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J. of Supercritical Fluids 97 (2015) 256–267

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids

j o ur na l ho me page: www.elsev ier .com/ locate /supf lu

apor liquid equilibrium prediction of carbon dioxide andydrocarbon systems using LSSVM algorithm

ohammad Mesbaha, Ebrahim Soroushb, Vahid Azaria, Moonyong Leec,lireza Bahadorid,∗, Samaneh Habibniab

Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, IranDepartment of Chemical Engineering, Sahand University of Technology, Tabriz, IranSchool of Chemical Engineering, Yeungnam University, Gyeungsan, Republic of KoreaSouthern Cross University, School of Environment, Science and Engineering, Lismore, NSW, Australia

r t i c l e i n f o

rticle history:eceived 1 November 2014eceived in revised form2 December 2014ccepted 13 December 2014vailable online 23 December 2014

eywords:upercritical phase equilibrium

a b s t r a c t

Many supercritical processes, like monomer separation depends crucially on VLE data. The need of sim-ple, robust and general method, which can overcome deficiencies of EOSs, especially in critical regions,is obvious. In this study, a mathematical algorithm based on Least-Squares Support Vector Machine(LSSVM) has been developed for simulating 425 VLE data of seven CO2/hydrocarbon binary mixtures insupercritical or near critical conditions. The target value, bubble point/dew point pressure, is consideredas a function of reduced temperature, hydrocarbon mole fraction and the hydrocarbons acentric factorand critical pressure. The proposed LSSVM model with its magnificent R2 of 0.9932 and AARD% of 3.61 isproving able to predict VLE data of CO2/hydrocarbon binary mixture in a very precise manner. In addi-

ubble pointew pointeast-Squares Support Vector MachineLSSVM)utlier diagnostics

tion, comparison of LSSVM with EOSs indicates its supremacy over conventional methods. A sensitivityanalysis, with three different methods, was performed on the independent variables in an effort to deter-mine the relative importance of each one. At the end with the aid of Leverage statistical algorithm, thestatistical validity of the model was guaranteed and proved that the majority of the data points are in theapplicability domain of the proposed LSSVM.

ensitivity analysis

. Introduction

One of the most important components, which is frequentlysed as SCF is CO2. Mixtures which contain supercritical CO2ave a wide range of industrial applications like, separationrocess design, crystal growth processes and supercritical fluidxtractions [1–3]. Carbon dioxide is abundant, separate easily, non-oxic, non-flammable, moderate critical conditions (TC = 304.25 K,C = 7.38 MPa) and low-cost. Through controlling pressure, temper-ture and using co-solvents, the characteristics of solvation coulde simply established and it becomes an excellent medium forxtraction [2,4].

Monomer separation is one of the most significant processesased on SC-CO2. This process crucially depends on the VLE dataf solute/SC-CO2 system [1]. Clearly, it is not conceivable for

xperimental measurement of solute/SC-CO2 phase behavior of allubstances in every temperature and pressure. In this regard, it is

∗ Corresponding author. Tel.: +61 2 6626 9412; fax: +61 266269857.E-mail address: Alireza.bahadori@scu.edu.au (A. Bahadori).

ttp://dx.doi.org/10.1016/j.supflu.2014.12.011896-8446/© 2014 Elsevier B.V. All rights reserved.

© 2014 Elsevier B.V. All rights reserved.

a vital need for finding a systematic general method, which canpredict phase behavior.

One of the most popular methods for predicting VLE data isthe equation of states (EOSs). Some empiricisms are involved inthis method because of the mixing rules and numerous parameteradjustments [5,6]. Using these equations is time consuming andtedious owing to several parameter adjustments and mixing rules,their also complexity and iterative nature [5,6].

Recently, artificial intelligence methods have found their wayin VLE data prediction [5–16]. Particularly artificial neural net-works (ANNs) have been recognized as a strong and precise method,providing outstanding VLE predictions. Albeit network randominitialization and stopping criteria variation throughout the param-eter optimization step, are great deficiencies of neural networks[17].

In this study, we follow up our previous work [16] aimed to usean innovative mathematical technique called Least Squares SupportVector Machine (LSSVM) for predicting phase behavior of seven

CO2/hydrocarbon binary mixtures. Through statistical parameters,the validity of the model and its sensitivity will be examined. At theend, with the use of the Leverage statistical algorithm, suspiciousdata are avoided.

M. Mesbah et al. / J. of Supercritica

Nomenclature

VLE vapor liquid equilibriumEOS equation of stateLSSVM Least-Squares Vector Support MachineAARD average absolute relative deviation,%R2 correlation coefficientSCF super critical fluidsANN artificial neural networksSVM Support Vector MachineTr,i reduced temperature of component iPc,i critical pressure of component iωi acentric factor of component iek regression error� relative weight of the summation of the regression

errors�2 squared bandwidthw regression weightT transpose matrixb linear regression intercept of the modelϕ Map from input space into feature spacexk input vectoryk output vector˛i Lagrange multipliersMSE mean square errorK(x, xk) Kernel functionRBF radial basis function

2

2

acsrbt

acbttptaoanom

rWoaa

ovt

(prep/pred − pexp)

H hat matrix

. Mathematical modeling

.1. Support vector machine

Finding a concrete mathematical relation between the variablesnd anticipated output depends on having a proper mathemati-al tool. Despite of high preciseness which neural networks havehown in different fields [12,15,18–22], non-reproducibility ofesults is a very serious deficiency of them. This problem is causedy the random initialization of the network and stopping criteriahrough model parameters optimization [23,24].

Support vector machine (SVM) is a fabulous mathematicallternative. This method is developed by the machine learningommunity [23,25,26]. SVM is recognized as a non-probabilisticinary linear classifier. This algorithm projects the input spectrao a higher/infinite dimension and solves classification problemshere. The main goal of this algorithm is finding an optimum hyper-lane, which has a minimum distance to experimental values. Inhis method, through a standard algorithm, finding a quick solutionnd converging to a global optimum is very promising. At the endf the training procedure, topology of the network will be specifiedutomatically. In addition, multiple adjustable parameters are notecessary and the probability of over fitting issue decreases. Usef convex optimization and excellent generalization performanceade this algorithm to outstrip ANNs [23,25].Suykens and Vandewalle developed a modification of SVM,

eferred to as Least-Squares Support Vector Machine (LSSVM).hile it has all the advantages of original SVM, solving a group

f linear equations has helped this algorithm to become simplernd rapid in comparison with traditional SVM which should solve

quadratic programming problem [23,24].

In LSSVM method, in order to have additional constrain during

ptimization, regression error as the deflection between predictedalues and experimental eligible data (targets) is defined. Althoughhe error is defined mathematically in the LSSVM, the value of

l Fluids 97 (2015) 256–267 257

regression error in traditional SVM method will be optimizedmeanwhile the calculation steps is going on [23,24,27].

2.2. Equations

In the LSSVM algorithm, the cost function, which should be min-imized, is signified as:

QLSSVM = 12

wT w + �

N∑k=1

e2k (1)

Which has the following constrain:

yk = wT ϕ(xk) + b + ek, k = 1, 2, 3, . . ., N (2)

In these equations w represents the linear regression (regres-sion weight), T is indicative of the transpose matrix, � denotes therelative weight of the regression errors summation compared tothe regression weight, ek is training objects regression error, b rep-resents the model linear regression intercept, and finally ϕ showsthe feature map.

With the use of Lagrange function, the regression weights gen-erally defined as follows [23,24,27]:

w =N∑

k=1

˛kxk (3)

where ˛k is defined as:

˛k = 2�ek (4)

With the assumption of linear regression between indepen-dent and dependent LSSVM variables, Eq. (2) can be re-written as[23,24,27]:

y =N∑

k=1

˛kxTk x + b (5)

With the following equation the Lagrange multipliers ˛k can becalculated [23,24,27]:

˛k = (yk − b)

xTkx + (2�)−1

(6)

Using Kernel function the former linear regression equation willtransformed into a nonlinear form [23,24,27]:

f (x) =N∑

k=1

˛kK(x, xk) + b (7)

In which K(x, xk) denote the Kernel function, formed by the innerproduct of the vectors �(x) and �(xk) [23,24,27]:

K(x, xk) = �(x)T · �(xk) (8)

Radial basis function (RBF) is the most common used expres-sion for computing the Kernel function and we used in this paper[23,24,27]:

K(x, xk) = exp(−||xk − x||2/�2) (9)

Here �2 plays the role of a decision variable. Its optimizationhandled by an external algorithm during models internal calcula-tions. The mean square error (MSE) definition for the LSSVM can bedescribed as follows:∑i=1

MSE = n

ns(10)

which p represents dew/bubble-point pressure, rep/predand exp specify the represented/predicted and experimental

2 critica

bp

2

Vgnhmcpe

Ptpt

TP

TR

58 M. Mesbah et al. / J. of Super

ubble/dew-point pressure data respectively and ns is the initialopulation number.

.3. Designing the model

In an attempt to construct the LSSVM model 425 experimentalLE data points of seven binary systems containing CO2 has beenathered from literature [1,28–31]. These hydrocarbons include-hexane [31], propyl acrylate [30], propyl methacrylate [30], 1-exene [1], 2-ethyl-1-butene [1], decafluorobutane [29], methylethacrylate [28]. Table 1 shows the properties of the mentioned

ompounds. The experimental VLE data contained 122 dew pointressures and 303 bubble points. The detailed information on thesexperimental data are provided in Table 2.

The LSSVM algorithm used in this study was developed by

elckmans et al. [27] and Suykens and Vandewalle [23]. In ordero construct a model, able to prognosticating the bubble/dewoint pressure of various CO2/hydrocarbon systems, acentric fac-or, hydrocarbon mole fraction, reduced temperature, and critical

able 1hysical characteristics of the studied components.

Component Tc (K) PC (bar)

Carbon dioxide (solvent) 304.1 73.8

1-Hexane 504 31.7

2-Ethyl-1-butene 512 31.6

n-Hexane 507.6 30.2

Propyl acrylate 568.9 32.5

Propyl methacrylate 598.9 29.1

Decafluorobutane 386.3 23.23

Methyl methacrylate 563.9 36.8

able 2ange of the data variables.

Component Temperature range (K) Pressure range (b

1-Hexane 313.15–393.15 17.4–120.6

2-Ethyl-1-butene 313.15–373.15 24.30–107.90

n-Hexane 298.15–313.15 9.71–76.57

Propyl acrylate 313.15–393.15 35.10–157.60

Propyl methacrylate 313.15–393.15 16.30–163.10

Decafluorobutane 263.15–353.15 9.63–68.63

Methyl methacrylate 313.15–378.65 11.40–135.50

Overall bubble point 263.15–393.15 9.63–163.10

1-Hexane 333.15–393.15 96.9–120.6

n-Hexane 298.15–313.15 9.71–76.57

Propyl acrylate 333.15–393.15 84.30–157.60

Propyl methacrylate 313.15–393.15 101.90–162.80

Decafluorobutane 263.15–353.15 9.63–68.63

Methyl methacrylate 313.15–378.65 77.20–134.80

Overall dew point 263.15–393.15 9.63–162.80

Overall 263.15–393.15 9.63–163.10

l Fluids 97 (2015) 256–267

pressure of hydrocarbon were assumed as the input variables:

Bubble/Dew point pressure

= f (Tr, Pc, ωhydrocarbon, xhydrocarbon, yhydrocarbon) (11)

It is worth noticing that for the determination of dew point pres-sure, xCO2 was assumed zero while in order to find bubble pointpressure, yCO2 was set to zero.

The VLE data set was divided into two parts. The first part iscalled “Training set” and it is used for model construction. The otherpart is known as “Testing set” and used to determine the degree ofprediction ability and validity of the suggested LSSVM. In this man-ner, in a way that comprise an isotherm from each binary system,80% of the whole data selected for Training set and the remained20% considered for Testing set.

At the end with by joining Coupled Simulated Annealing (CSA)

and a standard simplex algorithm, tuning parameters were deter-mined. The starting values are first found by CSA and then passedto the simplex algorithm for tuning the outcomes. The optimumparameters for the algorithm were determined as 541.70 for � and

� Chemical structures References

0.24 [36]

0.29 [1]

0.23 [1]

0.30 [31]

0.43 [30]

0.4 [30]

0.37 [29]

0.32 [28]

ar) Transition Number of data References

Bubble point 59 [1]Bubble point 27 [1]Bubble point 18 [31]Bubble point 46 [30]Bubble point 55 [30]Bubble point 74 [29]Bubble point 24 [28]Bubble point 303 –

Dew point 4 [1]Dew point 18 [31]Dew point 12 [30]Dew point 9 [30]Dew point 74 [29]Dew point 5 [28]Dew point 122 –

– 425 –

M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267 259

algo

0r

2

id[Taamafm[

H

Fig. 1. CSA-LSSVM

.0365 for �2. A typical schematic diagram for CSA-LSSVM algo-ithm is shown in Fig. 1.

.4. Outlier diagnostics

One of the most important aspects, in designing a mathemat-cal model is outlier diagnostics [32,33]. Outliers are individualata (or groups of data) which diverge from the bulk of data32,33]. Experimental errors are the main source of outliers [21].hese suspicious data may have negative impact on the modelnd reduce its accuracy [20,21]. Statistical based algorithm, knowns Leverage approach, is one of the most effective and robustethods for outlier diagnostics [33,34]. This method is based on

matrix whose elements are the deflection of model predictions

rom their corresponding experimental values. It is called as Hat

atrix [32,33]. Leverage or Hat indices are defined as follows32,33]:

= X(XtX)−1

Xt (12)

rithm schematic.

In this equation X is a matrix, consisted of n rows (each row rep-resenting a data) and k columns (each column indicates a modelparameter). Here t is a sign of transpose matrix. The diagonal ele-ments of the H matrix, indicate the Hat values in the feasible regionof the problem.

The suspicious data may graphically be identified by Williamsplot that uses the correlation of hat indices and standardized cross-validation residuals (R) [32,33]. With the help of Eq. (12), the Hvalues could be evaluated and used for sketching the Williams plot.Generally the warning Leverage value (H) is specified as 3p/n. Heren denotes the number of training points and p shows the number ofcorrelation input parameters [32,33]. A satisfactory “cut off” Lever-age value is generally 3 [32,33]. Therefore, the points that are inthe standard deviation range of ∓3 are accepted. If the bulk of datapoints lies within the range of 0 < H < H∗ and −3 < R < 3, then it con-firms that the model is statistically valid. The points which could

not predict by the model, are known as good high Leverage pointsand are located in the range of H ≤ H∗ and −3 < R < 3. The ones whichcan be considered as doubtful data are called outliers or bad highLeverage and are in the range of R < −3 or R > 3.

2 critical Fluids 97 (2015) 256–267

3

3

sitpFaitn(aobp

Table 3Statistical parameters of the developed LSSVM.

MSE AARD% R2 Number of data

All data 9.27 3.61 0.9932 425Train dataset 6.71 3.19 0.9948 340

60 M. Mesbah et al. / J. of Super

. Result and discussion

.1. Accuracy of the model

For a comprehensive investigation of model accuracy, bothtatistical and graphical measured are implemented. The graph-cal error analysis is performed through cross plot for checkinghe reliability of the suggested model and error distributionlot, which intended to investigate any possible error trend.igs. 2 and 3 provide a comparison between model predictionsnd their corresponding experimental values for Training and Test-ng sets. The bulk of data, accumulated about the 45◦ lines inhe cross plots (Figs. 2a and 3a) is an indication of the robust-ess of the proposed LSSVM. In addition, error distribution plotsFigs. 2b and 3b) indicate a reasonable error for the data set. For

more sensible visual comparison, Fig. 4 depicted p-x-y plotsf experimental equilibrium data for all seven CO2/hydrocarboninary systems and their corresponding predicted values by theroposed LSSVM at various temperatures. This figure simply

Fig. 2. Comparing model predictions for training data and their correspon

Test dataset 19.53 5.30 0.9880 85

suggests that between model predictions and experimental VLEdata an excellent agreement could be found.

LSSVM detailed statistical parameters, are provided in Table 3. Inaddition, Table 4 illustrates the detailed LSSVM statistical parame-ters for each binary system. This table proves that the proposedmodel is accurate, and can reproduce experimental values verywell. The minor deviations between equilibrium experimental dataand LSSVM predictions, is because of experimental uncertainties

and can satisfy engineering purposes (Table 5).

Comparing accuracy and performance of the suggested LSSVMwith the conventional EOSs will be noteworthy. Figs. 5–8 offer a

ding experimental data. (a) Scatter plot, (b) relative deviation plot.

M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267 261

Fig. 3. Comparing model predictions for testing data and their correspon

Table 4Detailed LSSVM statistical parameters for each CO2/hydrocarbon binary mixture.

Component Transition Numberof data

Error analysis

MSE AARD% R2

1-Hexane Bubble point 59 0.9292 1.1160 0.99862-Ethyl-1-butene Bubble point 27 0.3900 0.7449 0.9993n-Hexane Bubble point 18 15.4254 2.7311 0.9699Propyl acrylate Bubble point 46 4.8155 1.9267 0.9937Propyl methacrylate Bubble point 55 2.6297 1.6370 0.9982Decafluorobutane Bubble point 74 0.3808 1.3944 0.9987Methyl methacrylate Bubble point 24 5.2193 4.5623 0.9939Overall bubble point Bubble point 303 2.8469 1.7375 0.9976

1-Hexane Dew point 4 0.0786 0.2107 0.9991n-Hexane Dew point 18 51.4336 20.8761 0.8647Propyl acrylate Dew point 12 12.5314 1.7543 0.9756Propyl methacrylate Dew point 9 25.1816 2.5929 0.9362Decafluorobutane Dew point 74 21.8686 7.7134 0.9271Methyl methacrylate Dew point 5 31.5601 3.3068 0.9546Overall dew point Dew point 122 25.2394 8.2650 0.9856

Overall – 425

ding experimental data. (a) Scatter plot, (b) relative deviation plot.

clear visual comparison between suggested model and differentEOSs and indicate that LSSVM can prognosticate phase diagramof CO2 containing binary systems in a much accurate and pre-cise manner. In addition, the aforementioned figures demonstratesome interesting points. For example, in the CO2/propyl methacry-late (Fig. 5) and CO2/methyl methacrylate (Fig. 6) systems, despitethe PR acceptable predictions at low temperatures, notable devi-ations could be detected in high temperatures. Moreover, in theCO2/1-hexan binary system (Fig. 7), it is evident that PR equationparameter adjustment plays an important role in developing anaccurate model. Aside from SAFT obvious overestimate predictionsfor CO2/propyl acrylate binary (Fig. 8), the parameter adjustmentproblem is also the case for PR EOS in this system. In general, thetedious and iterative nature of these EOSs, the numerous parame-ter adjustments and their deficiencies in some physical situationsput them in a weak position compared to the proposed model.Above all, the supremacy of LSSVM comes evident when we talk

about a unifying general accurate model, which can be used forany system and in any physical conditions in an easy and quickmanner.

262 M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267

Fig. 4. Plot of the p-x-y data, including experimental values and LSSVM results at different temperatures in the binary system (a) CO2/1-hexane, (b) CO2/2-ethyl-1-butene[1], (c) CO2/n-hexane [31], (d) CO2/propyl acrylate [30], (e) CO2/propyl methacrylate [30], (f) CO2/decafluorobutane [29], (g) CO2/methyl methacrylate [28].

M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267 263

Table 5Comparing LSSVM predictions with the conventional EOSs.

T = 80 ◦C

1-hexane LSSVM MSE 1.18R2 0.9985

PR-EOS kij = 0.000, �ij = 0.000 MSE 121.22R2 0.9783

PR-EOS kij = 0.113, �ij = 0.050 MSE 6.26R2 0.9898

Propyl Acrylate LSSVM MSE 7.38R2 0.9843

PR-EOS kij = 0.000, �ij = 0.000 MSE 93.21R2 0.9348

PR-EOS kij = 0.045, �ij = −0.030 MSE 23.38R2 0.9859

SAFT-EOS kij = 0.000, �ij = 0.000 MSE 464.50R2 0.8419

SAFT-EOS kij = 0.090, �ij = −0.090 MSE 151.88R2 0.8563

T = 40 ◦C T = 60 ◦C T = 80 ◦C T = 100 ◦C T = 120 ◦CPropyl Methacrylate LSSVM MSE 3.30 2.94 2.18 2.72 1.32

R2 0.9958 0.9979 0.9992 0.9995 0.9997PR-EOS kij = 0.045, �ij = −0.030 MSE 50.91 117.51 109.35 63.57 145.48

R2 0.9530 0.9037 0.9404 0.9897 0.9499T = 40 ◦C T = 80 ◦C T = 105.5 ◦C

Methyl Methacrylate LSSVM MSE 6.63 5.36 17.842

SE

2

3

taf(ibftfa

Fs

RPR-EOS kij = 0.000, �ij = 0.000 M

R

.2. Outlier detection

The statistical parameters in Table 3 indicate that the devia-ion of LSSVM correlated values from actual experimental data isppropriate to be used in Leverage algorithm. Eq. (12) has been usedor finding the H values and through 3 (p + 1)/n warning Leveragesp + 1) have been determined. The Williams plots for each systems illustrated in Fig. 9. LSSVM statistical validity could be approvedecause of data accumulation between 0 ≤ H ≤ H∗ and −3 ≤ R ≤ 3

or each dataset. Fig. 9 denotes that majority of VLE data (morehan 96%) are in the applicability domain of proposed LSSVM. Yetor the CO2/n-hexane system (Fig. 9c) 9 points, for the CO2/propylcrylate system (Fig. 9d) 3 points and for the system (Fig. 9g)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

0 0.1 0. 2 0.3 0. 4

Pres

sure

(bar

)

ig. 5. Comparing the correlated results from the PR EOS (kij = 0.045, �ij = −0.030) [30]

ystem.

0.9895 0.9931 0.991812.29 159.52 107.840.9902 0.9958 0.9734

CO2/methyl methacrylate 4 points are recognized as bad high lever-ages. As mentioned before, these erroneous data can be consideredas doubtful data. In addition, the quality of the data used for modeldevelopment is different. The data which have lower H values andthe lower absolute values R, may be recognized as more reliableexperimental values.

3.3. Sensitivity analysis

A sensitivity analysis has been performed through the data set,for examining the sensitivity of the suggested LSSVM to the inde-pendent variables, or particular significance of each input/output

0.5 0. 6 0.7 0. 8 0.9 1x,y

Experimental (T=40 C)

Experimental (T=60 C)

Experimental (T=80 C)

Experimental (T=100 C)

Experimental (T=120 C)

LSSVM

PR

and LSSVM model with experimental values for CO2/propyl methacrylate binary

264 M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267

0

20

40

60

80

100

120

140

160

0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1

Pres

sure

(bar

)

x,y

Experimental (T=40 C)Experimental (T=80 C)Experimental (T=105.5 C)LSSVMPR

Fig. 6. Comparing the correlated results from the PR EOS (kij = 0.000, �ij = 0.000) [28] and LSSVM model with experimental values for CO2/methyl methacrylate binary system.

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1

Pres

sure

(bar

)

x,y

Exper imenta lLSSVMPR,Case1PR,Case2

Fig. 7. Comparing the correlated results from the PR EOS (Case 1: kij = 0.000, �ij = 0.000, Case 2: kij = 0.113, �ij = 0.050) [1] and LSSVM model with experimental values forCO2/1-hexane binary system (T = 80 ◦C).

0

20

40

60

80

100

120

140

0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0. 8 0.9 1

Pres

sure

(bar

)

x,y

Exper imenta lLSSVMPR,Case1PR,Case2SAFT ,Case1SAFT,Case2

Fig. 8. Comparing the correlated results from the PR EOS (Case 1: kij = 0.000, �ij = 0.000, Case 2: kij = 0.045, �ij = −0.030), SAFT EOS (Case1: kij = 0.000, �ij = 0.000, Case 2:kij = 0.090, �ij = −0.090) [30] and LSSVM model with experimental values for CO2/propyl acrylate binary system (T = 80 ◦C).

M. Mesbah et al. / J. of Supercritical Fluids 97 (2015) 256–267 265

F ed moC (f) CO

ce

t

ig. 9. Detection of probable outliers and the applicability domain of the presentO2/n-hexane [31], (d) CO2/propyl acrylate [30], (e) CO2/propyl methacrylate [30],

onnection. This analysis offers a valuable understanding of theffect of each parameter.

In this study for performing sensitivity analysis three differentechniques have been used: Pearson correlation, Spearman rank

del for the binary system (a) CO2/1-hexane [1], (b) CO2/2-ethyl-1-butene [1], (c)2/decafluorobutane [29], (g) CO2/methyl methacrylate [28].

correlation, Kendall’s Tau correlation. The linear or non-linear rela-tion of the input parameters and output assumptions is the maindifference in these techniques [35–38]. The correlation coefficientbetween the target variable, bubble point/dew point pressure and

266 M. Mesbah et al. / J. of Supercritica

-0.5

0

0.5

1

Pearson

Spearman rank

Kendall Tau

iTAwattmi

hopv

4

VmWocrbiis

iwdtoa

A

R

%

[

[

[

[

[

[

[

[

[

[

[

-1

Fig. 10. Relative variable significance on bubble/dew point pressure.

nput variables Tr, Pc, ω, xhydrocarbon, yhydrocarbon is shown in Fig. 10.he correlation coefficients can accept a value between +1 to −1.

value of +1 illustrates an increasing relation between variables,hile a value of −1 indicates a decreasing relation between vari-

bles and when correlation coefficient take the value of 0 it meanshat variables have no relationship. Increase in the absolute value ofhe correlation coefficient between any input and output variable

eans that the influence of that input in finding the target valuencreases.

Fig. 10 reveals the negative impact of reduced temperature andydrocarbon mole fraction, and it also shows the positive impactf critical pressure and acentric factor at finding bubble point/dewoint. This is in complete agreement with known effect of theseariables. Fig. 4 shows the model completely mimic this behavior.

. Conclusion

In this study a novel mathematical tool, Least-Squares Supportector Machine, is employed as a meta-learning technique for pro-oting phase behavior of seven CO2/hydrocarbon binary systems.ith the use of coupled simulated annealing (CSA) algorithm the

ptimum parameters of the model was determined. In contrary toonventional EOSs, which each system should be modeled sepa-ate, with the use of LSSVM advantages, a unifying model has beenuilt on a data set combined of seven different systems. The graph-

cal measures and statistical parameters of the model suggestedts high accuracy and comparing its result with EOSs indicate itsupremacy over conventional methods.

Leverage statistical algorithm indicates that the proposed models statistically valid. The majority of data set (more than 96%)

as in the LSSVM applicability domain; however, 16 outliers wereetected. At the end, sensitivity analysis showed that while reducedemperature and hydrocarbon mole fraction have a negative effectn predicting bubble/dew point pressure the critical pressure, andcentric factor of the hydrocarbons has a positive effect.

ppendix A.

Correlation factor (R2):

2 = 1 −∑N

i=1(Calc. (i)/Est. (i) − exp .(i))2∑Ni=1(Calc. (i)/Est. (i) − average (exp .(i)))2

(A1)

Average Absolute Relative Deviation (%AARD):

AARD = 100N

N∑i=1

|Calc. (i)/Est. (i) − exp .(i)|exp .(i)

(A2)[

l Fluids 97 (2015) 256–267

Mean Square Error (MSE):

MSE =∑N

i=1(Calc. (i)/Est. (i) − exp .(i))2

N(A3)

Standard Deviation (STD) [37,38]:

STD =

(1

N − 1

N∑i=1

(Calc. (i)/Est. (i) − average (Calc. (i)/Est. (i)))2

)1/2

(A4)

Relative Deviation:

Relative Deviation = Calc. (i)/Est. (i) − exp .(i)exp .(i)

(A5)

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