Ind. Eng. Chem. Res. 2009, 48, 2129–2134
2129
Estimation of Liquid-Liquid Equilibrium for a Quaternary System Using the GMDH Algorithm Shahbaz Zahr Reyhani,† Hossein Ghanadzadeh,*,† Luis Puigjaner,‡ and Francis Recances‡ Faculty of Engineering, UniVersity of Guilan, P.O. Box 41635-3756, Rasht, Iran, and Chemical Engineering Department, UniVersitat Polite`cnica de Catalunya, ETSEIB, Diagonal 647, Barcelona E-08028, Spain
Liquid-liquid equilibrium (LLE) data are important for designing and modeling of process equipment. In this article, the group method of data handling (GMDH) technique has been used to estimate LLE data of the quaternary system of corn oil + oleic acid + ethanol + water at 298.15 K. Using this method, a new model has been proposed that is suitable for use in predicting LLE data. The equilibrium data presented herein were predicted using the GMDH model. To evaluate the performance of the GMDH method, we compared the data predicted using the proposed model with experimental data. Moreover, we compared the deviation between the experimental data and the data estimated with the GMDH. We also used the deviations of some existing models for the comparison. The results show that the proposed model can be used as an efficient tool for estimating LLE data. 1. Introduction The importance of the availability of precise liquid-liquid equilibrium (LLE) data in the rational design of many chemical processes and separation operations has been the subject of much research in recent years. A large number of investigations have been carried out on LLE measurements, in order to understand and provide further information about the phase behavior of such systems. Usually, the presented equilibrium data are correlated using thermodynamic methods. These thermodynamic models have been successfully used for the correlation of several LLE systems.1 However, conventional thermodynamic methods for data prediction of complex systems are tedious and involve a certain amount of empiricism.2,3 Recently, new prediction methods were developed using artificial neural networks (ANNs) in order to avoid these limitations. ANNs are nonlinear and highly flexible models that have been successfully used in many fields to model complex nonlinear relationships. Hence, they offer potential to overcome the limitations of the traditional thermodynamic models and polynomial correlation methods for complicated systems, especially in estimating LLE and vapor-liquid equilibrium (VLE).3-8 ANNs can be viewed as universal function approximators, which means that, given enough data, they can approximate the underlying function with accuracy.9 Some ANNs have their origins in techniques for performing the exact interpolation of a set of data points in a multidimensional space.10 This type of mapping provides an interpolating function that passes exactly through every data point. However, the main disadvantage of such approaches is that the detected dependencies are hidden within the neural network structure.11 Moreover, the inherent complexity in the design of traditional neural networks in terms of understanding the most appropriate topology and coefficients has a great impact on their performance.12 Conversely, the group method of data handling (GMDH)13 is aimed at identifying the functional structure of a model hidden in the empirical data. The main idea of the GMDH is the use of feed-forward networks based on short-term polynomial transfer functions whose coefficients are obtained using regression combined with emulation of the self-organizing activity behind neural network (NN) structural learning.14 The GMDH was developed for complex systems for the modeling, prediction, * To whom correspondence should be addressed. E-mail: hggilani@ guilan.ac.ir. Tel.: +(98)-131-6690270. Fax: +(98)-131-6690271. † University of Guilan. ‡ Universitat Polite`cnica de Catalunya.
identification, and approximation of multivariate processes, diagnostics, pattern recognition, and clustering in data samples. It has been shown that, for inaccurate, noisy, or small data sets, the GMDH is the best optimal simplified model, with a higher accuracy and a simpler structure than typical full physical models. In this work, to avoid the limitations of ANNs, an LLE prediction method was developed using the GMDH algorithm. The aim of this proposed method is to predict LLE data of a quaternary system,15 namely, corn oil + oleic acid + ethanol + water, using the GMDH algorithm. Using existing data,15 the proposed network was trained. The trained network was used to predict the LLE data in the oil and alcohol phases. Then, the data predicted by the proposed model was compared with experimental data. Also, to investigate the reliability of the proposed method, root-mean-square deviations of results obtained by the NRTL, UNIQUAC, and GMDH models were compared. Phase diagrams for the studied quaternary system including both the experimental and predicted tie lines are presented. 2. Group Method of Data Handling (GMDH) The group method of data handling (GMDH) is a combinatorial multilayer algorithm in which a network of layers and nodes is generated using a number of inputs from the data stream being evaluated. The group method of data handling was first proposed by Ivakhnenko.13 The GMDH network topology has traditionally been determined using a layer-by-layer pruning process based on a preselected criterion of what constitutes the best nodes at each level. The goal is to obtain a mathematical model of the object under study. The GMDH adaptively creates models from data in the form of networks of optimized transfer functions in a repetitive generation of layers of alternative models of growing complexity and corresponding model validation and fitness selection until an optimal complex model that is not too simple and not too complex has been created. Neither the numbers of neurons and layers in the network nor the actual behavior of each created neuron is predefined. All of these parameters are adjusted during the process of self-organization by the process itself. As a result, an explicit analytical model representing relevant relationships between input and output variables is available immediately after modeling. This model contains the extracted knowledge applicable for interpretation, prediction, classification, or problem diagnosis.16
10.1021/ie801082s CCC: $40.75 2009 American Chemical Society Published on Web 01/05/2009
2130 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
layer (m + 1) produces a better result than layer m. When layer (m + 1) is found to be not as good as layer m, the process is halted. The formal definition of the problem is to find a function ˆf so that it can be approximately used instead of the actual function, f, in order to predict output yˆ for a given input vector X ) (x1 x2 x3... xn) as close as possible to its actual output y. Therefore, given M observation of multi-input-single-output data pairs (training data set), so that yi ) f(xi1,xi2,xi3,...,xin)
i ) 1, 2,..., M
(2)
it is possible to train a GMDH-type network to predict the output values yˆ using training data, i.e. yˆi ) ˆf(xi1,xi2,xi3,...,xin) Figure 1. Feed-forward GMDH-type neural network. Table 1. Overall Composition of Liquid-Liquid Equilibrium Data for the System Corn Oil (1) + Commercial Oleic Acid (2) + Solvent [Ethanol (3) + Water (4)] at 298.15 K15 100w3
i ) 1, 2,..., M
(3)
This equation is tested for fitness by determining the mean square error of the predicted yˆ and actual y values as shown in eq 4 using a set of testing data. This value should be minimized. M
∑ (yˆ - y )
100w1
100w2
100w4
47.98 47.21 43.46 39.25 35.65 29.85
5 wt % water in solvent 0.00 49.40 2.53 47.72 4.91 49.02 9.87 48.32 14.52 47.32 19.99 47.62
2.63 2.54 2.61 2.57 2.51 2.53
The general connection between input and output variables can be expressed by eq 1. For most applications, a quadratic equation of only two variables is used in the form
49.97 44.97 39.78 35.49 30.99
8 wt % water in solvent 0.00 46.03 5.39 45.67 9.81 46.38 14.59 45.93 19.77 45.30
4.00 3.97 4.03 3.99 3.94
50.07 47.94 45.85 41.49 34.15 30.04 24.59
12 wt % water in solvent 0.00 43.94 2.40 43.70 4.92 43.32 9.65 43.26 14.79 44.93 19.99 43.97 25.06 44.30
5.99 5.96 5.91 5.90 6.13 5.99 6.04
to predict the output y. A typical feed-forward GMDH-type network is shown in Figure 1. The coefficients ai in eq 5 are calculated using regression techniques,13,14 so that the difference between the actual output, y, and the calculated value, yˆ, for each pair of input variables (xi, xj) is minimized. Indeed, it can be seen that a tree of polynomials is constructed using the quadratic form given in eq 5 whose coefficients are obtained in a least-squares sense. In this way, the coefficients of each quadratic function Fi are obtained to optimally fit the output for the whole set of input-output data pairs, i.e.
50.35 48.27 44.10 39.94 34.70 29.66 25.22
18 wt % water in solvent 0.00 40.72 2.42 40.44 4.91 41.81 9.80 41.22 15.08 41.18 20.15 41.16 24.89 40.91
2
i
y ) a0 +
n
n
n
n
2
r
8.94 8.88 9.18 9.05 9.04 9.03 8.97
n
∑ax +∑∑a xx +∑∑∑a i i
i)1
ijlxixjxl + ...
ij i j
i)1 j)1
f min
yˆ ) F(xi,xj) ) a0 + a1xi + a2xj + a3xixj + a4xi2 + a5xj2
2.1. GMDH Algorithm. The traditional GMDH method13,14 is based on an underlying assumption that the data can be modeled by using an approximation of the Volterra series or Kolmorgorov-Gabor polynomial,17 as given by n
i
(4)
i)1
(1)
i)1 j)1 l)1
where xi, xj, and xl are the inputs; y is the output; and a0, ai, aij, and aijl are the coefficients of the polynomial functional nodes. A GMDH network can be represented as a set of neurons in which different pairs of neurons in each layer are connected through a quadratic polynomial and thus produce new neurons in the next layer.18 In the classical GMDH algorithm, all combinations of the inputs are generated and sent into the first layer of the network. The outputs from this layer are then classified and selected for input to be fed into the next layer, with all combinations of the selected outputs being sent into layer 2. This process is continued as long as each subsequent
∑ )
M i)1
[yi - Fi(·)]2
∑
M y2 i)1 i
f min
(5)
(6)
In the basic form of the GMDH algorithm, all possibilities of two independent variables out of total of n input variables are taken in order to construct the regression polynomial in the form of eq 5 that best fits the dependent observations (yi, i ) 1, 2,..., M) in a least-squares sense. Consequently
()
n n(n - 1) ) 2 2
(7)
neurons are built up in the second layer of the feed-forward network from the observations {(yi, xip, xiq), (i ) 1, 2,..., M)} for different p, q ∈ {1, 2,..., M}.12 In other words, it is now possible to construct M data triples {(yi, xip, xiq), (i ) 1, 2,..., M)} from observation using such p, q ∈ {1, 2,..., M} in the form of
[
x1p x1q l y1 x2p x2q l y2 ... ... ... ... xMp xMq l yM
]
(8)
Using the quadratic expression in the form of eq 5 for each row of M data triples, the following matrix equation can be readily obtained as Ba ) Y
(9)
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2131 Table 2. Experimental and Predicted Liquid-Liquid Equilibrium Data for the Studied System in the Alcohol and Oil Phases alcohol phase (II) 100w1 expt
100w2
calc
expt
calc
oil phase (I)
100w3 expt
100w4
calc
expt
calc
100w1 expt
calc
5 wt % water in solvent 5.99 5.79 91.63 91.99 5.34 5.87 87.79 84.64 5.37 5.66 84.23 84.67 5.03 5.28 75.20 75.30 4.48 4.66 65.77 65.18 3.86 3.65 50.11 50.95
100w2 expt
100w3
100w4
calc
expt
calc
expt
calc
0.00 2.24 4.64 9.35 13.87 19.29
0.00 2.38 4.53 9.24 13.96 18.99
8.07 9.65 10.74 14.89 19.70 28.51
7.73 12.53 10.45 14.94 20.08 28.03
0.30 0.33 0.39 0.56 0.67 2.09
0.28 0.45 0.34 0.53 0.77 2.03
1.61 2.33 1.61 4.33 7.35 16.72
1.42 2.19 2.08 4.21 7.42 16.50
0.00 2.40 5.11 10.26 15.11 20.25
0.00 2.54 4.89 10.01 14.83 20.76
92.39 89.93 87.91 80.39 73.06 59.17
92.79 89.40 87.36 80.50 73.09 59.09
0.66 1.34 1.71 2.57 5.14
0.77 1.18 1.44 2.85 5.40
0.00 4.54 8.73 13.82 19.33
0.00 4.77 8.91 13.71 19.24
88.38 83.36 79.45 73.76 66.49
88.24 83.26 79.29 73.61 66.40
10.96 10.76 10.11 9.86 9.03
8 wt % water in solvent 10.98 93.76 93.20 10.80 85.34 85.15 10.37 77.96 79.54 9.83 69.63 70.21 8.95 58.97 58.71
0.00 5,64 10.39 15.34 20.97
0.00 5.64 10.12 15.16 20.85
5.64 8.36 10.88 13.91 18.40
6.16 8.47 9.60 13.59 18.80
0.60 0.66 0.76 1.11 1.66
0.63 0.73 0.73 1.04 1.63
0.44 0.67 0.82 1.21 2.03 3.98 8.31
0.66 0.79 0.85 1.11 1.75 3.43 9.19
0.00 1.81 3.80 7.86 12.99 18.34 24.04
0.00 1.74 3.68 7.55 12.66 17.96 25.00
85.59 83.73 81.62 77.73 72.49 66.19 57.41
85.65 83.73 81.74 78.05 72.87 66.78 54.65
12 wt % water in solvent 13.97 13.69 94.57 94.85 13.80 13.74 90.56 90.39 13.76 13.73 86.09 85.83 13.21 13.28 78.14 78.37 12.49 12.72 69.08 69.10 11.48 11.83 59.72 58.39 10.24 11.17 48.27 43.92
0.00 2.71 5.65 10.97 16.54 21.67 26.35
0.00 2.88 5.86 11.42 16.58 22.46 26.28
5.10 6.08 7.59 10.13 13.37 17.08 22.83
4.81 6.21 7.63 9.46 13.33 17.48 26.34
0.34 0.65 0.66 0.77 1.01 1.53 2.55
0.34 0.52 0.68 0.74 0.99 1.67 3.46
0.20 0.19 0.21 0.12 0.07 0.64 3.32
0.07 0.19 0.07 0.12 0.31 1.15 2.84
0.00 1.43 2.84 6.08 10.30 14.94 19.77
0.00 1.23 2.74 6.06 10.34 15.17 20.18
79.52 77.88 76.69 73.56 69.60 65.56 59.94
79.27 77.65 76.69 73.58 69.94 65.22 59.68
18 wt % water in solvent 20.28 20.66 95.71 95.84 20.49 20.92 91.12 90.37 20.26 20.50 86.36 85.95 20.24 20.24 77.07 76.87 20.03 19.42 66.88 67.38 18.86 18.46 57.37 57.67 17.07 17.30 48.58 48.75
0.00 3.20 6.63 13.27 20.10 25.80 29.92
0.00 3.45 6.78 13.23 19.78 25.38 30.02
3.68 5.05 6.24 8.72 11.60 14.89 18.56
3.51 5.22 6.55 8.84 11.51 15.00 18.33
0.61 0.63 0.77 0.94 1.43 1.94 2.94
0.65 0.96 0.73 1.06 1.33 1.96 2.90
where a is the vector of unknown coefficients of the quadratic polynomial in eq 5 a ) {a0 a1 a2 a3 a4 a5}
(10)
and Y is the vector of output values from the observation Y ) {y1 y2 y3 ... yM}T
[
(11)
It can be readily seen that
B)
1 x1p 1 x2p
x1q x2q
x1px1q
x1p2
x2px2q
2
l l l l 1 xMp xMq xMpxMq
x1q2
x2p x2q2 l l xMp2 xMq2
]
(12)
The least-squares technique from multiple-regression analysis leads to the solution of the normal equations in the form a ) (BTB)-1BTY (13) which determines the vector of the best coefficients of the quadratic expression in eq 5 for the whole set of M data triples. 3. Prediction of LLE Using the GMDH-Type Network The proposed model is a feed-forward GMDH-type network and was constructed using an experimental data set from ref 15. This data set consists of 25 points for four different concentrations of water in the solvent. In Table 1, the overall experimental compositions of the mixtures are shown. Table 2 lists the experimental mass fractions of the components in the alcohol and oil phases. The data set was divided into two parts: 80% was used as training data, and 20% was used as test data. Each point in the training and test data consists of 13 values. The four mass fractions in overall compositions and water concentration in the solvent were normalized and used as inputs
Table 3. Polynomial Equations of the GMDH Model U1 ) 0.73798Z3 - 23.4792Z4 + 92.6359Z42 - 90.8835Z43 1.70331Z3Z4 + 47.2773Z1Z4 - 79.4317Z13Z4 - 98.4833Z1Z42 + 1.70020Z32Z42 - 17.4940Z33Z42 + 70.9298Z3Z43 + 1526.12Z12Z3Z42 + 173.498Z22Z3Z42 - 3657.72Z13Z3Z42 - 96.8414Z23Z3Z42 841.056Z1Z2Z3Z42 - 579.273Z1Z3Z43 - 250.126Z2Z3Z43 + 2210.67Z1Z2Z3Z43 + 0.82995Z33Z4 + 0.0007 U2 ) 1.15753Z3 - 7.53321Z1Z2Z3 U3 ) -4.73310 + 36.6241Z2 - 2.48921Z3 - 75.2730Z22 + 49.0331Z23 + 29.7946Z33 - 126.916Z53 + 36.3945Z3Z5 82.1899Z2Z3Z5 U4 ) -0.15983 + 6.57990Z1 - 52.0567Z12 - 0.52322Z32 + 148.862Z13 V1 ) 0.48270 + 3.91004Z2 - 3.78515Z3 - 70.0216Z12 - 7.50281Z22 + 177.149Z13 + 18.1188Z1Z2 + 29.9898Z1Z3 - 43.2306Z1Z2Z3 V2 ) 1.96267Z32 + 1.50010Z2Z3 + 15.4356Z2Z3Z5 V3 ) 3.28598 - 23.0236Z2 + 3.37477Z3 + 71.0456Z12 + 50.8963Z22 182.670Z13 - 32.5992Z23 - 20.0880Z33 - 18.0614Z1Z2 29.3338Z1Z3 + 31.7250Z1Z2Z3 V4 ) 0.81654 - 5.17926Z2 + 10.6367Z22 - 30.9816Z52 7.22017Z23 - 3.29721Z33 + 188.349Z53 + 3.01616Z2Z5 + 9.62013Z2Z3Z5
of the GMDH-type network, and the other eight values were used as desired outputs of the network, specifically, four mass fractions in the alcohol phase and four mass fractions in the oil phase. After the data set had been applied to the GMDH-type network, eight polynomial equations were obtained that for use in predicting the mass fractions in the alcohol and oil phases (Table 3). For example, the equations for predicting the mass fractions of acid in the alcohol and oil phases are U2 ) 1.15753Z3 - 7.53321Z1Z2Z3
(14)
V2 ) 1.96267Z3 + 1.50010Z2Z3 + 15.4356Z2Z3Z5
(15)
2
respectively, where Z1 is the water concentration in the solvent and Z2, Z3, and Z5 are the normalized mass fractions of oleic acid, ethanol, and water, respectively, in the overall composition. We used the GMDH model to calculate the mass fractions of the components in the alcohol and oil phases. The calculated
2132 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
Figure 2. System of corn oil (1) + oleic acid (2) + 5% aqueous solvent [ethanol (3) + water (4)] at 298.15 K: (b) experiment, ( · · · ) GMDH.
Figure 3. System of corn oil (1) + oleic acid (2) + 8% aqueous solvent [ethanol (3) + water (4)] at 298.15 K: (b) experiment, ( · · · ) GMDH.
values are presented in Table 2. Figures 2 and 3 show the experimental points and the tie lines predicted using the GMDH model for the systems corn oil + oleic acid + 5% aqueous ethanol and corn oil + oleic acid + 8% aqueous ethanol. The equilibrium diagrams are plotted in triangular coordinates. To represent the pseudoquaternary systems in triangular coordinates, ethanol + water was treated as a mixed solvent.15 These figures indicate that the GMDH model provided good estimations for both phases. Figure 4 presents the fatty acid distribution between the phases, and Figure 5 shows the experimental and estimated solvent selectivities. The distribution coefficient and solvent selectivity were calculated by eqs 16 and 17, respectively: kc )
wIIc wIc
k2 S) k1
(16)
(17)
Figure 4. Distribution diagram at 298.15 K for systems of corn oil (1) + oleic acid (2) + ethanol (3) + water (4): (0) 5 wt % aqueous ethanol, (2) 8 wt % aqueous ethanol, (O) 12 wt % aqueous ethanol, (9) 18 wt % aqueous ethanol, (s) GMDH.
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2133
Figure 5. Fatty acid distribution coefficient and selectivities for systems of corn oil (1) + oleic acid (2) + ethanol (3) + water (4): (- - -) k2 calculated by the GMDH model, ( · · · ) S calculated by the GMDH model, (∆) experimental k2, (O) experimental S. Table 4. Mean Deviations [∆w (%)] in Different Models 15
NRTL
15
system
GMDH
corn oil +oleic acid + 5% aqueous ethanol corn oil +oleic acid + 8% aqueous ethanol corn oil +oleic acid + 12% aqueous ethanol corn oil +oleic acid + 18% aqueous ethanol global deviation
0.68
1.27
UNIQUAC 1.39
0.39
0.82
0.79
0.92
0.71
0.79
0.27
0.81
0.79
0.57
0.90
0.94
Nomenclature
The root-mean-square deviations between experimental and predicted compositions in the two phases were calculated according to the equation
∆w ) 100
T
C
∑ ∑ [(w
I,ex I,calc 2 + c,t - wc,t
t
)
in relation to the experimental data was lower than 0.57%. Thus, the GMDH model is suitable for use in predicting LLE data. The quality of the model is related to the quality of data used for the training of the model. The agreements between the experimental and calculated data were generally found acceptable.
II,calc 2 (wII,ex )] c,t - wc,t
c
2TC (18)
where T is the total number of tie lines, C is the total number of components, w is the mass fraction, the subscripts c, t are component and tie line, respectively and the superscripts I and II stand for oil and alcoholic phases, respectively; ex and calc refer to experimental and calculated concentrations. The results are shown in Table 4. To investigate the reliability of the GMDH model, these deviations were also compared with the calculated deviations obtained from predictions using the NRTL and UNIQUAC models.15 According to the results, the GMDH model can provide a satisfactory method for estimating the LLE data. 4. Conclusions In this study, a GMDH model was designed using experimental liquid-liquid equilibrium data for the system of corn oil + oleic acid + ethanol + water at 298.15 K. The LLE data were predicted by the GMDH model and then compared with experimental data. Despite the complexity of the studied system, the GMDH model allowed a good prediction of the phase equilibrium. Also, the global deviation of the proposed model
a ) vector of coefficients of polynomial functions a ) coefficients of polynomial functional node B ) matrix of variables of polynomial functions C ) total number of components f ) actual function ˆf ) predicted function F ) two-variable quadratic function kc ) cth distribution coefficient M ) number of input-output data pairs n ) number of input variables r ) root-mean-square S ) solvent selectivity T ) total number of tie lines U ) mass fraction in the alcohol phase V ) mass fraction in the oil phase w ) mass fraction of components x ) input of polynomial functional node X ) vector of inputs y ) actual output of polynomial functional node yˆ ) predicted output of polynomial functional node Y ) vector of outputs from observation Z ) mass fraction in overall composition Greek Letter ∆w ) root-mean-square deviation Superscripts calc ) calculated ex ) experimental I ) oil phase II ) alcoholic phase
2134 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Subscripts c ) component c t ) tie line t
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ReceiVed for reView July 15, 2008 ReVised manuscript receiVed October 14, 2008 Accepted November 20, 2008 IE801082S
