Ahmad KhazaiePoul,M.Soleimani*,S.Salahi

1 PhD Candidate of Faculty of Water and Environmental Engineering,Shahid Beheshti University,Tehran,Iran

2 Amol University of Special Modern Technologies,Amol 46168-49767,Iran

3 Chemical Engineering Department,Islamic Azad University,Shahrood Branch,Shahrood,Iran

1.Introduction

In recent years,there has been a rising interest in the utilization of supercritical fluids as an alternative to the employment of organic solvents in many industrial applications,such as in chemical and biochemical reactions[1,2],extraction and purification processes[3,4],microand nano-scale particle production[2],textile industry[5],synthesis of fibers[6],production of polymers[7],and decaffeination of coffee and tea[8].

SC-CO2has many advantages related to water.For example,above the critical point the carbon dioxide has properties of both a liquid and a gas.In this way SC-CO2,has liquid like densities,which is advantageous for dissolving hydrophobic dyes,and gas-like low viscosities and diffusion properties,which can lead to shorter dyeing times compared to water[9,10],also SC-CO2eliminates colored wastewater and high drying energy costs[11].In other hand,working with SC-CO2requires high-pressure process which implies high investment costs for the machinery and the training of skilled staff[12].One way to reduce the working pressure is adding a little quantities of co-solvent,such as ethanol,that significantly increase the solubility in lower pressures[13,14].Although,solubility is one of the most important parameters for dye selection and also for optimization of process temperature and pressure[11],mechanism of dye solubility in the SC-CO2and cosolvent systems is highly complex and is dificult to model by means of conventional mathematical modeling[15].

As experimental studies are very expensive and time consuming,many researchers have tried to predictthe solubility of dye in supercritical carbon dioxide by classical equations of state and semi-empirical equations[16,17].Gordillo and coworkers[18]used both approach,classical equations of state and semi-empirical equation,to predict solubility of disperse blue 14 in supercritical carbon dioxide.Tamura and Shinoda[19]experimentally investigated binary and ternary solubilities of disperse dyes and their blend in SC-CO2.They also used empirical equation and modified Peng-Robinson-Stryjek-Vera equation of state to correlate the solubilities of the dyes.

Nonlinearity of dye solubility mechanism is mainly due to the interaction of more number of variables such as temperature,pressure and mixture density[11,20].Application of ANN has been considered as a promising toolbecause oftheir simplicity towards simulation,prediction and modeling.One of the characteristics of modeling based on ANNs is that it does not require the mathematical description of the phenomena involved in the process,and might therefore prove useful in simulating and up-scaling complex systems.So,itis preferable to use a nonparametric technique such as a neuralnetwork modelto make reliable prediction of dye solubility in the SC-CO2and co-solvent system[21].

Tabaraki and coworkers[22]used a wavelet neural network(WNN)model for solubility prediction of 25 anthraquinone dyes in SC-CO2.Gharagheizi and coworkers[23]reported application of ANN model for prediction of solubilities of 21 of the commonly used industrial solid compounds in supercritical carbon dioxide.There are rarely studies reported on employing neural network models for the prediction of dye solubility in the co-solvent systems.Accordingly,this article develops a robust ANN model to predict the solubility of three disperse dyes in a co-solvent mixture containing ethanol and SC-CO2.

2.Artificial Neural Networks

ANNs are known for their superior ability to learn and classify data.The inspiration of neural networks came from studies on the structure and function of the brain and nerve systems as well as the mechanism of learning and responding.The potential applications include prediction,classification,data association,data conceptualization,data filtering and optimization.

As mentioned before neurons are main building block of neural networks.In an ANN a neuron sums the weighted inputs from several connections and then output of neurons is produced by applying transfer function to the sum.There are many transfer function but the common transfer function is sigmoid and we used this transfer function.Sigmoid function can be expressed by the following equation:

In Eq.(1)ψ is the sum ofweighted inputs to each neuron and θ is the output of each neuron and ψ can be calculated from Eq.(2).

In Eq.(2)wijdenotes connection between node j of interlayer l to node i of interlayer l-1,bjis a bias term and n is number of neuron in each layer.In any interlayer l and neuron j input values integrate and generate ψj.

In order to minimize the difference between experimental data and calculated of neural network,mentioned process repeats for the total number of training data.After training,validation of neural network can be done by testing data.

Numerous types of the ANNs exist such as multi-layer perceptron(MLP),radial basis function(RBF)networks and recurrent neural networks(RNN).The type of network used in this work is the multi-layer perceptron network.Multi-layer perceptron networks are one of the most popular and successful neural network architectures,which are suited to a wide range of applications such as prediction and process modeling[24].

2.1.Preparation of dataset

Forty eightexperimentaldata sets which have been collected from a published paper[25],were used to develop the ANN model.Data specifications of model variables are summarized in Table 1.In other words three differenttypes of dyes such as disperse blue 79,disperse orange 3,and solvent brown 1 in a condition in which pressure varies from 15.92 till 30.11 MPa for three constant values of temperature equal to 353.2,373.2 and 393.2 K,were employed to their solubility be calculatedusing the model.Hence temperature and pressure and some other parameters such as critical temperature,critical pressure,acentric factor and molecular weight each as a function of type of dye,were included in the table.The other parameter is density which ranges from 326.6 to 753.8 kg·m-3and depends not only to the type of dye but also to the system conditions such as temperature and pressure.These parameters were selected based on the last research about solubility and have been devoted to the network as inputs[26].In Table 2 the molecular structures of the dyes are reported.

Table 1 Data statistics of model variables

Table 2 Different molecular structures of the dyes

In this work,all data are divided into three parts(training subset(70%of all data),validation subset(15%of all data)and testing subset(15%of all data)).To prevent larger number from overriding smaller number;all data are normalized.Normalization can be done by several equations.In present work,data is scaled between[0-1]by means of Eq.(3).Data preparation which has been illustrated in this section is the first step of a model development(see Fig.2).

2.2.ANN modeling

Programming,validation,training and testing of the ANN model were carried out by MATLAB 7.12.0.Also,all programs were run on a Pentium IV(CPU 2.7 GHz and 2 GB RAM)personal computer with windows 7 operating system.

2.3.Performance criteria

In this study in order to compare the results of proposed model,statistical parameters(as shown in Table 3)were utilized.In statistics,the mean absolute error(MAE)is a quantity used to measure how close forecasts or predictions are to the eventual outcomes.The smaller value of this index indicates higher accuracy of the model.The next parameter is RMSE which is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed that must be minimum if a good model is expected.Dr index is the mean value of predictions to the observed data.The greatervalue ofitthan one showsthe overestimation and the lowerone indicates the underestimation ofthe model,while the value ofone implies the perfect estimation of the model.The Nash-Sutcliffe model efficiency coef ficient is used to assess the predictive power of hydrological models.Nash-Sutcliffe efficiencies can range from-∞to 1.An efficiency of 1(N-S=1)corresponds to a perfect match of modeled discharge to the observed data.An efficiency of 0(N-S=0)indicates that the model predictions are as accurate as the mean of the observed data,whereas an efficiency less than zero(N-S＜0)occurs when the observed mean is a better predictor than the model or,in other words,when the residualvariance(described by the numeratorin the expression above),is larger than the data variance(described by the denominator).Essentially,the closer the model efficiency is to 1,the more accurate the model is.R or the Pearson product-moment correlation coefficient is a measure of the linear correlation(dependence)between two variables X and Y,giving a value between+1 and-1 inclusive,where 1 is the total positive correlation,0 is no correlation,and-1 is total negative correlation.

Table 3 Performance criteria used in this study

Fig.1.Overall structure of our proposed network.

Fig.2.Preparation steps of the ANN model.

3.Results and Discussion

3.1.ANN model

For the present study a multi-layer perceptron which is displayed in Fig.1 was employed.As shown on this figure,temperature,pressure,critical temperature,critical pressure,density,molecular weight and also acentric factor were devoted to the model to predict the value of solubility for three different dyes such as disperse orange 3,solvent brown 1 and disperse blue 79.Fig.2 illustrates the developmentprocess of the model schematically.The minimum value of the mean squared error(MSE)of the training,validation and testing data sets through trial and error was considered to define the best architecture of the ANN(MLP)model which contains a hidden layer with three neurons and tansig as transfer function.Levenberg-Marquardt algorithm which is the most common algorithm for training an artificial neural network model,was utilized.Table 4 shows the weights between inputs and hidden nodes(Win)and between hidden nodes and output(Wout)for the selected architecture.

The performance ofthe proposed architecture is compared considering training,validation and testing data in Fig.3.The diagonal line in the middle ofthe figure is the location of exact predictions and points in the figure show training,validation and testing data.The error of each prediction is relative to the distance between each point and the diagonal line,so disperse ofpoints near the diagonal line demonstrate the overall accuracy of the present model.

To make it abundantly clear,Table 5 depicts the model accuracy by means of different statistical parameters.As can be seen the values of correlation coefficient,Nash-Sutcliffe model efficiency coefficient and discrepancy ratio are 0.998,0.992 and 1.053 respectively.It should be noted that the more close these coefficients are to one,the model is more accurate.

Fig.4 also shows plot of error distribution in versus of dye solubility for proposed ANN model.As can be seen the values of relative error have a suitable distribution around zero line exceptforvery smallvalues of solubility,less than 0.00003 kg·m-3,for which error distribution ranges from 25%to 100%.Actually for the values of solubility in the vicinity of zero,the corresponding values of error are tangible which result in greater relative error even near to 100%.

Table 4Weights between input and hidden layers(W in)and between hidden and output layers(W out)

Fig.3.Comparison of estimated and observed values of dye solubility for training,validation and testing data.

Table 5 Statistical accuracy of ANN model for testing data

3.2.Sensitivity analyses

In the present study,to find out the degree of effectiveness of different input variables on the efficiency of the proposed model,two sensitivity analyses were conducted using Numeric Sensitivity Analysis(NSA)[27]and Garson equation independent and dependent to the magnitude of weights[28],respectively.

3.2.1.NSA

NSA operates based on the calculation of the slopes between the input and output variables,without considering the nature of the inputvariables[27].To use the NSAmethod,data sets should be ordered in ascending array based on the values of the considered input variable(Xi).Then a determined number Gofgroups ofapproximately equalsize is produced which the NSA index is calculated using Eq.(1).

In this equation）and）are the means of xicorresponding to the consecutive groups grand gr+1,andare the means of ykcorresponding to the same groups respectively.The expected value of the NSA index which is used to identify the correlation between xiand ykcan be acquired by means of Eq.(2).In this equation gGand g1show the last and the first group respectively.

In order to obtain the quantity of oscillations of the slope between xiand yk,standard deviation(SD)was applied which is presented by Eq.(3).

Since differentmeasurementscales ofinputand outputvariables are engaged calculating the NSA index,normalizing them should be considered as a really vital step in order to avoid feasible biases.Table 6 displays the results of NSA in which the index Mean is equal to E(NSAik(gr))and represents the in fluence of each input variable on dye solubility as the output of the model.The higher value of this index indicates the higher effect of the input on dye solubility while the minus value infers thatdye solubility has been decreased.For example a single step increasement in normalized pressure causes the normalized dye solubility to rise up to 0.505 averagely.

Fig.4.Error distribution for proposed ANN model.

Table 6 Results of sensitivity analysis using NSA

Fig.5.Relative importance of input variables using Garson(1991)method.

Itis seen thatdensity,pressure,molecularweightand acentric factor are categorized respectively as the first four factors of great importance.

3.2.2.Garson Equation(GE)

Garson proposed Eq.(4)which defines the relative in fluence(RI)of a special input variable on the output of the model based on the magnitude of the weights between its different layers.

whereis the sum of the connection weights between the N input neurons and the hidden neuron i,RIjis the relative importance ofthe i th inputvariable(xi)on the outputvariable(yk)and L is the number of hidden layer nodes.

Fig.5 shows the results ofGE in which Eq.(4)was devoted by the information of Table 4.As can be seen from the figure both the temperature and pressure which are the main components of the input variables have the in fluence of 18 and 9%respectively in relation to the other ones.Although the degree of these values is not high enough,but it should be noted that the density which is dependent on the temperature and pressure and simultaneously on the type of the dye material,has the contribution of 40%,indirectly represents the effect of temperature and pressure.The in fluence of other parameters which are material specifications reaches to 33%on the efficiency of the model to predict dye solubility.

3.3.Constructed model

Fig.6.Agreement between ANN outputs and experimental data as a function of pressure for disperse orange 3(molecular weight=242.24 kg·mol-1 and temperature=373.2 K).

Fig.7.Agreement between ANN outputs and experimental data as a function of pressure for disperse blue 79(molecular weight=639.43 kg·mol-1 and temperature=373.2 K).

Fig.8.Agreement between ANN outputs and experimental data as a function of pressure for solvent brown 1(molecular weight=262.31 kg·mol-1 and temperature=373.2 K).

After constructing the model some conditions of the system were considered and the model efficiency was checked.As shown in Figs.6-8,there is a good agreement between ANN outputs(solid line)and experimental data(square points).In these figures the solubility of disperse blue 79,disperse orange 3 and solvent brown 1 was plotted versus pressure while temperature is equal to 373.2 K.

In Figs.9-11,solubility was evaluated whereas both temperature and pressure were changed simultaneously and type of dye,actually its specifications such as molecular weight,acentric factor,critical temperature and critical pressure were considered constant.It should be noted that density which is another input variable is a function of type of dye and simultaneously temperature and pressure.Thus,these figures could be applied to estimate the value of solubility for different types of dyes in a situation in which both temperature and pressure vary which is an issue of great importance for an operator to define the usage policy in an especial condition to reach the desired level of solubility.

Fig.9.Simultaneous effect of temperature and pressure on solubility of disperse orange 3 in co-solvent of supercritical carbon dioxide and ethanol.

Fig.11.Simultaneous effect of temperature and pressure on solubility of disperse blue 79 in co-solvent of supercritical carbon dioxide and ethanol.

Fig.12.Comparison of solubility for different types of dyes,pressure varies while temperature has the constant value of 373.2 K.

Figs.12 as a 2D section of Figs.9-11,depicts a comparison between dye solubility versus pressure.Molecular weights of disperse orange 3,solvent brown 1 and disperse blue 79 are 242.24,262.31 and 936.43 kg·mol-1respectively while temperature has the constant value of 373.2 K.It is seen that molecular weight and solubility are inversely related.

4.Conclusions

In this work solubility of three dyes,disperse blue 79,disperse orange 3 and solvent brown 1 in a mixture containing 5.5 mol%of ethanol in carbon dioxide has been predicted by utilizing an optimized ANN with 7 inputs that consist of temperature,pressure,critical temperature,critical pressure,density,molecular weight and acentric factor.NSA and GE were used to find out the degree of effectiveness of different input variables on the efficiency of the proposed model.Sensitivity analysis results showed that density,pressure,molecular weight and acentric factor are categorized respectively as the first four factors of great importance.Statistical analysis indicated that our proposed ANN model has correlation coefficient,Nash-Sutcliffe model efficiency coefficient and discrepancy ratio about 0.998,0.992 and 1.053 respectively.

Appendix A.Statistical Analysis

Mean absolute error

Root Mean Squared Error

Discrepancy ratio

Nash-Sutcliffe model efficiency coefficient

Pearson product-moment correlation coefficient

where

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