Prediction of limiting activity coefficients for binary vapor-liquid equilibrium using neural networks

Accepted Manuscript Prediction of limiting activity coefficients for binary vapor-liquid equilibrium using neural networks Hesam Ahmadian Behrooz, R. Bozorgmehry Boozarjomehry PII:

S0378-3812(16)30537-4

DOI:

10.1016/j.fluid.2016.10.033

Reference:

FLUID 11310

To appear in:

Fluid Phase Equilibria

Received Date: 19 May 2016 Revised Date:

28 October 2016

Accepted Date: 28 October 2016

Please cite this article as: H. Ahmadian Behrooz, R.B. Boozarjomehry, Prediction of limiting activity coefficients for binary vapor-liquid equilibrium using neural networks, Fluid Phase Equilibria (2016), doi: 10.1016/j.fluid.2016.10.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Prediction of limiting activity coefficients for binary vapor-liquid equilibrium using neural networks Chemical Engineering Faculty, Sahand University of Technology, Tabriz, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

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Hesam Ahmadian Behrooz a,1, R.Bozorgmehry Boozarjomehryb,2

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Corresponding author. Tel.: (+9841)33459150, E-mail: [email protected] Tel.: (+9821)66166445, E-mail: [email protected]

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Solute

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ࢂ࡯ ࡹࢃ ࣆ

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࣓

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ࢀ࡯ ࡼ࡯ ࢂ࡯

Solvent ࡹࢃ

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Abstract The activity coefficient at infinite dilution is a representative of the limiting non-ideality of a solute in a mixture. Various methods for the prediction of infinite dilution activity coefficients

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(IDACs) have been developed. Artificial neural networks are powerful mapping tools for nonlinear function approximations. Accordingly, an artificial neural network model is proposed for the prediction of the IDACs of binary systems where the properties of the

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individual components are used as inputs to the network. The input parameters of the neural network are the mixture temperature, critical temperature, critical pressure, critical

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volume, molecular weight, dipole moment and the acentric factor of both solute and solvent. The output of the neural model is the natural logarithm of the activity coefficients at infinite dilution of the solute.

Two different approaches can be adopted, based on the available experimental in the

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literature: different solutes in a specific solvent and a series of solutes in various solvents. For the first case, the input parameters corresponding to the solvent can be eliminated from the network inputs without affecting the model performance. Five different examples were

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investigated to evaluate the performance of the proposed model where networks with minimum absolute average deviation were reported as the optimum structures.

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An artificial neural network trained with comprehensive dataset collected from literature with more than 1891 experimental data was used for IDAC prediction where the predicted results from the neural models are in close agreement with available experimental data with a squared correlation coefficient of R 2 = ( 0.9993,0.9976 and 0.9977 ) for training, validation and test data, respectively. The results were also compared with the predictions of the modified UNIFAC method for non-aqueous systems and COSMO-SAC method for aqueous

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systems where the NN model performs very well in comparison to physically-based models. Due to the convincing agreement between experimental values and predicted values for limiting activity coefficients, a multilayer perceptron form of the feed-forward

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neural networks with 7 or 13 neurons in the input layer (depending on the type of the available experimental data) and one hidden layer can be accurately used to predict the IDACs of a solute in a binary VLE system where only solute and solvent properties are

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required.

Keywords: Artificial neural network; limiting activity coefficient; binary systems; VLE;

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1. Introduction

The infinite dilution activity coefficient (IDAC ( γ ∞ )) is a thermodynamic property which is of great practical importance in separation processes, prediction of multicomponent vaporliquid equilibrium [1] and more particularly in calculating the parameters needed in the

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expressions for the excess Gibbs energy. This also verifies the great efforts made in both experimental techniques and theoretical models developed to accurate prediction of IDAC [2].

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IDAC of a solute in a solvent can be measured based on a number of experimental

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methods such as gas-liquid chromatography (GLC) [3], relative GLC [4], non-steady-state GLC method (NSGLC) [5], differential ebulliometry method [6], gas stripping method [7] and dew point method [8]. However, due to the expenses incurred in these experimental techniques and their time-consuming nature, different mathematical models for the prediction of IDACs [9] have been proposed including group contribution methods (such as UNIFAC (UNIQUAC functional group activity coefficients) [10], ASOG (analytical solution of groups) [11]), SPACE (Solvatochromic Parameters for Activity Coefficient Estimation)

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equation [12] , the MOSCED models [13], linear solvation energy relationships (LSER) [14] and free energy perturbation simulations [15]. Quantitative structure-property relationship (QSPR) models are an alternative approach of

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estimating IDAC where different types of molecular descriptors derived from the molecular structure are used to develop multiple linear regression and neural network models to predict IDAC [16]. Although in these models only the knowledge of the chemical structure is

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required without depending on any experimental properties [17], however, various ways of calculating physicochemical descriptors [18] is a challenging task.

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Artificial neural network (ANN) is one of the popular methods of dealing with nonlinear systems theoretic modeling, where the available experimental data can be used to develop NN models that can be used as a predictor.

Chow et al. [19] used the fragmented structural information to estimate the aqueous activity

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coefficients of aromatic organic compounds using a neural network approach. Giralt et al. [20] developed quantitative structure–property relations for the prediction of aqueous infinite dilution activity coefficient of organic compounds using the integration of self-

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organizing maps with a modified fuzzy ARTMAP neural system. Nami and Deyhimi [21] evaluated the performance of predicting IDACs for organic solutes in ionic liquids using a

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multilayer feed-forward ANN.

In order to be able to develop an accurate enough neuromorphic model, selection of appropriate input parameters is a critical issue. After the selection of the appropriate input parameters, a gathering of a rich enough database is the other important factor in the successful implementation of the ANN-based prediction methods. Accordingly, availability

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of a large amount of experimental data on IDACs of various solutes and solvents [2] is the main motivation for the development of predictive ANN models of the IDAC. In this work, IDACs are predicted using neural network approach. In Section 2 the details of

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the proposed methodology, including a brief overview of ANNs followed by the description of the proposed method as well as the data used in its development and validation are discussed in detail. In section 3, the proposed methodology is applied to various examples

2. Materials and Methods 2.1

Artificial neural networks

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experimental data and conclusions are drawn at the end.

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to evaluate the performance of the method in the prediction of IDACs against the

The thermodynamic behavior of a solute in a liquid phase solvent is affected by so many

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factors that make the development of a complete mathematical model impossible. Such complicated systems can be handled effectively with the ability of the ANNs in learning and recognizing complex and highly nonlinear functional relationships between inputs and

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outputs [22].

The interconnection of simple computational elements called neurons can represent an

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ANN which can mimic the computational abilities of biological systems [23,24]. ANNs are capable of performing many tasks [25] among which is the function approximation that can be used to map the input space to the output space. Multilayer perceptron (MLP) with feedforward network architecture is one of the most popular ANNs in use in this regard. This type of network consists of three main fully interconnected layer types including an input, one or more hidden layer(s) and an output layer as shown in Fig. 1. A neuron in each layer is connected to the neighborhood neurons as shown in Fig. 2. For the jth neuron, weighted

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sum of all the inputs ( xi ) plus the bias ( b j ) is passed through a transfer function (ϕ ) to

 n  y j = ϕ  ∑ w ji xi + b j   i =1 

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produce the output of the jth neuron ( y j ) according to the formula (1):

(1)

where wji = synaptic weight corresponding to ith synapse of jth neuron, xi = ith input signal

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to jth neuron; n = number of input signals to jth neuron.

Sigmoid type transfer function is the most common transfer function in the MLPs used by

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the neurons located in the hidden layer(s). The weights and biases are randomly initialized and their values are corrected in the training process of the NN using a learning algorithm [26]. A trained ANN (i.e. an ANN with appropriate weight and bias values) is capable of predicting the output corresponding to an unseen input within the range of the input data

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used in the training step. Accordingly, the range of the data used in the training step can affect the neural network prediction capabilities. 2.2

Neural network synthesis

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Two different approaches can be adopted for the selection of the type of data used in the synthesis of the ANN model for the prediction of IDAC:

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1. A specific solvent is chosen and a NN is trained for IDAC estimation of various solutes.

2. IDAC data corresponding to a wide range of solute and solvent combinations are considered in the model development. In the first approach, various solutes are considered in a specific solvent, therefore, it has limited applications. In order to improve the application domain of the proposed method,

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various solvents are considered as well in the second approach. Synthesis of an ANN model which is capable of predicting the limiting activity coefficient of a solute in a solvent at a specified temperature is performed in 3 main steps as follows.

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2.2.1 Data acquisition and analysis

Four examples were considered to evaluate the performance of a trained ANN model, based on the first approach. Accordingly, the IDAC of a series of solute in octane-1,8-

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diamine (1,8-ODA) [27] (Ex. 1), triacetin [28] (Ex. 2), Octadecanol [29] (Ex. 3) and N-

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formylmorpholine [30] (Ex. 4) over the temperature range from (328.21 to 348.25) K, (323 to 353) K, (333.15 to 353.15) K and (333.2 to 363.2) K were considered as the experimental data as shown in Table 1, Table 2, Table 3 and Table 4, respectively. A set of 1891 experimental data points of the IDAC of a series of solutes in various solvents was used to develop the neuromorphic model, based on the second approach (called Ex. 5).

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These data include experimental published data on the IDAC of binary systems from different literatures [1,28,31–50].

The data set used in the development of the neural model contains aqueous systems and

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both hydrocarbon and non-hydrocarbon compounds of various types including alcohols,

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amines, ketones, carboxylic acids, aldehydes, esters, ethers, nitriles and halogenated hydrocarbons in solute and solvent categories as well. The range of each input parameter for the experimental data points used for the development of the neural network model are summarized in Table 5, in which the wide range of each network input parameter can lead to the improved generality of the trained neural networks.

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In the training step of the ANNs, all data are normalized between -1 and 1 to avoid numerical problems due to very large or very small numbers.

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2.2.2 Neural network input After collecting and analyzing the available experimental data, the important task of specifying network input parameters should be accomplished. Selection of appropriate inputs for the NN model is one the key issues in the successful synthesis of the network for

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a specific application [23]. The prediction of the outputs with enough accuracy is greatly

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dependent on the ability of the trained network in implicit learning of the theoretical mapping between the inputs and outputs. Accordingly, the input parameters of the network (i.e. the independent variables of the model) should be informative enough which is an essential prerequisite for the development of such an ability in the trained network. The input parameters must be theoretically related to outputs. Accordingly, the theory

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behind the calculation of IDAC discussed by Miyano et al. [31] using the gas stripping method, which can be used as a basis for the selection of the most appropriate input parameters, is discussed as follows. This reasoning can be further supported by the fact

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that the Henry’s law constant is an appropriate representative of the intermolecular

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potential between different kinds of molecules [31]. The Henry's law constant is defined as the equation (2): H g = lim

xg → 0

f gV

(2)

xg

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V where fg is the vapor phase fugacity of the solute, xg is the mole fraction of the solute in

( )

the liquid phase and the subscript g indicates the solute. IDAC γ ∞

γ∞ =

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Henry’s law constants ( H g ) using equation (3):

can be related to the

Hg

(3)

ϕ Pgsat λ sat g

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sat sat where ϕ g is the fugacity coefficient of the pure solute at the saturation, Pg is the vapor

and is approximated by the equation (4):

(

)

 P − Pgsat v gL , sat     RT  

λ ≈ exp 

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pressure of the solute at the system temperature, and λ is the Poynting correction factor,

(4)

R is the ideal gas constant.

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L,sat where vg is the liquid molar volume of solute at saturation, P is the system pressure and

The virial equation of state can be used to calculate the thermodynamic properties

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V sat appearing in the equations (2)-(4) (i.e., f g , ϕ g and so forth). For example, the fugacity

coefficient of the solute at saturation is given by the equation (5):  BP sat C − B 2 = exp  g + 2  RT 

 Pgsat   RT

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ϕ

sat g

  

2

   

(5)

where B and C are the second and the third virial coefficients of the solute, respectively. The relationship between the parameters of the virial equation of state and intermolecular forces [51] can be further investigated using the extended theory of corresponding states [52,53] for the appropriate selection of the major input parameters. Appling the extended

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theory of corresponding states to the second virial coefficient, it is concluded that for all  BP  fluids with the same acentric factor (ω ) , the reduced second virial coefficient  c  is a  RT c 

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generalized function ( Fω ) of reduced temperature as shown in equation (6):

BPc T = Fω ( ) RT c Tc

(6)

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For example, Tsonopoulos [54] suggested equation (7) for polar and hydrogen-bonded

BPC T T T = F (0) ( ) + ω.F (1) ( ) + F (2) ( ) RTC TC TC TC

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fluids:

(7)

where F (0) and F (1) are the function of reduced temperature, however, F (2) is stated by

T a b )= − T T TC ( )6 ( )8 TC TC

µ R = 0.9869 ×105

µ 2 PC TC2

(8)

(9)

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F (2) (

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equation (8) where a and b are functions of reduce dipole moment define in equation (9).

In equation (9), µ (dipole moment), T C and PC have the units of debye, Kelvin and bar,

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respectively. The same discussion can also be applied to the reduced third virial coefficient, as there are several corresponding-states correlations available for the estimation of third virial coefficients of gas mixtures [55–58]. Based on the discussed theory for the calculation of IDAC, the selected input parameters are the critical temperature, critical pressure, critical volume, acentric factor and dipole moment for pure substances appearing in the equations (1) to (9). Because of the

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temperature dependence of the activity coefficient, it is considered as an input parameter as well. Another parameter that can affect the behavior of a mixture is the molecular weight of the

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components. Accordingly, molecular weight of the solute and solvent are chosen to be the two other input parameters needed in our ANN model. This selection is further verified by the following discussion. Gibbs excess energy

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models such as all UNIFAC and

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COSMO-RS [59] variants can be written as the sum of a combinatorial and residual

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contributions, essentially due to the differences in size and intermolecular forces between components, respectively. Accordingly, activity coefficient can be expressed as the sum of a combinatorial and a residual part as well [60]. Since the effect of intermolecular forces was discussed before, molecular weight of the solute and solvent which are proportional to the size of the molecules are chosen to be the other input parameters to represent the

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combinatorial part of the UNIFAC model for IDAC prediction. Based on the above discussion, the input of the ANN model can be summarized as:

critical volume

TC , critical pressure PC ,

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molecular weight MW , dipole moment µ , critical temperature

vC and acentric factor ω of the solute and molecular weight, dipole moment,

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critical temperature, critical pressure, critical volume and acentric factor of the solvent and system temperature. The output of the network is considered to be the natural logarithm of the IDAC of the solute.

It is also worth mentioning that, in the first approach, the solvent is fixed and experimental IDAC data for a series of solutes are considered. Accordingly, the network input data corresponding to the solvent can be eliminated from the network inputs which can lead to a

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  ∞   ln γ = f T , P , v , MW, ω , µ , T NN with 7 input parameters (i.e., solute C C C 4 2444 3  ). On the other hand,  144 solute  

(

)

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in the second approach, both solute and solvent can be of different natures to extend the applicability of the proposed methodology to cope with a wide range of binary systems. Accordingly,

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input

parameters

are

required

in

this

approach

(i.e.,

  ∞ ln γ solute = f  TC , PC , vC ,MW, ω, µ , TC , PC , vC , MW, ω, µ , T  ). This is also shown schematically in 3 14442444 3   14442444 solute solvent  

)

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(

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Fig. 1. 2.2.3 Neural network training

After deciding about the input and output parameters of the NN model and gathering the appropriate experimental data, the architecture of the network must be specified. In this

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study, networks with only one hidden layer were considered while the number of neurons within the hidden layer was varied based on the constructive approach [61]. In this approach, starting from a minimum number of neurons in the hidden layer, the number of

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neurons is increased until the trained network can satisfy the required accuracy. At the beginning of the training phase, the network weights and biases were initialized randomly,

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then the NN was trained by the Levenberg-Marquardt algorithm [62] while the performance of the networks was judged by the Percent Absolute Average Deviation (%AAD) and the root mean square (RMS) error which are defined as follows: 1 % ADD = N

N

∑ 100 × i =1

(

) ( ln ( γ )

ln γ i∞,exp − ln γ i∞, ANN

)

(10)

∞ i ,exp

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RMS =

1 N

∑ ( ln (γ N

i =1

∞ i ,exp

) − ln (γ

∞ i , ANN

))

2

(11)

γ i∞,exp and γ i∞, ANN are the experimental and

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where N is the number of data points, and predicted values of IDAC, respectively.

For each example, the data set was divided into three parts; training data set, validation

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data set and testing data set. The validation set is used as follows. The error corresponding to the validation set is calculated during the training process, however, this error is not used

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in the back propagation algorithm for weights and biases update. Typically, this error is decreasing in the initial phase of training, however, it begins to increase as the over-training or memorization occurs. Accordingly, the network weights and biases are saved at the minimum of the validation set error.

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A random sample of 70% and 15% of the total data were chosen for training and validation, respectively, to set up an ANN which learns training data and leads to the determination of the weight and bias values of the network.

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The remaining 15% of the total data were considered as the test data set to verify the generalization capability of ANN. The proposed model was applied to five examples

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mentioned above and the results are investigated in the results and discussion section. 3. Results and discussion

First, the four examples (i.e., Ex. 1 to Ex. 4) showing the application of the proposed ANN model according to approach 1 were investigated. The constructive approach was applied to train networks with a different number of neurons for each example and %AAD are compared to choose the minimum number of the neurons in the hidden layer. The results are three-layer networks with 7 neurons in the input layer and one neuron at output layer

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and the number of neurons in the hidden layers are shown in Table 6. Some properties of the optimum designed networks are also shown in Table 6 where the squared correlation coefficient ( R2 ) is shown for training, validation and testing data used for network training

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and testing as a measure of the correlation between the trained network estimations and the experimental data.

A satisfactory agreement between the predicted data by ANN and experimental data can

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be seen in all examples where experimental data are plotted against their predicted values in Figs. 3-6.

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It is also worth mentioning that these examples have two drawbacks. First, the solvent is fixed and second, a limited number of data are used. Accordingly, a NN trained based on these experimental data has a very limited application and prediction capability. In order to train a NN covering a wide range of applications, a rich database is provided

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and various networks are trained to find the optimum network. Finally, the result is a threelayer NN with 13, 20 and 1 neurons in input, hidden and output layers, respectively, which provides satisfactory results where experimental data are plotted against their predicted

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values as shown in Figs. 7-9 for training, validation and test data, respectively. As can be seen in Figs. 7-9, despite the wide range of the IDACs, ANN model is capable of

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covering this wide range and providing accurate enough results, where the value of R2 for training, validation and test data are 0.9993, 0.9976 and 0.9977, respectively. The %ADD of training, validation and test data are 15.33%, 3.43% and 28.77% with RMS errors of 0.0933, 0.1852 and 0.1670, respectively. Accordingly, an enhanced prediction capability is expected from the trained network between the ranges used for training the network, which originates from the richness and wide ranges of the data used in its training.

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UNIFAC is one of the widely used methods to estimate IDACs which has several variants [63]. According to the analysis done by Putnam et al. [64], who has investigated the prediction accuracy of various UNIFAC-type group-contribution models, modified UNIFAC

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(Dortmund) [65] was found to be the best predictive model for IDACs for non-aqueous binary mixtures. However, this method provides very poor estimates for aqueous systems and strongly overestimates the experimental data. On the other hand, Lin and Sandler [66]

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proposed a COSMO-RS (COnductor-like Screening MOdel for Real Solvents) [59] variation called COSMO-SAC (COSMO segment activity coefficient) as an alternative predictive

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method which is found to perform satisfactorily for aqueous systems [67]. Accordingly, to compare the models, the estimate of the IDACs of non-aqueous systems was also provided by the modified UNIFAC (Dortmund) method and in aqueous systems, COSMO-SAC was used to estimate the IDACs of the solutes, where the results are compared with

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experimental data in Fig. 10. It is worth mentioning that for a few number of solutes in aqueous systems, the predictions of COSMO-SAC model were very poor which were eliminated from the results.

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Accordingly, as shown in Fig. 10, for non-aqueous systems where the values of IDAC are typically well below 102, the model of modified UNIFAC (Dortmund) performs comparatively

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well, while for aqueous system where the IDAC might reach very large values ( e.g., the order of 108 for 1-Octadecanol in water), the model of COSMO-SAC performs relatively poor and extremely underestimated the experimental results. A more thorough discussion on the performance of the various UNIFAC variants and COSMO-SAC model can be found in [67,68].

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The %ADD corresponding to the prediction of physical-based method (i.e., modified UNIFAC and COSMO-SAC) is 74.43% with R 2 = 0.915 which shows that NN model is more successful in predicting IDACs in both aqueous and non-aqueous systems.

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4. Conclusion

A multilayer feed-forward neural network structure was proposed to predict the activity coefficient at infinite dilution of the solute in a binary system. The theory behind the

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calculation of IDAC was reviewed in order to provide the most relevant input parameters of the neural network. Accordingly, molecular weight, critical pressure, critical temperature,

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critical volume, dipole moment and acentric factor of the solute and temperature of the system are identified as the input parameters of the network in the cases where the experimental data are available for a fixed solvent and various solutes. If the experimental data for various solvents and solutes are available, then the molecular weight, critical

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pressure, critical temperature, critical volume, dipole moment and acentric factor of both solute and solvent along with the system temperature are considered as the input parameters of the network.

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For the fixed number of inputs and outputs, the number of neurons in the hidden layer was determined based on a constructive approach and selecting the network with the minimum

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mean relative error. Five examples were defined and the results of the NN were compared with both the experimental data and physical-based models which showed a satisfactory agreement. Therefore, the ANN model can be reliably used to estimate the IDAC of solutes in binary systems within the appropriate ranges of input parameters considered in the training of the network. It is also noted that the proposed method can be applied for both aqueous and non-aqueous systems where the IDAC can vary in a wide range (

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(

)

∞ −1.47 ≤ ln γ solute ≤ 23.38 ) containing both hydrocarbon and non-hydrocarbon compounds of

various types including alcohols, amines, ketones, carboxylic acids, aldehydes, esters,

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ethers, nitriles and halogenated hydrocarbons. 5. Appendix. Supplementary Information

Supplementary data related to this article can be found, in the online version, at http://dx.doi.org/***. Supplementary Information includes the experimental data used in Ex.

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5, trained network based on these data and the Matlab file that can be used to obtain the

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IDAC of a specific solute/solvent mixture at a specified temperature. The Excel file provides details about the used experimental data from the literature including the solutes, solvents and their CAS numbers, critical properties, molecular weights, dipole moments and acentric factors along with the system temperature and experimental IDAC values.

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References

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[1] J.P. Coutinho, E.A. Macedo, Infinite-dilution activity coefficients by comparative ebulliometry. Binary systems containing chloroform and diethylamine, Fluid Phase Equilibria. 95 (1994) 149–162. [2] J.C. Bastos, M.E. Soares, A.G. Medina, Infinite dilution activity coefficients predicted by UNIFAC group contribution, Ind. Eng. Chem. Res. 27 (1988) 1269–1277. doi:10.1021/ie00079a030. [3] C. Eckert, B. Newman, G. Nicolaides, T. Long, Measurement and application of limiting activity coefficients, AIChE J. 27 (1981) 33–40. [4] H. Orbey, S.I. Sandler, Relative measurements of activity coefficients at infinite dilution by gas chromatography, Ind. Eng. Chem. Res. 30 (1991) 2006–2011. [5] A.J. Belfer, D.C. Locke, I. Landau, Non-steady-state gas chromatography using capillary columns, Anal. Chem. 62 (1990) 347–349. [6] E.R. Thomas, B.A. Newman, T.C. Long, D.A. Wood, C.A. Eckert, Limiting activity coefficients of nonpolar and polar solutes in both volatile and nonvolatile solvents by gas chromatography, J. Chem. Eng. Data. 27 (1982) 399–405. [7] J.-C. Lerol, J.-C. Masson, H. Renon, J.-F. Fabries, H. Sannier, Accurate measurement of activity coefficient at infinite dilution by inert gas stripping and gas chromatography, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 139–144. [8] D.B. Trampe, C.A. Eckert, A dew point technique for limiting activity coefficients in nonionic solutions, AIChE J. 39 (1993) 1045–1050.

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[60] R. de P. Soares, R.P. Gerber, Functional-segment activity coefficient model. 1. Model formulation, Ind. Eng. Chem. Res. 52 (2013) 11159–11171. [61] R.B. Boozarjomehry, W.Y. Svrcek, Automatic design of neural network structures, Comput. Chem. Eng. 25 (2001) 1075–1088. doi:10.1016/S0098-1354(01)00680-9. [62] M.T. Hagan, M.B. Menhaj, Training feedforward networks with the Marquardt algorithm, Neural Netw. IEEE Trans. On. 5 (1994) 989–993. [63] E.C. Voutsas, D.P. Tassios, Prediction of infinite-dilution activity coefficients in binary mixtures with UNIFAC. A critical evaluation, Ind. Eng. Chem. Res. 35 (1996) 1438–1445. [64] R. Putnam, R. Taylor, A. Klamt, F. Eckert, M. Schiller, Prediction of infinite dilution activity coefficients using COSMO-RS, Ind. Eng. Chem. Res. 42 (2003) 3635–3641. [65] J. Gmehling, J. Li, M. Schiller, A modified UNIFAC model. 2. Present parameter matrix and results for different thermodynamic properties, Ind. Eng. Chem. Res. 32 (1993) 178–193. [66] S.-T. Lin, S.I. Sandler, A priori phase equilibrium prediction from a segment contribution solvation model, Ind. Eng. Chem. Res. 41 (2002) 899–913. [67] R.P. Gerber, R. de P. Soares, Prediction of infinite-dilution activity coefficients using UNIFAC and COSMO-SAC variants, Ind. Eng. Chem. Res. 49 (2010) 7488–7496. [68] S. Zhang, T. Hiaki, M. Hongo, K. Kojima, Prediction of infinite dilution activity coefficients in aqueous solutions by group contribution models. A critical evaluation, Fluid Phase Equilibria. 144 (1998) 97–112.

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Table 1 Experimental IDAC for the solutes in the solvent octane-1,8-diamine (1,8-ODA) at different temperatures [27]

γ∞

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T=348.25 K 3.61 3.72 3.95 4.13 4.36 2.65 2.77 2.93 3 3.14 0.37 0.44 0.61 0.69 1.48 1.73 0.51 2.02 1.37 0.86 0.93 1.05 1.59 2.08 1.72 3.36 7.18

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T=343.19 K 3.63 3.76 4 4.24 4.58 2.66 2.85 3.03 3.16 3.33 0.34 0.42 0.59 0.68 1.55 1.92 0.53 2.38 1.51 0.9 1 1.14 1.62 2.11 1.77 3.37 7.42

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T=338.23 K 3.68 3.83 4.09 4.4 4.82 2.7 2.94 3.21 3.38 3.58 0.32 0.41 0.58 0.68 1.65 2.13 0.55 2.73 1.66 0.95 1.08 1.25 1.67 2.17 1.86 3.41 7.79

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T=328.21 K 3.79 4.03 4.36 4.73 5.47 2.82 3.18 3.59 3.94 4.24 0.28 0.38 0.57 0.67 1.88 2.71 0.66 3.86 2.07 1.08 1.29 1.51 1.8 2.39 2.07 3.52 8.79

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solute n-pentane n-hexane n-heptane n-octane n-decane 1-pentene 1-hexene 1-heptene 1-octene 1-decene methanol ethanol propan-1-ol butan-1-ol 1-chlorobutane 1-chloropentane dichloromethane hexyl chloride carbon tetrachloride ethyl acetate propyl acetate butyl acetate benzene toluene ethylbenzene diethyl ether dibutyl ether

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Table 2 Experimental IDAC for the solutes in the solvent Triacetin at different temperatures [28]

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2.17 0.54 0.72 1.32

1.05 0.9

0.62 0.75 1.34 1.13 1.93 2.74 3.47 5.54 1.04 1.06

0.94

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T=353 K 8.9 7.04 4.81 4.6 1.21 1.59

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T=343 K 9.4 8.15 5.1 4.79 1.26 1.66 1.58 1.8 1.91 1.88 2.22

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T=323 K 11.84 9.28 5.79 5.21 1.21 1.63

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solute heptane hexane cyclohexane 1-hexene benzene toluene methanol ethanol 1-propanol 2-propanol 1-butanol 2-butanol chloroform 1,2-dichloroethane trichloroethylene chlorobenzene ethyl ether methyl tert-butyl ether isopropyl ether butyl ether ethyl acetate methyl ethyl ketone anisol acetone

1.59 1.7 1.69 2.06 1.78 0.62 0.75 1.3 1.35 1.96 2.7 3.4 5.66 1.1 1.1 3.43 0.98

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Table 3 Experimental IDAC for the solutes in the solvent Octadecanol at different temperatures [29]

γ∞ T=338.15 K

T=343.15 K

pentane hexane octane methanol ethanol propanol butanol dichloromethane chloropropane chlorobutane chloroform tetrachloride acetone THF ethyl acetate butyl acetate acetonitrile pentene heptene cyclohexane toluene diethyl ether

0.0292 0.0551 0.0513 0.0423 0.0433 0.0722 0.1537 0.0262 0.0331 0.0389 0.0227 0.0288 0.0185 0.0204 0.0445 0.0526 0.1076 0.0261 0.0431 0.0365 0.0372 0.0257

0.0272 0.0468 0.0497 0.0386 0.0396 0.0625 0.136 0.0233 0.0294 0.036 0.022 0.0285 0.0179 0.0188 0.0408 0.0468 0.0969 0.0245 0.0417 0.0337 0.0355 0.0235

0.0259 0.0411 0.0485 0.0357 0.0369 0.0597 0.1209 0.0212 0.0269 0.0338 0.0214 0.0283 0.0174 0.0177 0.0384 0.0428 0.0901 0.0234 0.0407 0.0315 0.0343 0.0219

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T=348.15 K 0.0243 0.0359 0.0473 0.033 0.0343 0.055 0.1101 0.0194 0.0245 0.0321 0.0208 0.028 0.017 0.0166 0.036 0.0391 0.0832 0.0222 0.0397 0.0294 0.0331 0.0204

T=353.15 K 0.0231 0.0315 0.0461 0.0306 0.032 0.051 0.0992 0.0178 0.0225 0.03 0.0202 0.0278 0.0166 0.0156 0.0335 0.0358 0.0766 0.0213 0.0388 0.0274 0.0318 0.019

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Table 4 Experimental IDAC for the solutes in the solvent N-formylmorpholine at different temperatures [30]

γ∞

TE D

AC C

T=363.2 K 7.54 8.51 10 10.59 12.02 4.99 5.57 6.28 6.7 1.63 1.77 1.92 2.16 4.55 5.41 6.8 6.06 0.511 0.618 0.548 0.632 1.1 1.42 1.61

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T=348.2 K 8.11 9.19 10.74 13.21 14.86 4.84 5.44 6.24 7.6 1.41 1.57 1.79 2.15 4.68 5.83 6.71 7.62 0.556 0.605 0.552 0.639 1.14 1.35 1.83

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T=333.2 K 8.32 9.51 11.43 13.76 16.67 4.75 5.51 6.37 8.39 1.27 1.47 1.72 2.07 5.16 6.32 7.51 8.23 0.482 0.606 0.537 0.627 1.1 1.39 1.73

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solute n-Pentane n-Hexane n-Heptane n-Octane n-Nonane 1-Pentene 1-Hexene 1-Heptene 1-Octene 1-Pentyne 1-Hexyne 1-Heptyne 1-Octyne Cyclopentane Cyclohexane Cycloheptane Cyclooctane Methanol Ethanol Propan-1-ol Propan-2-ol Benzene Toluene Ethyl benzene

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Maximum 863.00 22120.0 1.64 422.82 5.49 1.578 412.60

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Table 5 Range of IDAC and pure components properties in Ex. 5 Property Minimum Critical temperature/K 305.43 Critical pressure/kPa 868.0 3 Molar critical volume/(m /kmol) 0.0571 Molecular weight/(g/mol) 18.015 Dipole moment/Debye 0.000 Acentric factor 0.104 Temperature/K 249.84 ∞ Natural logarithm of IDAC ( ln ( γ solute )) -1.470

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Table 6 Results of the network synthesis for Ex. 1 to Ex. 4 octane-1,8-diamine [27]

triacetin [28]

Octadecanol [29]

N-formylmorpholine [30]

Ex. # No. of solutes No. data points No. neurons in the hidden layer train 2 validation R test train %ADD validation test train validation RMS test

Ex. 1 24 108

Ex. 2 27 64

Ex. 3 22 110

Ex. 4 24 72

5

4

4

4

0.9998 0.9984 0.9986 2.44% 4.78% 14.53% 0.014 0.055 0.049

0.9976 0.9954 0.9940 21.11% 8.40% 11.20% 0.049 0.098 0.107

0.9998 0.9992 0.9993 0.20% 0.29% 0.53% 0.009 0.014 0.023

0.9995 0.9945 0.9938 4.95% 12.84% 8.82% 0.037 0.098 0.115

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Input layer Hidden layer(s)

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Fig. 1 Schematic of MLP with feed-forward network architecture

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Fig. 3 Comparisons of the ANN-predicted values of IDACs and experimental values for training, validation and testing data set in Ex. 1

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Fig. 4 Comparisons of the ANN-predicted values of IDACs and experimental values for training, validation and testing data set in Ex. 2

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Fig. 5 Comparisons of the ANN-predicted values of IDACs and experimental values for training, validation and testing data set in Ex. 3

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Fig. 6 Comparisons of the ANN-predicted values of IDACs and experimental values for training, validation and testing data set in Ex. 4

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Fig. 7 Comparisons of the ANN-predicted values of IDACs and experimental values for training data set in Ex. 5

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Fig. 8 Comparisons of the ANN-predicted values of IDACs and experimental values for validation data set in Ex. 5

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Fig. 9 Comparisons of the ANN-predicted values of IDACs and experimental values for testing data set in Ex. 5

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Fig. 10 Comparisons of the modified UNIFAC/COSMO-SAC predicted values of IDACs and experimental values for data set in Ex. 5

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Infinite dilution activity coefficient (IDAC) of binary VLE systems are addressed

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A multilayer feed-forward neural network model for IDAC prediction is proposed

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Developed neural model only requires the solute and solvent properties

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