comparison the capability of artificial neural network (ann) and eos for

Fluid Phase Equilibria 308 (2011) 35 43

Contents lists available at ScienceDirect

Fluid Phase Equilibria

j our na l ho me page: www.elsev ier .co

Compa etwpredict car

M. LashkChemical Engin Shiraz

a r t i c l

Article history:Received 12 MReceived in reAccepted 3 JunAvailable online 12 June 2011

Keywords:Solid solubilitiesSupercritical carbon dioxideArticial neural networkCubic equationMixing rules

forwtic hy

supe and

validated by 343 testing data sets. The networks were different regarding to network parameters, suchas number of hidden layer, hidden neurons and training algorithm. Using validating data set, the networkthat is having the lowest absolute average relative deviation percent (AARD%), mean square error (MSE)and the highest regression coefcient (R2) is selected as an optimal conguration.

To verify the network generalization, 100 different data sets of 23 binary systems have been consid-

1. Introdu

Recentlychemical anexibility aout prior knhigher advatal correlati

ANNs hplicated anAbdolahi [4better than

CorresponE-mail add

0378-3812/$ doi:10.1016/j. of state ered. In the present work, 970 experimental data points of different works (up to now) which coversa wide range of temperatures and pressures have been used. Statistical analyses show that the arti-cial neural network (ANN) predictions have an excellent agreement (AARD% = 0.98, MSE = 2.8 105 andR2 = 0.99813) with the experimental data set.

Also, accuracy of the cubic PengRobinson (PR) and SoaveRedlichKwong (SRK) equations of state byusing six mixing rules, namely, the WongSandler (WS) rule, the OrbeySandler (OS) rule, the van derWaals one uid rule with one (VDW1) and two (VDW2) adjustable parameters, the covolume dependent(CVD) rule and the EsmaeilzadehAsadiLashkarbolooki (EAL) mixing rule for the prediction of solubilityof solids in supercritical carbon dioxide has been compared with a developed neural network model. Tobase this comparison on a fair basis, same experimental data points of 23 different compounds has beenused for both optimization of equations of state parameters and training, validation and testing of neuralnetwork. Results show that developed optimal ANN model is more accurate compared to the PR and SRKEOSs with mentioned mixing rules for the same compounds.

2011 Elsevier B.V. All rights reserved.

ction

, the use of ANNs in many technical elds such asd pharmaceutical areas has been increased due to theirnd ability to model linear and nonlinear systems with-owledge of an empirical model [1,2]. Thus, ANNs have antage over traditional tting methods and experimen-ons for some chemical applications [3].ave shown their strength in modeling of very com-d multi-variable dependent processes. Izadifar and] showed that the predictions of the ANNs model are

that of the mathematical model in the some cases.

ding author. Tel.: +98 711 2303071; fax: +98 711 6287294.ress: [email protected] (M.R. Rahimpour).

It should be noted that the application and capability of neuralnetworks for modeling of processes demands a large number ofexperimental data involving a used range of all variables.

The solubility is one of the most important physicochemicalproperties of chemical/pharmaceutical compounds.

Nowadays, supercritical uids have attracted great interest inthe development of alternate processes to substitute traditionalones such as solvent extraction, distillation, and wiped lm evapo-ration. Supercritical-uid technology offers numerous advantagescompared to conventional processes, such as high transfer rates,reduced number of unit operations, and lower operating costs [5,6].

Carbon dioxide is widely used in supercritical uid applicationsbecause it has mild critical conditions (Tc = 304.25 K, Pc = 7.38 MPa),is inexpensive, nonammable, nontoxic and readily available. Thesolubility of a solute in the supercritical uid is the most impor-tant thermo-physical property that needs to be determined andmodeled as a rst step to developing any supercritical uids appli-cation. In addition, experimental studies are very expensive and

see front matter 2011 Elsevier B.V. All rights reserved.uid.2011.06.002rison the capability of articial neural nion of solid solubilities in supercritical

arbolooki, B. Vaferi, M.R. Rahimpour

eering Department, School of Chemical and Petroleum Engineering, Shiraz University,

e i n f o

arch 2011vised form 29 May 2011e 2011

a b s t r a c t

The aim of this study is develop a feed-predict the solid solubilities of aromaaliphatic and aromatic alcohols in the

Different networks are consideredm/locate / f lu id

ork (ANN) and EOS forbon dioxide

71345, Iran

ard multi-layer perceptron neural network (MLPNN) model todrocarbons, aliphatic carboxylic acids, aromatic acids, heavyrcritical carbon dioxide.trained using 627 data sets; the accuracy of the network is

36 M. Lashkarbolooki et al. / Fluid Phase Equilibria 308 (2011) 35 43

Table 1Summary of the mixing rules used in this work.

Mixing rules Functional form Refs.

WongSandler a = RT QD1D [15,16]b = Q1Da =

i

j

xixj(b aRT

)ij

D =

i

xiai

biRT+ GexCRT

b aRT =

xixj(b aRT

)ij(

b aRT)

ij=(bi aiRT + bj

ajRT

) (1kij )2

OrbeySandler am = bRT

[Gex

CRT +n

i=1

xiai

biRT

][17]

bm =RTn

i=1

nj=1

xixj (b(a/RT))ij

RT(n

i=1xi (ai/bi )+(Gex/C))(

b aRT)

ij=(

bi+bj2

)(

aiajRT

)(1 kij)

CVD a =

xixjaij(

bbij

)mij, b =

xibi [18]

a =

(aiaj), b =

(bibj)

VDW1

VDW2 (1

time consumodynamicand effectivthe phase esimple EOSin chemicalids, etc. Duthe results important i

On the opredict thepredictionsmodels is ofound morearticial intconsumptio

ANNs arinformationof their relations are lo

ical ato an

of so tool his wg alglity inxideo EO

es. Ata seing m

atio

Table 2Summary of th

EOS and com

P = RTb (

S1 =

S2 =

(b aRT

)ij

=(b2 + (c1 + c

am = abRT =

C = 1c1c2 ln11

aThe kij is the a =

xixj(aiaj)0.5(1 kij), b =

xibi

a =

xixj(aiaj)0.5(1 kij), b =

xixj(

bi+bj2

)

ming. Many researchers have tried to predict the ther- properties by theoretical methods. EOS is an importante tool for calculations of thermodynamic properties andquilibrium of pure and uid mixtures. Accurate ands are widely used for theoretical and practical studies

process design, the petroleum industry, reservoir u-e to numerous industrial designs which are based onof these theoretical models, accuracy of them is a veryssue.ther hand, the conventional thermodynamics cannot

solubility of highly polar substances correctly and its have a large inaccuracy [7]. Application of the EOSften limited while articial intelligence (e.g. ANN) is

popularity for prediction of various processes. The

analytpared naturenative

In ttrainincapabibon diowith twing rul100 davalidat

2. Equ

elligence models have lower inaccuracy, cost, and time-n [8].e efcient tools which can be trained with experimental

to map input and output data regardless to complexitytion. Previous study showed that ANN model predic-w sensitive to noisy and incomplete data compared to

As discuand Soaveused to calcCO2, by usinand EAL. Ta

e EAL mixing rule.

bination rule PR

a+c1b)(+c2b) c1 = 1

2, c2 = 1 +

2

xixj(b aRT

)ij

b = S2S1[

1.51+2m 0.5

]xixj(b2 + (c1 + c2) aRT

)ij

m = abRT[ (bi(ai/RT))+(bj(aj/RT))2

](1 ka

ij) aij =

(aiaj)

2) aRT)

ij=(bij + (c1 + c2)

aijRT

)bij =

bi+bj2

N

i=1

xiai GexCRT

+c1+c2

binary interaction parameter.[13]

lij) [13]

pproaches, thus they are found more popularity com-alytical approaches [9,10]. Because of the nonlinearlubility, ANN method could be considered as an alter-for solubility modeling [11].ork, a feed-forward MLPNN with LevenbergMarquardtorithm has been developed in order to investigate its

prediction the solubilities of solids in supercritical car-. The proposed ANN model results have been comparedSs by using VDW1, VDW2, CVD, WS, OS, and EAL mix-

database containing experimental solubility data forts of 23 binary systems have been used in this study forodel results.

n of state and mixing rulesssed in this study, the cubic PengRobinson (PR) [12]RedlichKwong (SRK) [13] equations of state have beenulate the solid solubilities of 23 solutes in supercriticalg six mixing rules, namely, VDW1, VDW2, CVD, WS, OSble 1 shows a summary of the VDW1, VDW2, CVD, WS

SRK

c1 = 0, c2 = 1

b = S2S1[

21+m 1

]m = abRTaij =

(aiaj)

b3/4ij

=b3/4i

+b3/4j

2

M. Lashkarbolooki et al. / Fluid Phase Equilibria 308 (2011) 35 43 37

olids solubilities in supercritical carbon dioxide.

and OS mixEAL mixing

3. SolubiliEOS

The soludioxide is:

y2 =Psub2 (T

P

where Pperature, Ps2coefcient ocient of thevolume. In lated by thefrom the An

ln Psub = A +

The EOSsolid-superparametersusing the Mobjective fudeviations bcase, the fo

OF =Ni

(

The accuaverage rela

AARD% = 1N

where N

experimentwith the EO

thods

tici

icial moy by

maare cvariaern rful tombeeciesing led and thspecFig. 1. The schematic of used feed-forward MLPNN for prediction of s

ing rules and Table 2 shows a summary of the EOSs and rule [14].

ty of solid solute in supercritical uid by using

bility of the solid (component 2) in supercritical carbon

) sat.s2 (T)2(T, P, y)

exp

[vs2(P Psub2 (T))

RT

](1)

is pressure, R is the universal gas constant, T is tem-ub(T) is the sublimation pressure, sat.s2 is the fugacityf solute at saturation, 2(T, P, y) is the fugacity coef-

solid in the supercritical phase and vs2 is the solid molarthis work, the fugacity coefcient of the solid is calcu-

PR and SRK EOSs. The sublimation pressure is obtainedtoine equation:

B

T + C (2)

4. Me

4.1. Ar

Artmaticacenturused inworks multi-to pattpowerof a nuin a spprocesare callayer aone, re/Gex model was used to correlate the CO2 binarycritical equilibrium of the system [14]. The adjustable

in the models were tted to the all experimental dataarquardtLevenberg method [19] by minimizing annction (OF). The objective function is a measure of theetween the predicted and experimental values; in this

llowing objective function has been adopted:

yexp.s,i

ycal.s,i

yexp.s,i

)2(3)

racy of the calculations was evaluated by the absolutetive deviation percent dened as follows:

Ni

(yexp.s,i

ycal.s,i

yexp.s,i

)

100 (4)

is the number of solubility data points; yexp.s,i

is the ith

al value of the solubility; ycal.s,i

is the solubility calculatedS model.

0 10

8

107

106

105

104

103

102

101

MeanSq

uaredError

Fig. 2. Variatial neural networks

l neural networks (ANNs) are nonlinear learning mathe-dels that are designed in the second half of the twentiethsimulation of human brain procedures and have beenny scientic disciplines up to now [2023]. These net-apable to relate inputs and outputs of most nonlinearble systems with any complexity. ANNs can be appliedecognition, data processing and also are considered asols for function approximation. These networks consistr of simple processing units that are connected togetherd manner according to the type of the network. Theseunits have been inspired from biological neurons, andrticial neurons. Input signals always fed to the inputen transfer to neurons in the hidden layers and outputtively. It can be said that the neurons in the output layer200 40 0 60 0 80 0 1000Epoch

Mean Squared Error vs. Epoch ThruoghTraining Step

MSEGoal

ons of mean squared error with epoch during the training of ANN.


Table 3Physical properties of the studied components.

Component Tc (K) Pc (bar) Refs. vs2 (l/mol) Refs.

Carbon dioxide (solvent) 304.2 73.7 0.225 [26] Naphthalene [Phenanthren [Anthracene [Carbazole [Fluorene [Pyrene [2,6-Dimethy [2,7-Dimethy [Hexa methy [O-Hydroxy b [P-Hydroxy b [O-Methoxy [M-Methoxy [P-Methoxy p [1-Hexadeca [1-Octadecan [Palmitic acid [Stearic acid [2,5-Xylenol [3-4-Xylenol [Mandelic ac [Benzoin [Propyl-4-hy [

provide thefrom Eq. (5)

nj = f(

Nr=1

As can be fron are weweight coethat are adular weighthrough a f(f) of the nproposed fosigmoid, hytions [24]. I(7) are utilirespectively

f (x) = exp(exp(

f (x) =1 + e

The correlatan-sigmoidtransfer fun[0 1], respecnature of afto relate inptiability of which allowweights and

4.2. Data ac

After cotemperatursolutes, tembeen utilizeof our mode

s in ption

c, Pc

menthe n

numThusouldhe natic in su

ainin

ield eighto be 748.4 40.51 0.302 e 882.65 31.715 0.437

869.15 30.8 0.353 899.1 32.65 0.496 826.4 29.5 0.406 936 25.7 0.509

l naphthalene 777 31.8 0.420 l naphthalene 777 32.2 0.420 l benzene 758 24.4 0.515 enzoic acid 739 51.8 0.832 enzoic acid 739 51.8 0.832 phenyl acetic acid 819.5 32.5 0.8

phenyl acetic acid 788.9 32.5 0.73 henyl acetic acid 788.1 32.5 0.8

nol 761 14.9 0.748 ol 777 13.4 0.863

776 14.9 1.083 779 13.4 1.084 706.9 48 0.569

729.8 49 0.576 id 903.79 34.73 0.645

853.52 26.6 0.599 droxy benzoate 815.92 31.30 0.722

results of MLPNN. The output of a neuron is computed:

wjrxr + bj

)(5)

ound from Eq. (5), the input signals to the each neu-akened or strengthen through their multiplication tofcients (wjr). The biases (bj) are activation thresholdsded to the production of inputs (xr) and their partic-t coefcients. The net output of each neuron passesunction which is called activation or transfer functioneuron. Different types of transfer functions have beenr articial neural networks such as linear, logarithmicperbolic tangent sigmoid, and radial basis transfer func-n the present study, the dened functions in Eqs. (6) andzed as the transfer functions of input and output layer,:

of solidassum

S = f (TAs

work. Tby thetively. that shfore, tschemsolids

4.3. Tr

To yand whave tx) exp(x)x) + exp(x) (6)

1xp(x) (7)

tions indicated by Eqs. (6) and (7) are usually called and log-sigmoid transfer functions, respectively. Thesections compress input data into intervals [1 1] andtively. The nonlinearity, continuity and differentiabilityorementioned functions allow the neuromorphic modelut and output data with any complexity. The differen-

these transfer functions is an important characteristic,s the gradient-based training algorithms to update the

biases.

quisition and analysis

llecting 970 data sets from the literature; the criticale (Tc) and pressure (Pc) and acentric factor () of theperature (T) and pressure (P) of the binary system haved as inputs signals, which are the independent variablesl. The available correlations for prediction of solubility

during the input layerput one, thdesired outnetworks putilized in the neurom

The varicongurati

4.4. Selectio

The optby a trial ahidden layas applyingon Cybenkoestimate alstructure ebility in supnumber of optimizatio27,28] 0.111 [29]30] 0.182 [29]18] 0.1426 [18,3134]35] 0.1515 [18]18] 0.1393 [31,36]37] 0.1585 [31]38] 0.1392 [31]29,39] 0.136 [39,40]34] 0.1527 [31,36]41] 0.0957 [42]41] 0.0924 [42]43] 0.1238 [43]43] 0.1238 [43]43] 0.1238 [43]44] 0.2965 [45]44] 0.3330 [45]44] 0.2857 [46]44] 0.3024 [45,47]26,48] 0.1257 [48,49]26,48] 0.1243 [48]31,50,51] 0.1170 [52]31,50,51] 0.1620 [52]31,50,51] 0.1316 [53]

supercritical carbon dioxide are essentially based on the that solubility can be described as follows:

, , T, P) (8)

ioned earlier, the feed-forward MLPNN is used in thisumber of nodes in the input and output layers is denedber of independent and dependent variables, respec-, the number of output neurons is equal to the variables

be detected, i.e. solubility in the present study. There-etwork output consists of a one-element vector. Theof feed-forward MLPNN to predict the solubilities ofpercritical carbon dioxide is shown in Fig. 1.

g

a proper approximation result for ANN model, the biasess associated with links between neurons of the ANNoptimized with respect to some performance measure

training stage. At rst, the input signals are fed to the

of the ANN and cross through hidden layers and out-en the difference between the network results and theputs calculate and use as a criterion for updating ofarameters (i.e. weights and biases). 627 data sets are

training step and 343 data sets are used for validationorphic model.ation of the training error for the optimal neural networkon is shown in Fig. 2.

n of optimal conguration of ANN

imal conguration of the MLP network is determinednd error procedure through changing the number ofers, number of neurons in each hidden layer as well

different training algorithms. A MLP network based [25] has only one hidden layer which is capable tomost any type of nonlinear mapping. So, the networkmployed in this research for predicting the solid solu-ercritical carbon dioxide has only one hidden layer. Theneurons in the hidden layer is determined through ann procedure which minimizes some error indexes. The


appropriate number of neurons in hidden layer depends mainly onthree issues: (1) complexity of correlation between input and out-put data, (2) the number of available training and test data, and(3) the severity of noise imposed on the data sets. A low numberof neurons error, whilethe presentbeen estimtraining datequation:

MSE = 1n

ni=

5. Results

Extensivnents have supercriticathe values factor and ssolutes.

The soluthe ANN mmixing rulecarbons in a temperatapplied. Folated.

In all casrules, the Nmodel [54].based on twbinary syste

Table 4 ent neural number of the networsuitable R2,

Since thbias coefcthe networrandomly gvalues presand R2 whicguesses.

Accordintion for theneurons (boLevenberg M

The corrand the expThe perfectis shown byt to the pebetween thdata.

Also, theuated usingnot be usedshow that t1.812% andpredicted aFig. 4. It is (with) to th

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

0.08

0.09

Predicted Solubility

0 .1

lot of experimental solubility data vs. developed ANN prediction for training.

f topology studies to nd the optimal ANN conguration.

n neuron AARD%a test (%) MSEb train R2

2.834 1.24E03 0.963382.668 8.40E04 0.975212.282 7.60E04 0.977942.104 9.00E05 0.997711.940 4.10E05 0.999541.936 3.30E05 0.998651.812 1.20E06 0.999832.193 8.00E04 0.998442.224 1.00E04 0.99811

D% =Ni

( yexp.solid,iycalc.solid,iyexp.solid,i

1n) 100. = 1n

ni=1

(yexp.s,i

ycal.s,i

).

0 0.02 0.04 0.06 0.08 0.10

01

02

03

04

05

06

07

08

09

.1

Predicted Solubilit y

0

lot of experimental solubility data vs. developed ANN prediction for testing.

le 5 shows the comparison of the ANN model results, twond six mixing rules with the experimental data based one absolute relative deviation percent. AARD% of the devel-NN model was 0.98% while the best AARD% of EOSs wasmay cause a network unable to reach to the desired a large number of neurons may result in over tting. In

study, the number of neurons in the hidden layer hasated through minimizing AARD% and MSE of test anda sets, respectively. The MSE is dened by the following

1

(yexp.s,i

ycal.s,i ) (9)

and discussion

e experimental data of the solubilities of solid compo-been published, and the solubilities of some solids inl carbon dioxide are considered here. Table 3 showsof the physical properties (critical constants, acentricolid molar volume) of the supercritical solvent and the

bilites of binary systems have been calculated byodel and also the PR and SRK EOSs by using sixs. In total, 970 experimental data for 23 heavy hydro-a pressure range between 76.8 and 1156.5 bar, andure range between 308.15 and 368.15 K have beenr all of the components, the AARD% has been calcu-

es for the analytical calculation using EOSs and mixingRTL model was chosen as the excess Gibbs free energy

The results of the solid solubilities of six mixing ruleso EOSs (PR and SRK EOSs) for 100 data sets of 23 CO2ms were demonstrated by Esmaeilzadeh et al. [14].shows the AARD%, MSE and R2 calculated for differ-network congurations, differing with respect to theneurons in their hidden layer. As mentioned earlier,k with least error measures (i.e. MSE and AARD%), and

has been chosen as the optimal network conguration.e initial guess of the network parameters (weight andients) could affect the nal values of MSE and AARD%,k has been trained several times by applying differentenerated initial values of the network parameters. Theented in Table 4 are the best values of AARD%, MSEh obtained from the network training for several initial

g to Table 4, the optimal neural network congura- estimation of solubility has one hidden layer with tenld row). The feed-forward MLPNN has been trained byarquardt algorithm [24,55].

elation between the prediction results of the ANN modelerimental training data points is illustrated in Fig. 3.

t (ANN model prediction equal to experimental data) the dashed line. The close proximity of the best linearrfect t, as observed in Fig. 3, shows a good correlatione ANN model solubility and the experimental solubility

performance of the optimal ANN architecture was eval- another data set consisting of 343 data points which

in the training stage. The results of the test data sethe AARD% and MSE for the proposed model are about

2.7 105, respectively. The correlation between thend experimental test data of solubility is indicated inseen that our predicted results are in close agreemente experimental data.

0.0

0.0

0.0

0.0

0.0

0.0

0.0

Exp

erim

entalSo

lubility

Fig. 3. Pdata set

Table 4Results o

Hidde

2 4 6 78 910 11 12

a AAR

b MSE

0.

0.

0.

0.

0.

0.

0.

0.

0.

Exp

erim

entalSo

lubility

Fig. 4. Pdata set

TabEOSs aaveragoped A


Table 5AARD% for the solubility of pure component in supercritical CO2 with developed ANN and six different mixing rules using the PR and SRK EOSs.

Component T (K) P. Range (bar) Na Refs. EOS AARD%b

VDW1 VDW2 CVD WS OS EAL ANN

Naphthalene 308.15 86.8255.3 9 [56] PR 9.20 3.4.3 2.38 2.98 2.98 2.67 1.20SRK 6.72 4.34 3.94 4.06 4.06 3.92

308.2 98.2199.5 5 [56] PR 7.95 1.16 1.51 1.14 1.13 1.21 0.48SRK 6.03 1.44 1.46 1.24 1.24 1.19

328.15 82.2287.8 16 [56] PR 23.98 14.23 13.70 6.59 6.44 6.51 3.82SRK 21.6 17.24 17.96 10.54 11.39 11.74

328.2 92.3189.6 10 [56] PR 18.98 6.78 7.65 3.26 3.26 3.23 0.48SRK 20.24 9.20 10.17 5.49 5.48 5.40

333.55 108.4291.4 19 [56] PR 37.27 20.50 41.53 3.85 3.78 4.29 0.57SRK 27.01 24.07 46.18 8.30 8.28 10.51

338.05 151.8232.2 7 [56] PR 32.99 14.18 26.24 1.05 1.02 1.10 0.41SRK 30.60 16.78 28.73 1.08 1.41 1.04

Phenanthrene 308.15 100350 7 [35] PR 8.08 4.86 6.71 4.84 4.84 4.52 0.14SRK 6.02 6.04 10.62 5.97 5.97 5.80

308.15 101181 5 [57] PR 3.71 3.30 3.30 2.97 2.98 3.13 0.06SRK 3.46 3.64 3.51 2.94 2.92 3.56

308.2 78.3203.5 47 [57] PR 20.51 13.13 19.17 13.20 13.08 12.93 0.06SRK 19.23 14.31 17.93 13.94 13.94 13.56

313.1 100200 5 [58] PR 22.27 6.39 21.08 6.41 6.42 5.56 0.52SRK 20.79 7.12 19.67 6.70 6.68 5.83

318.15 101201 5 [59] PR 15.61 15.06 15.35 13.78 13.96 13.75 0.05SRK 15.47 15.34 15.31 14.19 6.11 14.18

318.2 95254 20 [60] PR 18.19 9.46 16.10 8.90 8.76 8.73 0.05SRK 16.52 10.57 14.41 10.01 10.1 9.79

323.15 104.3414.5 6 [51] PR 14.96 8.46 12.00 6.89 6.87 6.82 4.73SRK 11.36 10.22 8.48 10.02 10.32 9.31

323.2 89.4228.5 38 [60] PR 21.38 7.23 18.95 6.15 6.11 5.81 0.05SRK 20.95 9.50 23.61 7.84 9.37 7.48

328.2 90245 23 [60] PR 22.29 8.07 19.54 7.46 5.73 5.35 0.25SRK 22.14 9.04 19.30 8.35 8.64 9.70

Anthracene 308 101.55269.53 5 [61] PR 24.79 21.58 24.89 21.63 20.50 20.43 0.03SRK 27.22 20.48 27.36 19.98 19.44 18.22

308.15 100350 6 [57] PR 36.83 36.83 13.69 13.69 13.69 13.65 0.06SRK 38.77 38.77 13.44 13.48 13.48 13.40

308.15 102181 5 [58] PR 16.73 11.87 16.71 10.94 10.98 11.05 0.06SRK 15.53 12.40 15.51 11.44 11.43 11.69

313.1 100200 7 [62] PR 16.88 11.85 16.86 10.95 10.88 6.90 2.76SRK 15.68 12.39 15.66 11.39 11.41 7.71

313.15 100200 7 [62] PR 20.65 23.21 20.64 23.19 23.15 23.15 0.13SRK 23.55 24.90 23.52 24.96 24.85 24.86

313.15 80.4726.8 13 [61] PR 10.20 11.07 10.21 11.03 11.02 10.84 3.26SRK 8.86 9.71 8.87 9.73 9.73 9.50

318 101.55269.53 6 [62] PR 31.89 24.40 31.88 24.30 24.30 24.29 0.04SRK 38.98 24.29 38.96 24.24 24.23 24.21

318.15 84.4564.4 4 [58] PR 11.56 10.44 11.63 10.45 10.45 10.44 0.02SRK 12.54 10.14 12.62 10.18 10.18 10.13

318.15 117201 5 [51] PR 22.89 19.99 23.05 19.99 20.00 18.85 0.04SRK 29.31 19.06 29.51 18.91 18.92 18.49

323.2 90.6414.5 10 [62] PR 40.31 20.10 40.30 20.15 20.10 20.12 7.01SRK 51.25 21.83 51.23 21.91 21.84 21.89

323.25 89836.3 6 [62] PR 36.88 19.62 36.82 19.63 19.62 19.64 0.03SRK 40.49 20.56 40.47 20.57 20.56 20.60

328.15 94.789.09 5 [62] PR 34.67 18.49 34.64 18.45 18.46 18.51 0.04SRK 42.81 19.58 42.78 19.49 19.57 19.62

333.15 99.8948.8 7 [62] PR 33.88 17.24 33.86 17.23 17.24 17.28 0.05SRK 41.75 17.28 41.73 18.50 18.51 18.51

338.15 106.5999.1 7 [62] PR 33.15 16.24 33.09 16.27 16.24 16.31 0.06SRK 40.86 17.87 40.88 17.86 17.91 18.00

343.15 113.71053.2 7 [51] PR 10.29 11.54 10.15 8.67 8.70 8.68 0.07SRK 11.37 14.35 11.29 12.75 12.62 11.19

343.2 118.1414.5 9 [62] PR 34.42 14.79 34.41 14.74 14.82 14.89 0.09SRK 42.59 16.43 42.56 16.54 16.46 16.19

Anthracene 348.15 118.41105.4 7 [62] PR 34.73 14.21 34.67 14.02 14.11 14.09 0.07SRK 43.51 15.40 43.45 15.57 15.39 15.49

353.15 1241156.5 7 [62] PR 29.34 13.42 29.33 13.42 13.38 13.27 0.05SRK 37.29 14.08 37.21 14.11 14.11 14.18

358.15 129930.3 6 [57] PR 36.83 36.83 13.69 13.69 13.69 13.65 0.03SRK 38.77 38.77 13.44 13.48 13.48 13.40

363.15 132.6975.7 6 [62] PR 28.46 12.36 28.42 12.37 12.37 12.41 0.04SRK 36.36 12.94 36.30 12.91 12.91 13.04

368.15 1371020.5 5 [62] PR 29.50 13.76 29.38 13.67 13.67 13.88 2.91SRK 37.11 15.74 37.06 15.65 15.66 15.95

Carbazole 308.15 103201 5 [57] PR 16.64 9.39 16.66 1.40 1.35 1.34 0.88SRK 18.27 9.68 18.65 2.25 3.89 1.48


Table 5 (Continued)



313.1 100200 7 [58] PR 6.18 6.25 6.18 3.94 3.95 5.07 0.62SRK 6.65 6.63 6.66 4.36 4.36 5.64

318.15 111201 5 [58] PR 18.27 17.43 18.28 5.15 5.53 6.05 0.72SRK 19.13 18.00 19.14 14.22 12.19 6.62

Fluorene 308.2 83.7414.5 6 [51] PR 29.94 4.51 31.22 5.66 5.44 6.10 1.99SRK 38.63 6.35 39.59 5.77 5.68 6.11

308.2 78.3203.5 47 [35] PR 13.10 10.01 13.67 10.13 10.09 9.96 4.82SRK 14.65 10.71 15.18 10.04 9.58 10.87

313.1 100200 5 [58] PR 5.70 4.97 4.97 4.93 4.93 4.35 0.30SRK 5.32 5.54 4.49 5.50 5.52 4.80

318.2 85254 21 [35] PR 17.72 6.67 20.06 6.68 6.67 6.83 0.53SRK 19.36 6.94 21.60 6.72 6.70 7.27

323.15 83.7414.5 8 [51] PR 40.23 13.75 43.76 13.61 13.61 13.08 0.25SRK 42.97 11.10 46.14 10.79 10.82 10.76

323.2 89.4228.5 38 [35] PR 14.59 7.85 17.06 7.54 7.57 7.46 0.44SRK 14.33 6.20 16.91 6.06 6.05 6.02

328.2 85245 24 [35] PR 21.84 10.45 25.25 9.96 9.98 9.92 2.91SRK 20.18 7.91 23.60 7.71 7.71 7.72

343.15 83.7483.4 8 [51] PR 59.02 18.19 68.87 16.52 16.35 13.73 0.82SRK 61.11 12.51 70.17 10.93 10.92 10.43

Pyrene 308.2 83.6483.4 7 [51] PR 39.70 16.87 39.65 13.06 13.00 13.15 0.13SRK 45.51 19.45 45.77 19.35 19.17 19.81

308.2 80.4203.5 45 [35] PR 12.38 12.71 12.46 12.12 11.86 12.81 0.18SRK 13.71 13.59 13.81 13.79 13.76 13.47

318.2 95254 20 [35] PR 9.46 10.09 9.53 10.33 10.52 10.47 0.04SRK 11.50 11.30 11.63 11.57 11.45 11.31

323.15 104.3483.4 7 [51] PR 39.23 9.37 40.24 8.97 8.52 9.31 0.08SRK 47.86 12.39 48.94 13.07 12.80 12.88

323.2 101.1228.5 35 [35] PR 4.90 6.22 4.84 6.13 6.01 5.97 0.29SRK 5.82 7.71 5.75 7.58 7.63 7.21

343.15 104.3483.4 8 [51] PR 56.80 25.02 59.27 24.95 24.93 25.04 7.32SRK 59.39 22.88 61.74 22.73 22.86 23.33

328.2 105.245 20 [35] PR 6.02 7.53 5.95 7.00 6.71 6.54 0.13SRK 7.23 9.43 7.15 9.27 9.29 8.66

2,6-Dimethylnaphthalene

308.2 79146 4 [39] PR 32.08 21.50 30.80 20.49 20.55 19.75 0.39SRK 32.09 21.60 30.98 21.39 21.40 20.70

328.2 100127 4 [39] PR 10.51 3.26 9.34 2.79 2.97 2.66 0.38SRK 13.12 2.59 11.72 4.08 4.30 2.26

2,7-Dimethylnaphthalene

308.2 88242 5 [39] PR 15.97 7.95 12.40 7.72 7.59 6.13 0.17SRK 13.33 9.15 10.52 8.78 8.36 7.13

328.2 100249 5 [39] PR 24.43 11.46 15.61 7.39 6.20 5.92 1.08SRK 23.72 15.21 15.16 11.89 12.79 10.33

Hexa methyl benzene 308.15 150350 4 [63] PR 14.57 3.93 14.02 3.84 3.84 3.69 0.06SRK 21.20 4.48 20.16 4.27 4.22 4.20

323.2 76.8345.6 9 [51] PR 14.86 15.66 15.04 15.37 15.36 13.91 0.16SRK 19.07 19.61 20.01 19.28 19.31 17.87

343.2 83.7483.5 10 [51] PR 12.20 7.68 17.11 6.80 6.78 5.90 1.27SRK 16.52 13.47 19.80 12.53 12.43 12.31

O-Hydroxy benzoicacid

318.15 81.1202.6 12 [64] PR 24.33 5.94 24.05 5.93 5.93 5.92 2.96SRK 24.15 6.96 23.88 6.81 6.81 6.80

328.15 101.3202.6 11 [64] PR 19.86 3.79 19.46 3.71 3.70 3.70 0.01SRK 19.76 5.02 19.32 4.95 4.95 4.93

O-Hydroxy benzoicacid

328.15 101.3202.6 6 [41] PR 24.93 1.68 24.52 1.64 1.64 1.66 0.29SRK 24.89 3.38 24.45 3.26 3.27 3.17

P-Hydroxy benzoic acid 318.15 101.3202.6 6 [41] PR 20.49 5.53 20.49 5.53 5.53 5.52 0.19SRK 19.64 5.94 19.64 5.93 5.93 5.93

328.15 101.3202.6 6 [41] PR 16.36 4.62 16.36 4.11 4.10 4.62 0.08SRK 16.44 3.32 16.44 3.32 3.32 3.31

O-Methoxy phenylacetic acid

308.2 122337 5 [43] PR 40.11 4.92 40.41 5.53 4.82 4.50 0.06SRK 46.11 4.82 46.39 5.93 4.79 4.50

M-Methoxy phenylacetic acid

308.2 117339 5 [43] PR 20.32 3.60 25.26 4.11 3.89 2.26 0.21SRK 25.93 4.07 30.74 3.32 4.33 4.84

P-Methoxy phenylacetic acid

308.2 121339 5 [43] PR 64.18 4.21 57.65 4.83 4.70 3.88 2.32SRK 63.13 4.43 64.46 4.63 4.62 4.21

1-Hexadecanol 308 92203.7 7 [65] PR 13.04 9.13 16.20 9.09 9.09 5.87 0.83SRK 15.73 10.16 19.25 10.13 10.05 5.84

318 152.1415.1 7 [66] PR 42.22 3.40 44.29 4.19 4.12 1.77 0.45SRK 51.71 7.20 50.82 7.14 6.91 1.42

318 85.5201.3 6 [65] PR 24.54 24.51 27.02 24.24 24.36 17.78 0.14SRK 25.78 25.62 29.53 25.39 25.45 16.72

323 85.52002.6 6 [65] PR 25.26 26.94 31.24 24.79 23.54 17.02 6.23SRK 27.27 28.17 34.02 27.17 25.65 17.76

328 86.4203 6 [65] PR 29.65 32.44 31.52 29.67 27.56 19.31 0.34SRK 32.30 33.56 33.84 31.30 30.98 16.01


Table 5 (Continued)



6 7 2 2

1-Octadecan 2 1 2 0 4 7

1-Octadecan 2 1

Palmitic acid

Stearic acid

2,5-Xylenol

3-4-Xylenol

Mandelic ac

Benzoin

Benzoin

Propyl-4-hybenzoate

a N is the nu

b AARD% =

9.41% whichthat the neis the best supercritica

It is obsecritical CO2superior. ThSRK EOSs:

CVD > VdW328 141.8415.9 5 [66] PR 80.3SRK 87.6

338 147.1373 6 [66] PR 80.3SRK 84.3

ol 308 86.2199.6 6 [65] PR 33.0SRK 35.0

318 86.2200.3 6 [65] PR 26.2SRK 27.7

318 152437.9 4 [67] PR 9.4SRK 25.3

ol 328 139.9447.7 7 [65] PR 94.0SRK 95.1328 87.6199.3 6 [65] PR 28.66 SRK 30.12

338 84.5204.4 6 [65] PR 32.09 SRK 34.26

338 145.8452.8 6 [67] PR 107.93 SRK 109.41

318 142.1360.6 5 [66] PR 46.57 SRK 55.13

328 144.1573.5 7 [66] PR 147.37 SRK 104.81

338 142.5574.8 7 [66] PR 249.07 SRK 96.46

318 145.4361.5 6 [67] PR 12.56 SRK 9.37

328 154.8467.5 6 [67] PR 79.08 SRK 99.79

338 161.5463.8 5 [67] PR 81.91 SRK 93.41

308.15 87267 7 [49] PR 30.06 SRK 28.72

308.15 82262 7 [48] PR 26.00 SRK 25.04

id 308.15 101228.5 7 [68] PR 42.56 SRK 45.48

318.15 102.3225.7 7 [68] PR 36.34 SRK 26.22

328.15 104.4230.6 7 [68] PR 29.43 SRK 31.64

308.15 121.6236.1 6 [68] PR 2.83 SRK 4.47

318.15 111.3244.3 7 [68] PR 3.11 SRK 4.34

328.15 114.8244.3 6 [68] PR 11.06 SRK 9.42

droxy 308.15 94.1220.9 7 [68] PR 16.13 SRK 18.46

318.15 96.8214.7 7 [68] PR 14.72 SRK 16.35

328.15 105.1220.2 7 [68] PR 19.83 SRK 21.51

308.15368.1576.81156.5 970 PR 30.61 SRK 31.12

mber of data points.Ni

( yexp.solid,iycalc.solid,iyexp.solid,i

1n) 100.

obtained by PR EOS and EAL mixing rule. It is observedw ANN model has the minimum prediction error andmodel for the prediction of the solubility of solids inl CO2.rved that, for prediction of the solid solubility in super-, using the six mixing rules, the EAL mixing rule ise following conclusion can be obtained for the PR and

1 > VdW2 > WS > OS > EAL

In additiof solubilitythan that w

6. Conclus

Accuracsolubility oide. The ANSRK EOSs by12.04 98.32 10.02 10.07 11.68 0.2933.54 106.79 13.69 13.08 10.652.34 108.27 2.34 2.21 1.69 0.367.56 114.38 7.47 6.33 2.28

25.69 32.77 26.80 27.10 12.73 0.4426.56 34.37 26.61 27.63 13.6925.27 25.95 24.74 24.34 20.93 6.8326.22 26.83 26.10 25.50 20.513.71 9.04 3.56 3.56 0.57 0.894.36 20.95 4.07 4.01 0.59

12.09 111.85 8.51 9.04 5.90 0.2619.86 120.81 17.24 16.89 6.42

29.18 27.23 27.63 26.38 16.31 0.5731.03 29.38 29.70 28.67 16.6432.60 32.68 31.35 29.60 25.25 0.3534.60 35.08 33.13 32.25 26.068.16 162.34 2.73 2.18 3.68 0.07

26.16 170.96 18.46 15.05 2.116.56 52.42 2.50 2.41 0.79 0.24

11.61 64.14 9.47 6.52 0.669.67 192.22 20.23 18.81 12.38 9.85

34.57 216.60 31.92 32.68 21.8241.57 440.83 50.56 48.87 26.64 0.56

105.60 463.95 64.72 52.00 31.897.29 11.34 6.89 6.85 3.26 0.519.02 7.67 8.42 8.49 4.08

16.46 106.58 16.55 16.46 16.83 1.3213.09 127.41 10.64 10.51 10.3513.65 115.82 9.29 9.54 5.54 0.0319.60 130.73 9.40 9.83 5.283.17 24.43 3.57 3.53 2.74 0.044.39 22.95 4.66 4.65 3.896.15 21.86 6.04 6.04 5.62 0.077.04 20.43 7.09 7.04 6.62

11.49 45.73 7.00 7.03 5.58 0.1512.14 48.85 8.19 8.07 5.755.49 40.86 4.70 4.72 4.41 0.135.48 44.32 4.66 4.65 4.10

10.71 36.46 8.57 8.34 8.36 0.9111.51 38.09 9.37 9.46 8.182.62 3.01 2.58 2.59 2.26 0.152.67 4.63 2.60 3.00 2.213.25 3.23 3.15 3.18 3.06 0.143.38 4.66 3.31 3.31 3.245.75 10.59 5.70 5.71 5.53 0.536.81 8.90 6.73 6.74 6.562.07 16.51 2.08 2.08 2.03 0.412.31 18.86 2.38 2.39 2.368.49 15.00 8.81 8.73 4.85 0.668.88 16.65 9.07 9.00 5.02

18.91 20.06 19.98 20.12 8.43 2.3119.72 21.76 20.76 20.86 10.5811.90 34.70 10.90 10.75 9.41 0.8314.21 38.18 12.45 12.22 10.23

on, it was found that for any mixing rule, the prediction of solids in supercritical CO2 with the PR EOS is betterith the SRK EOS.

ions

y of ANN and two EOSs were compared for prediction off 23 different compounds in supercritical carbon diox-N model was developed and compared with the PR and

using VDW1, VDW2, CVD, WS, OS, and EAL mixing rules


based on the 970 experimental data points from the literature. Theoptimal neural network conguration for the estimation of solubil-ity has one hidden layer with ten neurons. The feed-forward MLPNNhas been trained by Levenberg Marquardt algorithm. The AARD% ofoverall data set is 0.98% for the ANN method while the best EOSsAARD% is 9.41% which obtained by PR EOS and EAL mixing rule.In other words, the results of the ANN model are very closer tothe experimental data points than the EOSs prediction. Our resultsshow that the new ANN model is a reliable and useful model forthe modeling of solubility of solids in supercritical uids in highlynon-ideal systems, and is a useful tool for the analysis and designof supercritical technology.

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Comparison the capability of artificial neural network (ANN) and EOS for prediction of solid solubilities in supercritical...1 Introduction2 Equation of state and mixing rules3 Solubility of solid solute in supercritical fluid by using EOS4 Methods4.1 Artificial neural networks4.2 Data acquisition and analysis4.3 Training4.4 Selection of optimal configuration of ANN

5 Results and discussion6 ConclusionsReferences

comparison the capability of artificial neural network (ann) and eos for

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