-15.5704
28.74576
2
3.15344
3 4 5
AN M ED PT CE AC
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ππ
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ππ
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Temperature K
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ππ
Synaptic weight Molecular weight of ionic liquid
Neuron
ACCEPTED MANUSCRIPT Table 5- Synaptic weights and bias values of hidden layers and output layer in ANN model.
8
58.34032
2.855059
99.47121
-42.8351
-12.6215
-4.20298
-4.79229
-6.86615
-84.3053
0.599832
-0.04915
1.573251
-0.31896
-0.04692
-7.36406
0.312275
-2.80256
-1.90396
0.497526
56.45419
-13.2935
-0.14388
-22.9751
-2.0302
-3.39146
-0.2451
-0.54653
-11.3916
18.04934
0.182459
-0.05438
1.554223
2.029688
-0.06993
-7.35237
0.304924
-2.79466
-1.97959
7.108374
-3.68504
17.17956
-2.98179
-11.7064
-4.41652
-4.72018
0.129705
0.105782
1.906867
-0.12221
-2.16568
-0.15375
0.175739
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7
-56.6214
-0.34206
-13.1039
0.391238
11.74196
-11.5618
0.128815
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-1.79275
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6
6
AN
5
5
M
4
4
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3
3
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2
2
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Second layer neurons
1
Bias
Output layer Synaptic Bias weight
1
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Neuron
First layer neurons Synaptic weight
1.03789
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Table 6- Comparison of intelligent models for the prediction of solute distribution coefficient.
SCE
912.6581
PSO GA
Test
Ξ΅
c
R
1.8267
-
-
1000.0521
1.6555
-
951.2953
1.6340
-
SCE
-
0.1235
PSO
-
GA -
2
Train STD
R
0.9964
0.028
0.4666
-
0.9492
0.109
-
0.9488
0.109
0.0014
25.8262
0.9867
0.054
0.4668
0.1491
0.0001
20.1723
0.9731
0.083
-
0.1541
0.0002
28.7472
0.9880
-
-
-
-
0.9925
RMSE
STD
0.9997
0.010
0.5811
0.4840
0.9939
0.041
0.5257
0.4840
0.9939
0.041
0.5256
0.9994
0.014
0.5830
0.5046
0.9999
0.006
0.5329
0.056
0.5111
0.9987
0.019
0.5283
0.041
0.4733
0.9996
0.012
0.5839
IP
CR
US AN M
2
T
RMSE
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ANN
π
2
PT
SVM
Ξ³
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LSSVM
Model Parameters
Optimization technique
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Model
ACCEPTED MANUSCRIPT Highlights ο The phase behavior of benzene/alkane/IL ternary systems are predicted. ο The ANN, SVM and LSSVM models have been applied as intelligent models.
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ο 664 data points for C6 to C16 and 28 different ionic liquids are collected.
Fariborz Shaahmadi, Mohammad Amin Anbaz PII: DOI: Reference:
S0167-7322(16)32771-4 doi: 10.1016/j.molliq.2017.02.108 MOLLIQ 7021
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 September 2016 6 February 2017 25 February 2017
Please cite this article as: Fariborz Shaahmadi, Mohammad Amin Anbaz , The prediction of liquid-liquid equilibria for benzene/alkane/ionic liquids mixtures using intelligent models. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.02.108
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ACCEPTED MANUSCRIPT
The Prediction of Liquid-Liquid Equilibria for
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Benzene/Alkane/Ionic Liquids Mixtures Using Intelligent
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Models
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Fariborz Shaahmadi*, Mohammad Amin Anbaz Gas Engineering Department, Petroleum University of Technology (PUT), Post Box: 63431,
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Ahwaz, Iran
*
Corresponding author. E-mail addresses: [email protected].
ACCEPTED MANUSCRIPT Abstract Liquid-liquid extraction of aromatics from aliphatic hydrocarbons is a main process in petrochemical industry. Therefore, accurate predicting the liquid-liquid equilibria (LLE) for ternary systems of aromatic/alkane/ionic liquid can result in better liquid-liquid extraction. In this
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study, three intelligence methods including artificial neural network (ANN), support vector
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machine (SVM) and least square support vector machine (LSSVM) have been applied to predict the thermodynamic phase behavior of LLE for benzene/alkane/ionic liquid ternary systems.
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Optimization techniques such as particle swarm optimization (PSO), genetic algorithm (GA) and shuffled complex evolution (SCE) have been used to obtain adjustable parameters of SVM and
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LSSVM models. The results of prediction operation demonstrate that there was good agreement
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between the estimation of intelligent models and the experimental data of LLE.
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Key words: Liquid-liquid extraction, Ternary mixture, Artificial neural network, Support vector
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machine
ACCEPTED MANUSCRIPT
1- Introduction The separation of mixtures of aromatic and aliphatic hydrocarbons is a main problem in
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petrochemical industry, because these components have near boiling points and their
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combinations form azeotropic mixtures [1]. Liquid-liquid extraction is widely used as an
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industrial separation process for azeotropic mixtures unfit for distillation. There are traditional organic solvents employed industrially for liquid-liquid extraction such as sulfolane, N-methyl
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pyrrolidone (NMP), N-formyl morpholine, ethylene glycols and propylene carbonate [2-7].
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However, these solvents have some problems such as volatility, flammability and toxicity. Ionic liquids can be environmentally considered as a proper alternative for the traditional solvents in
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liquid-liquid extraction because of their properties such as negligible vapor pressure (non-
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volatility), non-flammable, high chemical and thermal stability.
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In recent years, the use of ionic liquids (IL) has been gradually increased as solvents for the liquid-liquid extraction of aromatic hydrocarbons from their mixtures with aliphatic hydrocarbons
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[8-15]. Consequently, there are a considerable number of publications conducted by researchers
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around the world reporting experimental liquidβliquid equilibrium (LLE) data for ternary systems of aromatic/aliphatic hydrocarbon/ionic liquid to develop the liquid-liquid extraction process. Most of these studies used the thermodynamic models such as NRTL and UNIQUAC to correlate the experimental LLE data of ternary systems [16-19]. These models usually required several tuning parameters adjusted according to actual experimental data of each ternary system and cannot be applied as a predictive tool for other ternary systems. It should be noted that these limited models are not accurate and reliable without adjusting their parameters for each system.
ACCEPTED MANUSCRIPT Bonilla-Petriciolet et al. [20] utilized the optimization tools such as particle swarm optimization (PSO) to adjust the fitting parameters of thermodynamic models for phase stability and equilibrium calculations through global minimizing of Gibbs energy. It becomes a necessity to have a simple and robust model to accurately predict the thermodynamic
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phase behavior of LLE for aromatic/alkane/ionic liquid systems. Therefore, intelligence methods
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can be used as appropriate alternatives for the conceptual models, physically based models, and conventional statistical models. Such intelligence methods include artificial neural network
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(ANN), support vector machine (SVM), least square support vector machine (LSSVM), etc. ANN model is a computational system based on the operation of biological neural networks. ANN
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method has become popular as feasible tool in a variety of fields of study in recent years [21-28].
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Torrecilla et al. [29] used ANN method for the prediction of the LLE data of only four toluene/heptane/ionic liquid systems at 313.2 and 348.2 K. They concluded that ANN was
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successfully used as a predictive tool for ternary LLE data. Another intelligence model is SVM,
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proposed by Vapnik [30] increasingly gained popularity to be employed in various engineering and science disciplines. According to SVM models, the results are obtained by quadratic
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programming (QP). Although several studies successfully used the SVM method in various
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applications [31-34], this method has not been applied to evaluate the LLE data of these ternary systems. Finally, the LSSVM method, introduced by Suykens and Vandewalle [35] as a modified version of conventional SVM, has the idea of the equality constraints which is the major advantage of LSSVM over the original SVM . In this study, the thermodynamic phase behavior of LLE of benzene/alkane/ionic liquid ternary systems has been predicted by intelligence methods including ANN, SVM and LSSVM models. These methods have been constructed by wide range of experimental LLE data gathered from
ACCEPTED MANUSCRIPT literature (664 data points) for this ternary system with 28 different ionic liquids. The solute distribution coefficient (π½) of benzene/alkane/ionic liquid systems has been estimated as a main parameter to evaluate the reliability of the ionic liquids as solvents for liquid-liquid extraction.
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2- Model development
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2-1. Data acquisition
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The reliable experimental data are required to have an accurate and precise model. Therefore, the
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experimental LLE data of benzene + Alkane + ionic liquid system at atmospheric pressure are gathered from published studies in the literature over the years. In this study, a data set consisting
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of 664 data points for 28 different ionic liquids and n-alkanes range in carbon number from C6 to C16 have been collected as shown in Table 1 [36-63]. The IUPAC names and chemical formulas of
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studied ionic liquids have been presented in Table 2. Table 3 shows the parameter ranges of input
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and output data in which the input data are feed compositions (π1 , π2 and π3 ), temperature,
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molecular weight of alkane and properties of ionic liquid (ππ , ππ , ππ and π), and the output parameter is the solute distribution coefficient of benzene/alkane/ionic liquid system. The
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[64, 65].
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properties of ionic liquids were obtained by group contribution method presented by Valderama
2-2. Artificial neural network ANNs, considered as mathematical tools, can learn from an experience, improve its performance and adopt to the environment changes. ANNs are known for their capability of performing nonlinear statistical modeling, relating input and output parameters and generalizing the results. In this study, multilayer perceptron (MLP) neural network as one of the most powerful types of
ACCEPTED MANUSCRIPT ANNs has been used to estimate the thermodynamic phase behavior of LLE for benzene/alkane/ionic liquid ternary systems. MLP neural networks originally are made of three different types of layers including input layer, hidden layer(s), and output layer. Figure 1 shows the structure of the MLP neural network
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considered in this study. As can be seen from Figure 1, each layer is composed of some fully
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interconnected processing units called neurons. It should be mentioned that the number of neurons in the input and output layers are corresponded to number of input and output data, respectively.
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The number of neurons in hidden layers was obtained by a trial and error method. Input layer received all input signals, and then these signals are transmitted to other neurons for processing in
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hidden layers. Then, the output layer received the information from hidden layers. Synaptic
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weights and biases are the parameters of this type of network. These parameters are adjusted by using network error as a benchmark through receiving the input data within the training algorithm.
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MLP neural network was trained by an iterative optimization procedure starting with random
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guesses on synaptic weights and biases. This training process continues to reach a minimum value
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in squared weights and errors over several iterations. 2-3. Support vector machine (SVM)
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SVM model is an effective strategy and analytical tool developed from the machine-learning community for classification and regression purposes. According to this method, consider an assumed train sample of N data samples in which x is input vector with a dimension of N Γ n (n is the number of input parameters) and π¦π is the relevant target sample. The aim of the SVM method is to find an regressed function π(x) as follows:
ACCEPTED MANUSCRIPT π(x) = π€ π π(x) + π
(1)
Where π€ π is weight vector, π(x) represents the kernel function, and b is a bias term. In order to calculate π€ π and b, Vapnik [30] proposed minimization of the following cost function: π
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1 πΆππ π‘ ππ’πππ‘πππ = π€ π + π β(ππ β ππβ ) 2
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π=1
(3)
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π = 1, 2, β¦ , π π = 1, 2, β¦ , π π = 1, 2, β¦ , π
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To satisfy constraints: π¦π β π€ π π(xk ) β π β€ π + ππ {π€ π π(xk ) + π β π¦π β€ π + ππ ππ . ππβ β₯ 0
(2)
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Where xk and π¦π represent the input vector at the train sample k and the relevant target sample, respectively. The Ξ΅ determines the fixed precision of the function approximation and ππ (or ππβ ) is
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slack variable. The variable c is the deviation from the desired Ξ΅ in Eq. (3) considered as the
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tuning parameter of the SVM. Lagrangian is utilized to minimize the cost function as follows: π
π
π
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1 πΏ(π. πβ ) = β β (ππ β ππβ )(ππΌ β ππΌβ )πΎ(xk , xI ) β π β(ππ β ππβ ) + β π¦π (ππ β ππβ ) 2 π
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π.πΌ=1
β(ππ β ππβ ) = 0 π=1
.
ππ . ππβ β [0, π]
πΎ(xk . xI ) = π(xk )π π(xI )
π = 1, 2, β¦ , π
π=1
(4)
π=1
(5)
(6)
Where ππ and ππβ are Lagrangian multipliers. Finally, the resulting form of SVM method is obtained as follows:
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π(x) = β (ππ β ππβ ) πΎ(x, xk ) + π
(7)
π.πΌ=1
The unknowns (ππ , ππβ , and b) can be obtained by solving a quadratic programming problem.
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Therefore, the tuning parameters of SVM model are Ξ΅, c and a parameter of kernel function.
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2-4. Least-squares support vector machine (LSSVM)
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The least square modification of SVM (LSSVM) is developed by Suykens and Vandewalle [35]
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through introducing a system of equations instead of a nonlinear quadratic programming in original SVM. The method applied in the LSSVM model is briefly introduced as following. A
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new optimization issue is applied in LSSVM as follows:
(8)
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k=1
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π
1 1 πΆππ π‘ ππ’πππ‘πππ = π€ π π€ + πΎ β ππ2 2 2
Where Ξ³ is the factor of regularization considered as a tuning parameter in LSSVM method and
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πk represents the error of LSSVM in training phase. An equality constraint is established to
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minimize the equation (8): π¦k = π€ π π(xk ) + π + πk
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(9)
The Lagrangian is applied to locate the solutions of the optimization in equation (8) along with its constraint defined in Eq. (9) as follows: π
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1 1 πΏ(π€. π. π. π) = π€ π π€ + πΎ β ππ2 β β ππ (π€ π π(xk ) + π + ππ β π¦π ) 2 2 π=1
π=1
(10)
ACCEPTED MANUSCRIPT Where ππ are support values or Lagrange multipliers. In order to acquire the solution, the derivatives of Eq. (10) should be equated to zero. Therefore, the following equations are obtained: π
ππΏ = 0 β π€ = β ππ π(xk ) ππ€ ππΏ = 0 β β ππ = 0 ππ π=1
(11)
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ππΏ = 0 β ππ = πΎππ π = 1,2, β¦ , π πππ ππΏ = 0 β π€ π π(xk ) + π + ππ β π¦π = 0 π = 1,2, β¦ , π {πππ
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π=1
π
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Eq. (11) shows that there are 2N + 2 equations and 2N + 2 unknown parameters (ππ , ππ , π€, and
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b). Thus, the parameters of LSSVM can be obtained by solving the system of equations defined in
0 1π
π 1ππ 0 ] [ ]=[ ] β1 π¦ Ξ©+πΎ πΌ πΌ
(12)
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[
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Eq. (11). The system of linear equations can be rewritten in matrix form as:
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Where π¦ = [π¦1 β¦ π¦π ]π , 1π = [1 β¦ 1]π , πΌ = [πΌ1 β¦ πΌπ ]π , I is an identity matrix, and Ξ© = π(xπ )π in which π(xπ ) = πΎ(xπ , xl ) β π, π = 1,2, β¦ , π. πΎ(xπ . xl ) is the kernel function taking into
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account the Mercer limitation [66]. The radial basis function (RBF) Kernel has been used in this study as follows:
πΎ(x. xk ) = exp(
β|| xk β x||2 ) π2
(13)
ACCEPTED MANUSCRIPT where π 2 is squared bandwidth which is optimized through an external optimization technique during the training process as shown in Figure 2. The resulting formulation of LSSVM model for function estimation is then written as follows: π
π¦(x) = β πΌπ πΎ(x, xk ) + π
(14)
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π=1
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Finally, it should be noted that LSSVM model is exactly influenced by two adaptable parameters
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π 2 and regularization parameter (Ξ³). 2-5. Data normalization
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Data must be well processed and scaled prior to input to the intelligent models. Therefore, all the
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inputs and output were normalized between 0 and 1, because higher valued input variables may be likely to suppress the impact of the smaller ones. The following equation has been utilized for
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normalization as below:
(15)
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x xπ = ( ) Γ 0.8 + 0.1 1.5 Γ xπππ₯
Where xπ represents normalized data, xπππ₯ is the maximum value of the data and x is the real
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data. It should be noted that the normalization of data has no effect on the model results. At the end, these values were returned to their original values. 2-6. Computational procedure The database was randomly divided into two subsets involving train and test sets to construct the studied models. In this study, 80% of all data points were randomly selected for train set to generate the model structure. The remaining 20% of data points have been applied for test set to
ACCEPTED MANUSCRIPT investigate modelsβ prediction validity and capability. As previously mentioned, nine input parameters have been applied including temperature, composition of three components, molecular weight of alkane, critical temperature of IL, critical pressure of IL, molecular weight of IL and the acentric factor of IL. These input data have been used to develop the intelligent models. In this
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study, the structure of ANN model was determined through evaluating different networks. The
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tuning parameters of SVM (c, Ξ΅, and π 2 ) and LSSVM (Ξ³ and π 2 ) have been adjusted with the
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experimental train data using non-traditional optimization techniques such as PSO, genetic
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algorithm (GA) and shuffled complex evolution (SCE) [67]. SVM and LSSVM models have been constructed after finding the optimum parameters. The prediction data were obtained by replacing
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the test data sample into these models. The validity of models can be evaluated through these
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predictions.
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2-7. Accuracy of intelligent models
Several statistical and graphical error analyses have been utilized to check the precision and
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reliability of studied intelligent models such as coefficient of determination (π 2 ), root mean
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square error (RMSE), and standard deviation error (STD). The formulation of these statistical parameters are as follows:
π πππΈ = (
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2 βπ π=1(π½ππππ (π) β π½πΈπ₯π (π)) π =1β π Μ (π))2 βπ=1(π½ππππ (π) β π½πΈπ₯π 2
βπ π=1 (π½ππππ (π) β π½πΈπ₯π (π)) π
(16)
2
)0.5
(17)
ACCEPTED MANUSCRIPT π
Μ (π))2 (π½ππππ (π) β π½πΈπ₯π πππ· = β( )0.5 π
(18)
π=1
Where π½πΈπ₯π and π½ππππ are the experimental and predicted solute distribution coefficient, Μ respectively. And π½πΈπ₯π is the mean of the experimental data values of solute distribution
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coefficient.
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In this study, crossplot and error distribution curves have been applied to visualize the validity of studied intelligent models. In crossplot curve, the predicted values of intelligent models are
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sketched versus the experimental values. The consistency of the model can be obtained by
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concentrating the data points around the 45 Β° line which indicates the perfect model line.
3- Results and discussion
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distribution around the zero error line.
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According to error distribution curve, the error trend can be observed by showing error
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Three intelligent models have been applied to evaluate the phase behavior of ternary LLE
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systems. The feed compositions, temperature, molecular weight of alkane and properties of ionic liquid have been considered as correlating parameters of the solute distribution coefficient for
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benzene/alkane/ionic liquid system. The optimum structure of ANN model was obtained by try and error method through evaluating different networks. The results showed that the optimum structure had two hidden layers with 8 and 6 neurons, respectively. The structure of studied ANN model has been shown in Figure 1. Tables 4 and 5 also show the parameters of synaptic weights and bias for three types of layers in this structure of AAN model. Three optimization techniques (PSO, GA and SCE) are applied to adjust the parameters of SVM and LSSVM models. The results of modelβs predictions have been presented in Table 6. According to Table 6, SCE
ACCEPTED MANUSCRIPT technique had better results in comparison with PSO and GA methods for parameter optimization. It can be observed from Table 6 that the π 2 values for train sets were 0.9996, 0.9994 and 0.9997 for ANN, SCE-SVM and SCE-LSSVM models, respectively. Moreover, the π 2 values of test sets were 0.9925, 0.9867 and 0.9964 for ANN, SCE-SVM and SCE-LSSVM models, respectively. It
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can be inferred from the values of these statistical parameters in Table 6 that the solute
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distribution coefficient has been successfully estimated with studied intelligent models. The
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optimized values of c, π and π 2 for SVM model obtained by SCE were 25.8262, 0.0014 and 0.1235, respectively. The optimized LSSVM parameters also obtained by SCE were 1.8267 and
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912.6581 for π 2 and πΎ, respectively. The schematic representations of SCE-SVM and SCE-
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LSSVM have been shown in Figure 2. Figures 3-6 have been presented to visually evaluate the accuracy of each model in various π½ ranges based on the best parameters obtained by SCE
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technique for SVM and LSSVM models. Figure 3, a crossplot curve, shows π½ππ₯π versus π½πππ for
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all three intelligent models. The concentration of the data points was almost around the 45β¦ line
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for all three models. The ANN and SCE-SVM outputs show a small deviation from 45Β° line for the values of π½ in the range of 0-1.5 and 1.5-2, respectively. However, a small deviation from
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experimental values around π½ =4 has been observed for SCE-LSSVM model. The error
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distribution curves (Figures 4 and 5) were applied to evaluate the trend of error for various π½ ranges in details. As it can be seen from Figure 4, the error deviation ranges were β0.27 π‘π 0.2, β0.4 π‘π 0.2 and β0.2 π‘π 0.1 for ANN, SCE-SVM and SCE-LSSVM models, respectively. The deviation respect to the real (experimental) value can be seen in Figure 5. The relative deviation for ANN model was in π½ range of 0.05 π‘π 0.1 and 0.1 π‘π 0.2 which all of them belonged to π½ < 2 and the model was in good agreement with experimental data for π½ > 2. As can be seen from Figures 3-5, the most of modelsβ deviations were occurred in the range of π½ < 2 for test set.
ACCEPTED MANUSCRIPT However, the results of SCE-LSSVM model showed a major deviation around π½ = 4 for both test and train sets. The one reason for these deviations may be the limited experimental data around π½ = 4. The final results of intelligent models for all literature data have been shown in Figure 6. According to Figure 6, the phase behavior of LLE for benzene/alkane/ionic liquid ternary systems
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have been successfully predicted by intelligent models. Wide literature data, best selected input
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parameters and proper parameters for defining ionic liquids can be the main reasons for the good
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accuracy of these models. Finally, Figure 7 shows the LLE prediction of intelligent models for
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benzene/nonane/[Bmim][NTf2] ternary system presented by Dominguez et al. [38]. The composition of both IL-rich and alkane-rich phases were obtained by assuming that the
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composition of ionic liquid was zero in alkane-rich phase.
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4- Conclusion
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The implement of computer-based algorithms including ANN, SVM and LSSVM was used to estimate the phase behavior of LLE for benzene/alkane/ionic liquid ternary systems. The wide
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range of experimental data from the literature was utilized to construct the models for prediction
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of the LLE for various alkanes and ionic liquids. The optimized parameters of SVM and LSSVM models have been adjusted by three optimization tools (PSO, GA and SCE). The results in form of
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both graphical and statistical error parameters showed good consistency between experimental data and predicted values. The SCE-LSSVM model presented better consistency with experimental data according to higher correlation coefficient of π 2 = 0.9997 and lower values of π πππΈ = 0.010 and πππ· = 0.5811 in comparison with ANN and SCE-SVM models. Finally, it can be concluded that all constructed intelligent methods can be sued as invaluable tools for liquid-liquid extraction of aromatic hydrocarbons from their mixtures with aliphatic hydrocarbons.
ACCEPTED MANUSCRIPT References [1] J.G. Villaluenga, A. Tabe-Mohammadi, A review on the separation of benzene/cyclohexane mixtures by pervaporation processes, Journal of Membrane Science, 169 (2000) 159-174. [2] I. Ashour, S.I. Abu-Eishah, Liquid-liquid equilibria of ternary and six-component systems
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Journal of Chemical & Engineering Data, 51 (2006) 1717-1722.
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including cyclohexane, benzene, toluene, ethylbenzene, cumene, and sulfolane at 303.15 K,
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[3] A.S. Al-Jimaz, M.S. Fandary, K.H.E. Alkhaldi, J.A. Al-Kandary, M.A. Fahim, Extraction of aromatics from middle distillate using N-methyl-2-pyrrolidone: experiment, modeling, and
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optimization, Industrial & engineering chemistry research, 46 (2007) 5686-5696.
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[4] J. Mahmoudi, M.N. Lotfollahi, (Liquid+ liquid) equilibria of (sulfolane+ benzene+nhexane),(N-formylmorpholine+ benzene+n-hexane), and (sulfolane+N-formylmorpholine+
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benzene+n-hexane) at temperatures ranging from (298.15 to 318.15) K: Experimental results and
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correlation, The Journal of Chemical Thermodynamics, 42 (2010) 466-471. [5] W. Wang, Z. Gou, S. Zhu, Liquid-liquid equilibria for aromatics extraction systems with
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tetraethylene glycol, Journal of Chemical & Engineering Data, 43 (1998) 81-83.
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[6] T.A. Al-Sahhaf, E. Kapetanovic, Measurement and prediction of phase equilibria in the extraction of aromatics from naphtha reformate by tetraethylene glycol, Fluid Phase Equilibria,
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M
pressure, The Journal of Chemical Thermodynamics, 41 (2009) 1215-1221.
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atmospheric pressure, Journal of Chemical & Engineering Data, 51 (2006) 988-991.
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using 1-ethyl-3-methylpyridinium ethylsulfate ionic liquid at several temperatures and
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atmospheric pressure: Effect of the size of the aliphatic hydrocarbons, The Journal of Chemical Thermodynamics, 42 (2010) 104-109.
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ACCEPTED MANUSCRIPT Nomenclature Abbreviations Artificial neural network
GA
Genetic algorithm
IL
Ionic liquid
LLE
Liquid-liquid equilibria
LSSVM
Least square support vector machine
MLP
Multilayer perceptron
PSO
Particle swarm optimization
QP
Quadratic programming
RBF
Radial basis function
RMSE
Root mean square error
SCE
Shuffled complex evolution
SVM
Variables
Standard deviation error Support vector machine
CR US
AN
M
ED
PT
CE
AC
STD
IP
T
ANN
ACCEPTED MANUSCRIPT Lagrangian multiplier
C
Adjustable parameter of SVM model
πk
Error of LSSVM in training phase
ππ
Critical pressure of ionic liquid
ππ
Critical temperature of ionic liquid
π€π
Weight vector
xk
Input vector at the train sample k
xπ
Normalized data
π¦π
Target vector at the train sample k
π1
Feed composition of benzene
π2
Feed composition of alkane
π3
Feed composition of ionic liquid
CR US
AN
M
ED
PT
CE
AC
Greek symbols π½
IP
T
ππ (ππβ )
Solute distribution coefficient
Ξ³
Regularization parameter
Ξ΅
Adjustable parameter of SVM model
ππ (ππβ )
Slack variable
ACCEPTED MANUSCRIPT π2
Squared bandwidth
π(x)
Kernel function
π
Acentric factor of ionic liquid
Max
Maximum value
Pred
Predicted
CR
Experimental
AC
CE
PT
ED
M
AN
US
Exp
IP
T
Subscripts
ACCEPTED MANUSCRIPT Figure captions Figure 1. Structure of studied MLP neural network. Figure 2. Schematic presentation of SCE-SVM and SCE-LSSVM models.
T
Figure 3. Crossplot of predicted solute distribution coefficient data points versus experimental
CR
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data points for LSSVM, SVM and ANN models at both train and test sets.
Figure 4. The error distribution between experimental and predicted solute distribution coefficient
US
data point for LSSVM, SVM and ANN models.
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Figure 5. The relative error distribution between experimental and predicted solute distribution coefficient data point for LSSVM, SVM and ANN models.
M
Figure 6. Comparison between the experimental data and the results of intelligent models for all
ED
data points.
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Figure 7. Tie-lines of the ternary system of Benzene/Nonane/[Bmim][NTf2] at T=298.15 K and
CE
atmospheric pressure. Dashed lines and empty squares indicate the experimental tie-lines and solid lines and fill points indicate the tie-lines calculated by intelligent model: a) LSSVM model;
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b) SVM model and c) ANN model
ACCEPTED MANUSCRIPT
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M
AN
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CR
IP
T
Figure 1-
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CR
IP
T
Figure 2-
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AN
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IP
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Figure 3-
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AN
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Figure 4-
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AN
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IP
T
Figure 5-
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PT
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AN
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IP
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Figure 6-
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PT
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M
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US
CR
IP
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Figure 7- a)
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PT
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M
AN
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T
Figure 7- b)
ACCEPTED MANUSCRIPT
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Figure 7- c)
ACCEPTED MANUSCRIPT Table 1- List of experimental data points used in this study. Molecular weight of ionic liquid min max
Reference
Number of data points
min
max
[36]
20
298.15
298.15
250.32
[37]
20
298.15
298.15
[38]
31
298.15
[39]
3
[40]
18
[41]
max
4
8 ,9
0.539
0.673
419.36
419.36
2
7 ,8
0.861
1.721
298.15
405.34
419.36
2, 23
8 ,9
0.818
1.619
298
298.15
247.31
764
28
6
0.564
5.773
313
313
384.32
388.31
16, 26
6
0.474
1.238
7
298.15
298.15
288.29
288.29
6
7
1.069
6.277
[42]
63
283.15
303.15
247.31
247.31
15
6 ,7
0.59
1
[43]
24
298.2
328.2
236.29
236.29
13
6
0.432
0.694
[44]
8
313.2
313.2
407.31
407.31
1
7
0.691
1.455
[45]
10
298.15
298.15
284.18
284.18
3
8 ,11
0.174
0.88
[46]
8
298.15
298.15
340.29
340.29
25
8 ,11
0.326
0.895
[47]
17
298.2
298.2
320.45
394.53
14, 20
7 ,16
0.102
1.199
[48]
36
298.15
298.15
419.36
531.58
6
0.816
2.211
[49]
36
283.15
298.15
233.28
233.28
17
6 ,7
0.385
0.602
[50]
35
283.15
298.15
247.31
247.31
15
8 ,9
0.474
0.843
[51]
34
298.15
298.15
236.29
AN
2, 10, 11, 24
236.29
13
6 ,7 ,8 ,9
0.449
0.608
[52]
52
298.2
298.2
254.08
312.24
18, 19
7 ,12 ,16
0.266
0.922
[53]
58
298.15
298.15
201.22
257.33
5, 22
6 ,7 ,8
0.418
1.159
[54]
13
298.2
298.2
230.78
230.78
21
7 ,12 ,16
0.456
0.867
[55]
14
313.2
333.2
237.05
237.05
9
6
0.635
1.364
[56]
24
298.2
328.2
250.32
250.32
4
6
0.619
0.899
[57]
28
303.2
328.2
265.12
265.12
7
6
0.561
0.83
[58]
18
303.2
328.2
240.3
240.3
8
6
0.801
1.811
[59]
9
298.15
298.15
284.18
284.18
3
6
3.387
4.556
[60]
16
298.15
313.15
391.31
391.31
12
6
0.757
1.371
[61]
35
283.15
298.15
233.28
233.28
17
8 ,9
0.376
0.468
[62]
19
303.15
328.15
216.28
216.28
27
6
0.655
1.455
[63]
8
298.15
298.15
247.31
247.31
15
7
0.564
5.773
IP
CR
US
M
ED
T
min
250.32
AC
Carbon number
PT
π·
Ionic liquid index
CE
Temperature, K
ACCEPTED MANUSCRIPT Table 2- Names and chemical formulas for ionic liquids of ternary systems used in this study. Index number
IUPAC name
Chemical formula
1
C8H11F6N3O5S2
1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
2
C10H15F6N3O4S2
1-butyl-3-methylimidazolium hexafluorophosphate
3
C8H15F6N2P
1-butyl-3-methylimidazolium methylsulfate
4
C9H18N2O4S
T
1-(2-hydroxyethyl)-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
5
IP
1-butyl-3-methylimidazolium nitrate
C8H15N3O3
6
C9H15F3N2O3S
7
C14H16BN5
8
C14H16N4
9
C10H16BF4N
10
C16H27F6N3O4S2
1-dodecyl-3-methyl-1H-imidazolium bis(trifluoromethylsulfonyl)amide
11
C18H31F6N3O4S2
1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
12
C8H11F6N3O4S2
1-ethyl-3-methylimidazolium ethyl sulfate
13
C8H16N2O4S
14
C14H28N2O4S
15
C10H17NO4S
1-ethylpyridinium bis[(trifluoromethyl)sulfonyl]imide
16
C9H10F6N2O4S2
1-ethylpyridinium ethylsulfate
17
C9H15NO4S
1-hexyl-3-methylimidazolium hexafluorophosphate
18
C10H19F6N2P
1-hexyl-3-methylimidazolium tetrafluoroborate
19
C10H19BF4N2
1-methyl-3-octylimidazolium 2-(2-methoxyethoxy)ethyl sulfate
20
C17H34N2O6S
1-methyl-3-octylimidazolium chloride
21
C12H23ClN2
1-methyl-3-octylimidazolium nitrate
22
C12H23N3O3
1-methyl-3-propylimidazolium bis[(trifluoromethyl)sulfonyl]imide
23
C9H13F6N3O4S2
1-octyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide
24
C14H23F6N3O4S2
1-octyl-3-methylimidazolium hexafluorophosphate
25
C12H23F6N2P
2-hydroxyethyl-N,N,N-trimethylammonium bis(trifluoromethylsulfonyl)imide
26
C7H14F6N2O5S2
N-butyl-3-methylpyridinium dicyanamide
27
C12H16N4
trihexyl(tetradecyl)phosphonium bis[(trifluoromethyl)sulfonyl]imide
28
C34H68F6NO4PS2
CR
1-butyl-3-methylimidazolium trifluoromethanesulfonate 1-butyl-3-methylpyridinium tetracyanoborate
1-butyl-4-methylpyridinium tetrafluoroborate
M
AN
1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)amide
AC
CE
PT
ED
1-ethyl-3-methylimidazolium octyl sulfate 1-ethyl-3-methylpyridinium ethylsulfate
US
1-butyl-3-methylpyridinium tricyanomethanide
ACCEPTED MANUSCRIPT
Table 3. Type and range of studied data points. Maximum
Temperature, K
283.15
333.2
Z1
0.0155
0.977
Z2
0
0.937
Z3 Molecular weight of ionic liquid Tc ,K
0.0215
764
625.8 8.5
40.6
0.1671
1.3276
86.18
226.44
Ξ²
0.1015
6.2771
ED
M
US
AN
Output
PT
201.22
Molecular weight of alkane
Ο
CE
0.4881
1586.7
Pc, bar
AC
T
Minimum
IP
Inputs
Property
CR
Type of data
ACCEPTED MANUSCRIPT Table 4- Synaptic weights and bias values of input layer in ANN model.
π»π
ππ
π
Molecular weight of alkane
Bias
28.19173
18.68931
-0.83718
-23.4028
8.695186
-4.25175
0.885245
-10.7624
7.836228
9.942562
-0.54197
-13.7927
19.13606
7.517467
0.14734
15.88987
-17.9302
-1.1312
62.64396
57.59585
28.29098
-2.09597
15.3728
-10.2341
-3.25386
-1.03335
-43.7159
-3.02793
12.84845
5.703048
10.45933
3.727307
-20.7846
33.92001
6.999725
6.968783
-22.8848
0.445422
1.351546
-0.83988
2.541374
0.061344
0.040065
-0.41822
-0.13339
-0.2638
-1.14714
6
-1.86944
23.63197
25.1401
10.05138
3.134528
-6.10607
-7.16134
-2.30675
-0.87074
-12.7862
7
0.284779
-8.86003
-4.66846
-8.49389
2.603556
-7.01541
5.521464
2.470344
0.975593
6.713943
8
3.08028
23.47227
20.22212
11.08549
0.713912
-1.89859
0.939572
0.560179
-0.42201
-17.582
1
-15.5704
28.74576
2
3.15344
3 4 5
AN M ED PT CE AC
IP
ππ
CR
ππ
US
Temperature K
T
ππ
Synaptic weight Molecular weight of ionic liquid
Neuron
ACCEPTED MANUSCRIPT Table 5- Synaptic weights and bias values of hidden layers and output layer in ANN model.
8
58.34032
2.855059
99.47121
-42.8351
-12.6215
-4.20298
-4.79229
-6.86615
-84.3053
0.599832
-0.04915
1.573251
-0.31896
-0.04692
-7.36406
0.312275
-2.80256
-1.90396
0.497526
56.45419
-13.2935
-0.14388
-22.9751
-2.0302
-3.39146
-0.2451
-0.54653
-11.3916
18.04934
0.182459
-0.05438
1.554223
2.029688
-0.06993
-7.35237
0.304924
-2.79466
-1.97959
7.108374
-3.68504
17.17956
-2.98179
-11.7064
-4.41652
-4.72018
0.129705
0.105782
1.906867
-0.12221
-2.16568
-0.15375
0.175739
T
7
-56.6214
-0.34206
-13.1039
0.391238
11.74196
-11.5618
0.128815
IP
-1.79275
CR
US
6
6
AN
5
5
M
4
4
ED
3
3
PT
2
2
CE
Second layer neurons
1
Bias
Output layer Synaptic Bias weight
1
AC
Neuron
First layer neurons Synaptic weight
1.03789
ACCEPTED MANUSCRIPT
Table 6- Comparison of intelligent models for the prediction of solute distribution coefficient.
SCE
912.6581
PSO GA
Test
Ξ΅
c
R
1.8267
-
-
1000.0521
1.6555
-
951.2953
1.6340
-
SCE
-
0.1235
PSO
-
GA -
2
Train STD
R
0.9964
0.028
0.4666
-
0.9492
0.109
-
0.9488
0.109
0.0014
25.8262
0.9867
0.054
0.4668
0.1491
0.0001
20.1723
0.9731
0.083
-
0.1541
0.0002
28.7472
0.9880
-
-
-
-
0.9925
RMSE
STD
0.9997
0.010
0.5811
0.4840
0.9939
0.041
0.5257
0.4840
0.9939
0.041
0.5256
0.9994
0.014
0.5830
0.5046
0.9999
0.006
0.5329
0.056
0.5111
0.9987
0.019
0.5283
0.041
0.4733
0.9996
0.012
0.5839
IP
CR
US AN M
2
T
RMSE
ED
ANN
π
2
PT
SVM
Ξ³
CE
LSSVM
Model Parameters
Optimization technique
AC
Model
ACCEPTED MANUSCRIPT Highlights ο The phase behavior of benzene/alkane/IL ternary systems are predicted. ο The ANN, SVM and LSSVM models have been applied as intelligent models.
AC
CE
PT
ED
M
AN
US
CR
IP
T
ο 664 data points for C6 to C16 and 28 different ionic liquids are collected.