estimation of vapour liquid equilibria of binary systems, carbon dioxide–ethyl caproate, ethyl...

Fluid Phase Equilibria 235 (2005) 92–98

Estimation of vapour liquid equilibria of binary systems, carbondioxide–ethyl caproate, ethyl caprylate and ethyl caprate

using artificial neural networks

Swati Mohanty∗

Regional Research Laboratory, Bhubaneswar 751013, India

Received 9 October 2004; received in revised form 27 June 2005; accepted 2 July 2005

Abstract

Vapour liquid equilibrium (VLE) data are important for designing and modeling of process equipments. Since it is not always possible tocarry out experiments at all possible temperatures and pressures, generally thermodynamic models based on equations of state are used forestimation of VLE. In this paper, an alternate tool, i.e. the artificial neural network technique has been applied for estimation of VLE for threeb ction. Thet lute deviationf ole fraction.T©

K

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mmidbpFtliuiac

epen-porteder-sure,a-tivechutirour

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inary systems viz. carbon dioxide–ethyl caproate, ethyl caprylate and ethyl caprate which are of importance in supercritical extraemperature range in which these models are valid is 308.2–328.2 K and the pressure range is 1.6–9.2 MPa. The average absoor all the three systems in the estimation of liquid phase mole fraction was 3% or less and less than 0.02% for the vapour phase mhe error was less compared to that estimated by SRK or Peng Robinsons equation of state.2005 Elsevier B.V. All rights reserved.

eywords:Vapour liquid equilibria; Artificial neural networks; Carbon dioxide; Esters

. Introduction

Vapour liquid equilibrium data are required for design,odelling and control of process equipments. Conventionalethod of estimating the vapour liquid equilibrium (VLE)

s based on equations of state (EoS). These EoS althougherived from strong theoretical principles also involve a num-er of adjustable parameters in terms of binary interactionarameters, as well as parameters in mixing rule equations.urthermore, the binary interaction parameters that are func-

ions of both temperature and composition need to be calcu-ated at every temperature at which the VLE is required. Theterative method of estimation of VLE using EoS makes itnsuitable for real time control. The development of numer-

cal tools, such as neural networks, has paved the way forlternative methods to estimate the VLE[1–6]. It has attractedonsiderable interest because of its ability to capture with

∗ Tel.: +91 2581635/251638; fax: +91 2581637.E-mail address:swati [email protected].

relative ease the non-linear relationship between the inddent and dependent variables. Several authors have reapplication of ANN for estimation of thermodynamic propties such as estimation of viscosity, density, vapour prescompressibility factor and VLE. A ANN model for estimtion of vapour pressure from aerosol composition, relahumidity and temperature has been reported by Potuand Wexler[7]. Chouai et al.[8] have used a ANN model foestimating the compressibility factor for the liquid and vapphase as a function of temperature and pressure for srefrigerants. ANN has also been used for estimating the sfactors as a function of temperature and density for a nuof refrigerants that can be used in the extended correspostate model[9,10]. Lagier and Richon[11] have used ANNmodel for estimation of compressibility factor and denas a function of pressure and temperature for some rerants. Although a number of papers have been publwith experimental data for vapour liquid equilibriumvarious systems and estimation of VLE using conventithermodynamic models, not many have used this techn

378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2005.07.003

S. Mohanty / Fluid Phase Equilibria 235 (2005) 92–98 93

for estimating the VLE. A ANN based group contributionmethod for estimation of liquid phase activity co-efficienthave been suggested by Petersen et al.[1] that can be usedfor estimation of VLE. A multilayer perceptron with a singlehidden layer has been used by Guimaraes and McGreavy[2]for estimating the VLE of benzene–hexane system. Sharmaet al.[3] have used the multi-layer perceptron model to esti-mate the VLE for the methane–ethane and ammonia–watersystems. They have also highlighted the advantage of ANNover conventional EoS for estimating the VLE systems con-taining polar compounds. Ganguly[4] on the other hand, hasused the radial basis function to estimate the VLE for severalbinary and ternary systems. Urata et al.[5] have estimated theVLE using two multi-layer perceptrons. The input parame-ters for the first ANN are normal boiling point divided bymolecular weight, density and dipole moment for both thecomponents and the output is a negative or positive sign. Thesecond ANN has an extra input of mole fraction of one ofthe components in the liquid phase in addition to the inputsof the first ANN. The output from the second ANN is log-arithm of the activity coefficient for that component. Usingthe logarithmic activity coefficients, vapour liquid composi-tion and equilibrium temperature were estimated. Mohanty[6] has used a single multilayer perceptron for estimating theVLE of carbon dioxide–difluoromethane system.

In this paper, attempt has been made to use artificial neu-r bond andc ethylc duc-t andf se ofi iox-i andp peri-m dataa s3 from1 lsh an beu surer

2. Artificial neural network theory

The driving force behind the development of the ANNmodels is the biological neural network, a complex struc-ture, which is the information processing system for a livingbeing. Thus ANN mimics a human brain for solving com-plex problems, which may be otherwise difficult to solveusing available mathematical techniques. The advantage ofusing a ANN model is that it does not require any other dataexcept the input and output data. Once the model has beenadequately trained, the input data is sufficient to estimate theoutput. The other advantage is a single model can be used toget multiple outputs. From its initiation in the early fortiestill today there are hundreds of ANN architecture developed,however, there are a few such as multi-layer perceptron andradial basis function that are more popular and find wideapplications. Details have been dealt with elsewhere[13,14],therefore only a brief description of multilayer perceptronneural network that belongs to the feed forward neural net-work architecture in general has been described.

2.1. The multi-layer perceptron (MLP) network

This type of network is composed of an input layer, an out-put layer and one or more hidden layers (Fig. 1). Bias termi oly-n tputl out-p ddenl arei th oft soci-a ayeri thei

z

ans-f en

TD etwork

S a)in li

C00.

C00

C00

al networks for estimating the VLE for the systems carioxide–ethyl caproate, carbon dioxide–ethyl caprylatearbon dioxide–ethyl caprate. While ethyl caproate andaprate find application in organic synthesis or the proion of essential oil, ethyl caprylate is used in cosmeticsood industry. These systems are of importance becauncreasing interest in supercritical extraction. Carbon dde being non-toxic and having low critical temperatureressure is an ideal choice as a supercritical fluid. Exental VLE data for this system is scarce and the only

vailable is that reported by Hue et al.[12] at temperature08.2, 318.2 and 328.2 K and over the pressure range.6 to 9.2 MPa. (Table 1). The weights for the ANN modeave been determined for all the three systems that csed for estimating the VLE in the temperature and presanges listed inTable 1.

able 1ata source and range used for development of the artificial neural n

ystem Temperature (K) Pressure (MP

arbon dioxide–ethylcaproate 308.2 1.699–6.462318.2 1.699–7.823328.2 1.733–9.218

arbon dioxide–ethylcaprylate 308.2 1.75–7.177318.2 1.699–7.823328.2 1.699–9.218

arbon dioxide–ethylcaprate 308.2 1.665–7.109318.2 1.699–7.891328.2 1.699–9.218

n each layer is analogous to the constant term of any pomial. The number of neurons in the input and the ou

ayer depends on the respective number of input andut parameters taken into consideration. However, the hi

ayer may contain zero or more neurons. All the layersnterconnected as shown in the figure and the strenghese interconnections is determined by the weights asted with them. The output from a neuron in the hidden l

s the transformation of the weighted sum of output fromnput layers and is given as

j = g

(d∑

i=0

wjipi

)(1)

The output from the neuron in the output layer is the trormation of the weighted sum of output from the hidd

model

Mole fraction CO2

quid phaseMole fraction CO2

in vapour phaseNo. of datapoints

Reference

0.2823–0.843 0.9994–0.999 8 [12].2301–0.8541 0.9992–0.9976 10209–0.8463 0.9987–0.9963 12

0.2786–0.8904 0.9997–0.9994 9 [12].2407–0.8063 0.9997–0.9992 10.2088–0. 8156 0.9996–0.9985 12

0.258–0.8454 0.9998 9 [12].2297–0.7775 0.9998–0.9996 11.2068–0.778 0.9998–0.9992 10

94 S. Mohanty / Fluid Phase Equilibria 235 (2005) 92–98

Fig. 1. Multilayer perceptron with one hidden layer.

layer and is given as

qk = g̃

n∑

j=0

w̃jizi

(2)

wherepi is the ith output from the input layer,zj is the jthoutput from the hidden layerwij is the weight in the firstlayer connecting neuroni in the input layer to neuronj in thehidden layer,̃wkj is the weight in the second layer connectingneuronj in the hidden layer to the neuronk in the output layerandg and g̃ are the transformation functions. The transfor-mation function is usually a sigmoidal function with the mostcommon being,

g(a) = tanha = ea − e−a

ea + e−a(3)

The other commonly used function is,

g(a) = 1

1 + e−a(4)

One of the reasons for using these transformation func-tions is the ease of evaluating the derivatives that is requiredfor minimization of the error function.

3. Neural network model

emsv and

ethyl caprate is based on the experimental data reported byHwu et al.[12] in the temperature range 308.2–328.2 K andpressure in the range 1.6–9.2 MPa. The summary of the datais shown inTable 1. All neural networks take numeric inputand produce numeric output. The transformation function ofa neuron is typically chosen so that it can accept input inany range, and produce output in a strictly limited range.Although the input can be in any range, there is a saturationeffect so that the unit is only sensitive to inputs within a fairlylimited range. Numeric values have to be scaled into a rangethat is appropriate for the network. Typically, raw variablevalues are scaled linearly. In the present study a linear scalingof the data is done so that each variable has a zero mean andunit standard deviation and is given by Eq.(5).

scaled value= actual value× m + c (5)

wherem andc for the three systems are given inTable 2.The two input parameters to the multi-layer perceptron arethe pressure and temperature and the two output parametersare the mole fraction carbon dioxide in the vapour and liquidphases. For all the three systems, the VLE data at temper-atures 308.2 and 328.2 K and their corresponding pressureranges given inTable 1were used for training the networkas these were the only data available in literature. In thisstudy the transformation function given in Eq.(3), whichw df earf usedf ur-i ntala andv Thed htsa ue, inw ardsti theg ed int t byv odelw thatg layerw sys-t ns int pry-l was

TC on dio

S aprate

1234 −

The neural network models for the three binary systiz. carbon dioxide with ethyl caproate, ethyl caprylate,

able 2onstants in Eq.(5) for the system carbon dioxide–ethyl caproate, carb

l no. Variable Carbon dioxide–ethyl caproate

c m

Temperature −15.410 0.0500Pressure −0.226 0.1330x1 −0.327 1.5649y1 −321.39 322.58

as found to give better results compared to Eq.(4), was useor the hidden layer and for the output layer it was a linunction. The standard back propagation algorithm wasor training the network and optimizing the weights. Dng the training period, the error between the experimend estimated mole fraction carbon dioxide in the liquidapour phases is minimised by optimising the weights.erivatives of the error function with respect to the weigre estimated using the error back propagation techniqhich the error in the output layer is propagated backw

o estimate the derivatives in the lower layer[13]. The min-mization of the error function is then carried out usingradient descent method in which the weights are mov

he direction of negative gradient. Training is carried ouarying the number of neurons in the hidden layer. The mith the minimum number of neurons in the hidden layerives the desired accuracy is selected. A single hiddenas found to be sufficient for all the three cases. For the

em carbon dioxide–ethyl caproate, the number of neurohe hidden layer was eight, for carbon dioxide–ethyl caate it was five and for carbon dioxide–ethyl caprate it

xide–ethyl caprylate and carbon dioxide–ethyl caprate

Carbon dioxide–ethyl caprylate Carbon dioxide–ethyl c

c m c m

−15.410 0.0500 −15.41 0.050−0.226 0.1330 −0.22 0.132−0.306 1.4671 −0.32 1.566832.083 833.33 −1665.33 1666.67


six. The model was validated with the data set at 318.2 K forall the three systems.

4. Results and discussion

Although a large data set increases the validity of the ANNmodel and improves its performance, in the present study dueto paucity of data, a small data set was taken and the estima-tion of VLE by the model for the validation data set was quitesatisfactory. The preprocessing factors for the system carbondioxide–ethyl caproate, carbon dioxide–ethyl caprylate andcarbon dioxide–ethyl caprate are given inTable 2. The opti-mized weights for the different layers for the system carbondioxide–ethyl caproate, carbon dioxide–ethyl caprylate andcarbon dioxide–ethyl caprate are given inTables 3–5, respec-tively. The part before the decimal point represents the layernumber and the part after the decimal point is the neuronnumber. The neuron number in the horizontal row representsthe neuron to which the neuron number in the vertical columnis connected. From this table the weight for the connectionbetween any two neurons can be obtained. Using Eqs.(1)–(3)one can estimate the mole fraction carbon dioxide in the liq-uid and vapour phases. Prior to using the equations the inputto the model should be scaled using Eq.(5). Similarly, theo t thea tedm thet 8.2 Ka

Fig. 2. Comparison of experimental and estimated mole fraction car-bon dioxide in the liquid phase for the training data set for the sys-tems: (a) CO2–ethyl caproate at temperature 308.2 K in the pressurerange 1.699–6.462 MPa, and temperature 328.2 K in the pressure range1.733–9.218 MPa; (b) CO2–ethyl caprylate at temperature 308.2 K in thepressure range 1.75–7.177 MPa, and temperature 328.2 K in the pressurerange 1.699–9.218 MPa; and (c) CO2–ethyl caprate at temperature 308.2 Kin the pressure range 1.665–7.109 MPa, and temperature 328.2 K in the pres-sure range 1.699–9.218 MPa.

Fig. 3 shows the experimental and estimated mole fractioncarbon dioxide in the vapour phase for all the three systemsfor the training data set at the same temperatures and pres-sures.Fig. 4shows the experimental and the estimated molefraction carbon dioxide in the liquid phase for all the threesystems for the validation data set at temperature 318.2 Kand the corresponding pressure range as given inTable 1,andFig. 5 shows the experimental and the estimated mole

TW e (1)–ethyl caproate (2)

2.6 2.7 2.8 3.1 3.2

B 59344 0.194096 1.01801 0.513011 0.277473 0.1204861 60248 0.031769 0.16592 0.7498811 060642 0.953680 −0.96620 0.0359452 0.478530 0.4388142 −0.332397 −0.7420222 0.426127 0.2181772 −0.646615 −0.1432202 −0.171724 0.3604722 0.660803 0.1603542 −0.403605 0.7632042 −0.540601 −0.380630

TW n dioxide (1)–ethyl caprylate (2)

L 2.4 2.5 3.1 3.2

B 294761 80541 762222 222 72 6

utput obtained from the model is to be de-scaled to gectual values.Fig. 2shows the experimental and the estimaole fraction carbon dioxide in the liquid phase for all

hree systems for the training data set i.e. at 308.2 and 32nd in the corresponding pressure ranges as given inTable 1.

able 3eights for the hidden and output layers for the system carbon dioxid

2.1 2.2 2.3 2.4 2.5

ias 0.190831 −1.502339 −1.174430 −0.26897 −0.4.1 0.196031 0.549792 0.029331 −0.00215 −0.1.2 0.486165 1.042663 0.560635 −1.00166 0..1.2.3.4.5.6.7.8

able 4eights for the hidden and output layers for the system carbon carbo

ayer/neuron number 2.1 2.2 2.3

ias 0.366536 0.32296 0.2.1 0.015884 −1.12342 0.12.2 −0.820210 −0.13723 0.33.1.2.3.4.5

3.21393 −0.072951 −0.190196 −0.8208030.26868 0.642640

−2.78710 0.420363−0.855552 −0.283201

0.637935 0.145521.006784 −0.4978460.289834 2.003890.566427 0.18942


Table 5Weights for the hidden and output layers for the system carbon carbon dioxide (1)–ethyl caprate (2)

2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2

Bias 0.057529 −1.036878 0.162198 −0.612877 1.23953 0.58494 0.900210 0.163501.1 0.476137 −0.107133 0.525031 0.323086 −0.32440 −0.390251.2 0.160051 −0.116891 −0.878496 0.519768 −0.97963 −1.863352.1 −0.217015 −0.434962.2 0.566084 0.262382.3 −0.457225 0.694122.4 −0.090759 −0.978722.5 −0.014356 1.041332.6 −0.555817 −1.00271

Fig. 3. Comparison of experimental and estimated mole fraction car-bon dioxide in the vapour phase for the training data set for the sys-tems: (a) CO2–ethyl caproate at temperature 308.2 K in the pressurerange 1.699–6.462 MPa, and temperature 328.2 K in the pressure range1.733–9.218 MPa; (b) CO2–ethyl caprylate at temperature 308.2 K in thepressure range 1.75–7.177 MPa, and temperature 328.2 K in the pressurerange 1.699–9.218 MPa; and (c) CO2–ethyl caprate at temperature 308.2 Kin the pressure range 1.665–7.109 MPa, and temperature 328.2 K in the pres-sure range 1.699–9.218 MPa.

Fig. 4. Comparison of experimental and estimated mole fraction carbondioxide in the liquid phase for the validation data set for the systems:(a) CO2–ethyl caproate at temperature 318.2 K in the pressure range1.699–7.823 MPa; (b) CO2–ethyl caprylate at temperature 318.2 K in thepressure range 1.699–7.823 MPa; and (c) CO2–ethyl caprate at temperature318.2 K in the pressure range 1.699–7.891 MPa.

Fig. 5. Comparison of experimental and estimated mole fraction carbondioxide in the vapour phase for the validation data set for the systems:(a) CO2–ethyl caproate at temperature 318.2 K in the pressure range1.699–7.823 MPa; (b) CO2–ethyl caprylate at temperature 318.2 K in thepressure range 1.699–7.823 MPa; and (c) CO2–ethyl caprate at temperature318.2 K in the pressure range 1.699–7.891 MPa.

fraction carbon dioxide in the vapour phase for all the threesystems for the validation data set at sane temperatures andpressures. TheR2 value for the mole fraction carbon diox-ide in the liquid phase for the system carbon dioxide–ethylcaproate is 0.9981 both for the training and validation datasets showing that the model has captured the features quiteaccurately. In the case of mole fraction carbon dioxide inthe vapour phase theR2 is 0.8723 and 0.8348 for the train-ing and validation data set, respectively. Since the spread ofthe vapour phase composition is very small, even a smalldeviation lowers theR2 value considerably. Similarly, for thesystem carbon dioxide–ethyl caprylate, theR2 value for themole fraction carbon dioxide in the liquid phase is 0.9883and 0.997 for the training and validation data set, respec-tively, whereas, for the mole fraction carbon dioxide in thevapour phase, theR2 is 0.7797 and 0.8103 for the trainingand validation data set, respectively. The reasons for this lowvalue ofR2 has been explained earlier. For the system car-bon dioxide–ethyl caprate theR2 value for the mole fractioncarbon dioxide in the liquid phase is 0.9961 and 0.9951 thetraining and validation data set, respectively. TheR2 valuefor the mole fraction carbon dioxide in the vapour phase is0.8916 and 0.8823 for the training and validation data set,respectively. The absolute average percent deviation taking


Table 6Predicted VLE for the sytem carbon dioxide- ethyl caproate, ethyl caprylate and ethyl caprate

Temperature (K) Pressure(MPa)

CO2–ethyl caproate CO2–ethyl caprylate CO2–ethyl caprate

CO2 in theliquid phase

CO2 in thevapour phase





320 1.8 0.2309 0.9991 0.2519 0.9996 0.2270 0.9998320 2 0.2562 0.9991 0.2711 0.9996 0.2472 0.9998320 2.5 0.3184 0.9991 0.3197 0.9996 0.2998 0.9998320 3 0.3785 0.9991 0.3687 0.9996 0.3545 0.9998320 3.5 0.4364 0.9990 0.4178 0.9996 0.4101 0.9998320 4 0.4918 0.9990 0.4670 0.9996 0.4651 0.9998320 4.5 0.5448 0.9989 0.5158 0.9996 0.5183 0.9998320 5 0.5950 0.9988 0.5639 0.9996 0.5686 0.9998320 5.5 0.6427 0.9987 0.6109 0.9996 0.6153 0.9998320 6 0.6877 0.9985 0.6564 0.9996 0.6581 0.9998320 6.5 0.7302 0.9984 0.6997 0.9995 0.6966 0.9997320 7 0.7702 0.9982 0.7401 0.9995 0.7311 0.9997320 7.5 0.8079 0.9980 0.7768 0.9994 0.7618 0.9996320 8 0.8434 0.9978 0.8089 0.9992 0.7890 0.9996

into consideration both the training and validation data set is1.65 for estimation of liquid phase mole fraction and 0.0168for the vapour phase for the system carbon dioxide–ethylcaproate, 3.0 and 0.005, respectively for the system carbondioxide–ethyl caprylate, and 2.69 and 0.0027, respectively forthe system carbon dioxide–ethyl caprate. The percent devia-tion estimated by SRK and Peng Robinson equation of statefor the vapour phase mole fraction as reported by Hwu et al.[12] for the system carbon dioxide–ethyl caproate are 0.04and 0.03, respectively, for the system carbon dioxide–ethylcaprylate it is 0.02 by both the methods and for the system car-bon dioxide–ethyl caprate it is 0.02 by both the methods. In allthe cases it can be seen that the neural network models havebeen able to estimate the composition more accurately thanthe conventional methods. The models were also used to pre-dict the VLE at temperature 320 K and pressures between 1.8and 8.0 MPa for all the three systems (Table 6). The weightsthat were optimized during the training period were used inthese models for predicting the VLE. In spite of limited data,it is seen that the models have been able to extract the fea-tures of the system quite accurately. The reason could bethat the system is not highly non-linear in the range consid-ered and the data are evenly spread over the entire range.While considering a wider range of temperatures it is nec-essary to have more data in that range. Close data sets arerequired in the region where it is highly non-linear. Whilei od-e reado datas nsid-e lt ina atisti-c izet odeld oree , the

model could be improved to be applicable for a much widerrange.

5. Conclusions

In this work, artificial neural network models have beendeveloped for the three binary systems, carbon dioxide–ethylcaproate, ethyl caprylate and ethyl caprate, to estimatethe vapour liquid equilibria in the temperature range,308.2–328.2 K and the pressure range, 1.6–9.2 MPa. Theweights have been optimized so as to minimize the errorbetween the estimated and experimental VLE. The weightsfor the models have been tabulated for all the three systems,that can be used for predicting the VLE at any tempera-ture and pressure in the range listed inTable 1. The modelswere able to estimate the vapour liquid equlibria satisfacto-rily. The percent deviation in estimating the vapour phasemole fraction was found to be lower when using ANN modelthan using Soave–Redlich–Kwong or Peng Robinson’s equa-tion of state. The weights thus optimized during the trainingperiod can be used in ANN models for predicting the VLEof the three systems at any temperature and pressure in therange considered in this paper. Development of ANN modelfor estimating VLE is less cumbersome than methods basedon EoS. It does not require parameters such as the criti-c tionp ionalm arlyr linearr odeli con-s ighlys realt ox,i ich

t is believed that a huge data set is required for ANN mls, it is basically depends on how well the data is spver the range and the quality of the data. Having largeets concentrated at two extreme ends of the range cored and not in the intermediate region may not resuvery good model. Researchers have exploited the st

al methods such as the ‘design of experiment’, to minimhe number of data sets required for neural network mevelopment for process parameter optimization. If mxperimental data are available for the present system

al properties of the components or the binary interacarameters, nor the mixing rules as required by conventethods. Binary interaction parameters may not be line

elated to the temperature and hence assumption ofelation may lead to erroneous results. Once the ANN ms trained estimation of VLE is a one step process. Thisiderably saves computational time. Hence, it may be huitable to use in place of conventional methods forime process control. Since ANN works like a black bt can be applied to any type of binary mixture for wh


the VLE data is available irrespective of the type of the sys-tem. However, the major disadvantage of this technique isthat it can be used only in the range in which it has beentrained, as it is empirical in nature. The model is as goodas the quality of data used for the training of the model. Asno model is the best model, the model can be improved ifmore data are available in a wider range of temperature andpressure.

List of symbolsa weighted sum of input to the neurons for different

layersc constant term for scaling input or outputd number of neurons in the input layerg, g̃ transformation functionm multiplication factor for scaling the input or outputn number of neurons in the hidden layerp output from the input layerq output from the output layerr number of neurons in the output layerw weight for the hidden layerw̃ weight for the output layerx liquid phase mole fractiony vapour phase mole fractionz output from the hidden layer

Si1

Acknowledgement

The author acknowledges the Director, Regional ResearchLaboratory, Bhubaneswar, for permission to publish thispaper.

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estimation of vapour liquid equilibria of binary systems, carbon dioxide–ethyl caproate, ethyl...

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