ARTICLE pubs.acs.org/IECR
LiquidLiquid Equilibrium Calculation for Ternary Aqueous Mixtures of Ethanol and Acetic Acid with 2-Ethyl-1-hexanol Using the GMDH-Type Neural Network H. Ghanadzadeh,*,† S. Fallahi,‡ and M. Ganji‡ † ‡
Department of Chemical Engineering, University of Guilan, Rasht, Iran Department of Mathematics, Faculty of Sciences, University of Guilan, Rasht, Iran ABSTRACT: A GMDH-type neural network was used to calculate liquid phase equilibrium data for the (water + ethanol or acetic acid + 2-ethyl-1-hexanol) ternary systems in the temperature range of 298.2313.2 K. Using this method, a new model was proposed that is suitable for predicting the liquidliquid equilibrium data. The proposed model was “trained” before the requested calculation. The data set was divided into two parts: 70% were used as data for “training” and 30% were used as a test set, which were randomly extracted from the database. After the training on the inputoutput process, the predicted values were compared with experimental values to evaluate the performance of the group method of data handling neural network method.
1. INTRODUCTION Liquidliquid equilibrium (LLE) studies have been the subject of much interest in recent years.18 In particular, LLE investigations for ternary mixtures are important in evaluation of industrial solvent extraction units. Precise LLE data are needed for efficient separation operations, which can be obtained from direct measurements917 or by the use of different thermodynamic methods.18 Traditional activity coefficients based thermodynamic models have been successfully used to describe several LLE systems. The nonrandom two-liquid (NRTL) model of Renon and Prausnitz19 and the universal quasi-chemical (UNIQUAC) method of Abrams and Prausnitz20 models have been widely used to correlate LLE data for the many multicomponent mixtures,2123 and a group contribution method (UNIFAC)24 has been generally used to predict the LLE systems. Recently, new prediction methods were developed using artificial neural networks (ANNs). ANNs are nonlinear and highly flexible models that have been successfully used in many complicated systems for estimating vaporliquid equilibria (VLE) and LLE data.2532 The ANNs can be considered as universal function approximators. Giving enough data, they can approximate the underlying function with accuracy.31,32 However, the main disadvantage of traditional NNs is that the detected dependencies are hidden within the NN structure.33 Conversely, the group method of data handling (GMDH)34 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.35 The GMDH was developed in complex systems for modeling, prediction, identification, and approximation. It has been shown that the GMDH r 2011 American Chemical Society
is the best optimal model for inaccurate, noisy, or small data sets. It has higher accuracy and simpler structure as compared with typical full physical models. In this work, a model for LLE data prediction was developed using the GMDH algorithm. The aim of this method is to predict LLE data for [water + ethanol or acetic acid + 2-ethyl-1-hexanol (2EH)] ternary systems. Using existing experimental data,36,37 the proposed network was trained. The trained network was used to predict the LLE data in the aqueous and organic phases, then the predicted data were compared with the experimental data, which have been previously reported. To investigate the reliability of the proposed method, the accuracy of the model was determined using coefficient of determination (R2), mean square error (MSE), root-mean-square error (RMSE), and mean absolute deviation (MAD).
2. GROUP METHOD OF DATA HANDLING Using the GMDH algorithm, a model can be represented as a set of neurons in which different pairs of them in each layer are connected through a quadratic polynomial. Therefore, they produce new neurons in the next layer.33 Such representation can be used in modeling to map inputs to outputs. The formal definition of the identification problem is to find a function, ^f , that can be approximately used instead of the actual one, f, to predict output ^y for a given input vector X = (x1, x2, x3, 3 3 3 , xn). For a given number of observations (M) of multiinput, single output system we have yi ¼ f ðxi1 , xi2 , xi3 , :::, xin Þði ¼ 1, 2, 3, :::, MÞ
ð1Þ
Received: July 5, 2010 Accepted: July 26, 2011 Revised: June 26, 2011 Published: July 26, 2011 10158
dx.doi.org/10.1021/ie101425w | Ind. Eng. Chem. Res. 2011, 50, 10158–10167
Industrial & Engineering Chemistry Research
ARTICLE
Table 1. Experimental, UNIQUAC-Correlated, and GMDH-Estimated Tie-Line Data in the Aqueous and Organic Phases for (Water + Ethanol + 2EH) at 298.2313.2 Ka water-rich phase (aqueous phase) x1 (water)
solvent-rich phase (organic phase)
x2 (ethanol)
x1 (water)
exp
UNIQ
GMDH
exp
UNIQ
GMDH
0.9772
0.9799
0.9756
0.0226
0.0198
0.0214
0.9043 0.8861
0.9062 0.8869
0.8994 0.8808
0.0995 0.1137
0.0963 0.0816
0.0958 0.1102
0.8650
0.8548
0.8605
0.1248
0.1229
0.8556
0.8749
0.8596
0.1441
0.1241
0.8275
0.8305
0.8336
0.1421
0.7798
0.772
0.7800
0.9781
0.9790
0.9045
exp
x2 (ethanol)
UNIQ
GMDH
exp
UNIQ
GMDH
0.2100
0.1812
0.2075
0.1690
0.1668
0.1783
0.2260 0.2340
0.1966 0.2385
0.2175 0.2537
0.2302 0.2911
0.2019 0.2858
0.2268 0.3027
0.1262
0.2814
0.2728
0.2951
0.3510
0.3202
0.3431
0.1487
0.3360
0.3120
0.3211
0.3709
0.3644
0.3909
0.1433
0.1406
0.4050
0.4211
0.3517
0.3914
0.3818
0.3859
0.169
0.1659
0.1692
0.5285
0.5299
0.5190
0.4100
0.3822
0.3823
0.9764
0.0213
0.0203
0.0181
0.2241
0.2240
0.2305
0.1890
0.2241
0.1869
0.9059
0.9046
0.0949
0.0931
0.0947
0.2370
0.2889
0.2367
0.3337
0.3699
0.3346
0.8873 0.8679
0.8862 0.8726
0.8873 0.8664
0.1119 0.1311
0.1126 0.1259
0.1110 0.1305
0.2665 0.2942
0.3070 0.3196
0.2678 0.3027
0.3617 0.3876
0.4020 0.4191
0.3646 0.3887
0.8524
0.8549
0.8507
0.1463
0.1432
0.1457
0.3241
0.3360
0.3338
0.4036
0.4361
0.4042
0.8302
0.8304
0.8348
0.1683
0.1669
0.1663
0.3946
0.3584
0.3836
0.4178
0.4518
0.4306
0.7902
0.7934
0.7878
0.2071
0.2025
0.2046
0.4576
0.3911
0.4636
0.4151
0.4631
0.4221
0.9782
0.9797
0.9798
0.0211
0.0199
0.0244
0.1919
0.2253
0.1849
0.1606
0.1219
0.1541
0.9176
0.9046
0.9164
0.0818
0.0944
0.0834
0.2302
0.2598
0.2440
0.3442
0.3695
0.3326
0.8958
0.8847
0.8941
0.1034
0.1139
0.1048
0.2616
0.2807
0.2761
0.3815
0.4016
0.3692
0.8791 0.8598
0.8711 0.8534
0.8776 0.8576
0.1198 0.139
0.1273 0.1446
0.1207 0.1417
0.2851 0.3213
0.3202 0.3365
0.2961 0.3391
0.3958 0.4297
0.4187 0.4358
0.3880 0.4110
0.8295
0.8294
0.8315
0.169
0.1679
0.1707
0.3949
0.3589
0.3942
0.4377
0.4515
0.4298
0.7891
0.7916
0.7865
0.2082
0.2041
0.2099
0.458
0.4216
0.4732
0.4451
0.4627
0.4263
0.9777
0.9785
0.9802
0.0216
0.0250
0.0235
0.1985
0.2001
0.1949
0.1691
0.1683
0.1690
0.9096
0.9046
0.9121
0.0898
0.0943
0.0900
0.2423
0.2890
0.2317
0.3173
0.3097
0.3274
0.8889
0.8848
0.8899
0.1103
0.1138
0.1101
0.2683
0.3069
0.2625
0.3512
0.3920
0.3627
0.8723 0.8542
0.8712 0.8537
0.8731 0.8549
0.1266 0.1445
0.1271 0.1443
0.1256 0.1434
0.2990 0.3355
0.3194 0.3357
0.2928 0.3292
0.3723 0.3930
0.3891 0.3998
0.3872 0.4084
0.8295
0.8201
0.8316
0.1690
0.1598
0.1697
0.3955
0.4010
0.3849
0.4189
0.4191
0.4196
0.7890
0.7980
0.7886
0.2048
0.2101
0.2035
0.4550
0.4512
0.4672
0.4175
0.4362
0.4273
T = 298.2 K
T = 303.2 K
T = 308.2 K
T = 313.2 K
a
The experimental and UNIQUAC correlated data were taken from our previous report.36
It is now possible to train a GMDH-type NN to predict the output values ^yi for any given input vector X = (xi1, xi2, xi3, ..., xin), that is, ^yi ¼ ^f ðxi1 , xi2 , xi3 , :::, xin Þði ¼ 1, 2, 3, :::, MÞ
M
h i2 ^f ðxi1 , xi2 , :::, xi Þ yi f min
y ¼ ao þ
ð2Þ
To determine a GMDH-type NN, the square of the differences between the actual output and the predicted one is minimized.
∑ i¼1
Volterra functional series34 in the form of
ð3Þ
The general connection between the inputs and the output variables can be expressed by a complicated discrete form of the
n
n
n
n
n
n
∑ ai xi þ i∑¼ 1 j∑¼ 1 aijxi xj þ i∑¼ 1 j∑¼ 1 k∑¼ 1 aijk xi xj xk þ ::: i¼1 ð4Þ 34
This is known as the KolmogorovGabor polynomial. The general form of mathematical description can be represented by a system of partial quadratic polynomials consisting of only two variables (neurons) in the form of ^y ¼ G xi , xj ¼ a0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 xi 4 þ a5 xj 2 :::
ð5Þ
In this way, such a partial quadratic description is recursively used in a network of connected neurons to build the general 10159
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Industrial & Engineering Chemistry Research
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Table 2. Experimental, UNIFAC-Predicted, and GMDH-Estimated Tie-Line Data in the Aqueous and Organic Phases for (Water + Acetic acid + 2EH) at 298.2313.2 Ka aqueous phase
organic phase
mole fraction
mole fraction
x1 (water)
x2 (acetic acid)
x1 (water)
exp
UNIFAC
GMDH
exp
UNIFAC
GMDH
exp
UNIFAC
0.9996 0.9739
0.9997 0.9735
0.9994 0.9742
0.0000 0.0257
0.0000 0.0264
0.0001 0.0252
0.1081 0.1252
0.1181 0.1265
0.9482
0.9444
0.9483
0.0514
0.0555
0.0515
0.1431
0.8982
0.8934
0.8982
0.1016
0.1065
0.1019
0.1729
0.8821
0.8782
0.8822
0.1177
0.1216
0.1178
0.8529
0.8527
0.8525
0.1468
0.1471
0.8217
0.8239
0.8213
0.1776
0.7957
0.8000
0.7961
0.6835
0.6870
0.9996
x2 (acetic acid) GMDH
exp
UNIFAC
GMDH
0.1073 0.1224
0.0000 0.0936
0.0000 0.1495
0.0000 0.0946
0.1453
0.1423
0.1796
0.2092
0.1781
0.1777
0.1691
0.2878
0.2822
0.2860
0.1820
0.1874
0.1763
0.3113
0.2990
0.3108
0.1468
0.1960
0.2035
0.1939
0.3520
0.3230
0.3502
0.1737
0.1769
0.2192
0.2271
0.2192
0.3811
0.3511
0.3837
0.2033
0.1951
0.2024
0.2790
0.2834
0.2741
0.4230
0.3944
0.4278
0.6835
0.2966
0.2899
0.2952
0.3970
0.4035
0.3807
0.4398
0.4225
0.4428
0.9996
0.9996
0.0000
0.0000
0.0001
0.1081
0.1087
0.1071
0.0000
0.0000
0.0000
0.9830
0.9809
0.9827
0.0168
0.0190
0.0173
0.1269
0.1254
0.1354
0.0934
0.1211
0.0917
0.9483
0.9489
0.9482
0.0516
0.0507
0.0518
0.1459
0.1450
0.1487
0.1868
0.2015
0.1854
0.8986
0.8986
0.8990
0.1013
0.1014
0.1013
0.1763
0.1764
0.1717
0.2868
0.2902
0.2878
0.8826
0.8811
0.8831
0.1172
0.1188
0.1170
0.1848
0.1854
0.1783
0.3107
0.3096
0.3125
0.8533
0.8505
0.8538
0.1463
0.1493
0.1458
0.1995
0.2011
0.1935
0.3476
0.3387
0.3509
0.8224
0.8221
0.8211
0.1769
0.1776
0.1775
0.2116
0.2168
0.2230
0.3969
0.3629
0.3898
0.7962 0.6960
0.7976 0.6577
0.7955 0.6962
0.2027 0.2916
0.1999 0.3135
0.2031 0.2927
0.2640 0.3731
0.2677 0.3787
0.2705 0.3738
0.4350 0.4788
0.4164 0.4440
0.4304 0.4681
0.9996
0.9996
0.9997
0.0000
0.0000
0.0000
0.1081
0.1087
0.1070
0.0000
0.0000
0.0003
0.9811
0.9849
0.9813
0.0186
0.0112
0.0188
0.1285
0.1266
0.1323
0.0933
0.1262
0.0933
0.9484
0.9546
0.9487
0.0515
0.0394
0.0514
0.1492
0.1479
0.1501
0.1864
0.2068
0.1878
0.8989
0.9071
0.8997
0.1009
0.0845
0.1006
0.1797
0.1797
0.1733
0.2863
0.2898
0.2908
0.8830
0.8917
0.8840
0.1167
0.0992
0.1162
0.1881
0.1887
0.1798
0.3099
0.3079
0.3156
0.8538
0.8634
0.8548
0.1458
0.1263
0.1449
0.2029
0.2048
0.1953
0.3468
0.3355
0.3553
0.8232 0.7968
0.8335 0.8058
0.8231 0.7976
0.1761 0.2001
0.1547 0.1810
0.1762 0.2011
0.2360 0.2730
0.2206 0.2784
0.2372 0.2664
0.4027 0.4398
0.3582 0.4131
0.4042 0.4326
0.6750
0.6837
0.6755
0.3128
0.2923
0.3125
0.3967
0.4023
0.4005
0.4679
0.4488
0.4773
0.9996
0.9993
0.9998
0.0000
0.0000
0.0000
0.1081
0.1140
0.10707
0.0000
0.0000
0.0008
0.9793
0.9526
0.9798
0.0205
0.0473
0.0200
0.1301
0.1311
0.129953
0.0929
0.1488
0.0965
0.9486
0.9214
0.9478
0.0513
0.0785
0.0518
0.1280
0.1535
0.134169
0.1858
0.2267
0.1782
0.8992
0.8946
0.8997
0.1006
0.1053
0.1002
0.1420
0.1671
0.140987
0.2700
0.2634
0.2702
0.8835 0.8508
0.8740 0.8513
0.8823 0.8504
0.1162 0.1487
0.1259 0.1482
0.1175 0.1489
0.1685 0.1820
0.1775 0.1997
0.180179 0.190435
0.3320 0.3600
0.2874 0.3293
0.3221 0.3598
0.8319
0.8333
0.8313
0.1675
0.1652
0.1674
0.2022
0.2117
0.211123
0.3830
0.3477
0.3867
0.7661
0.7652
0.7670
0.2243
0.2238
0.2246
0.2579
0.2614
0.260131
0.4150
0.4017
0.4166
0.6815
0.7017
0.6808
0.2896
0.2685
0.2901
0.3770
0.3790
0.387344
0.4190
0.4185
0.4165
T = 298.2 K
T = 303.2 K
T = 308.2 K
T = 313.2 K
a
The experimental and UNIFAC-predicted data were taken from our previous report37.
mathematical relation of the inputs and output variables given in eq 4. The coefficients ai in eq 5 are calculated using regression techniques. It can be seen that a tree of polynomials is constructed using the quadratic form given in eq 5. In this way, the coefficients of each quadratic function, Gi, are obtained to optimally fit the
output data in the whole set of inputoutput data pairs. M 2 yi G i i¼1 E¼ f min M
∑
10160
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dx.doi.org/10.1021/ie101425w |Ind. Eng. Chem. Res. 2011, 50, 10158–10167
Industrial & Engineering Chemistry Research
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Table 3. Polynomial Equations of the GMDH Model for the System (Water + Ethanol + 2EH) x11 (mole fraction of water in aqueous phase) y1 = 0.0024 + 0.0047x1 0.0300x4 + 0.00001x12 0.9498x42 + 0.0069x1x4 y2 = 9.0572 21.2865x1 22.8416x3 + 12.6978x12 + 16.6970x32 + 27.1212x3x1 y3 = 0.4071 + 0.628x3 3.6353x2 + 0.6353x32 + 12.9171x22 1.2622x3x2 y4 = 0.00001 0.0002y1 + 0.0017y2 + 0.0923y12 + 0.00001y22 + 0.0030y1y2 x11 = 0.0070 0.7810y3 + 1.8114y4 + 84.3831y32 + 82.2158y42 166.6555y3y4 x21 (mole fraction of alcohol in aqueous phase) y1 = 0.0048 + 0.0072x2 + 0.0612x3 + 0.00001x22 + 1.9339x32 + 0.0013x2x3 y2 = 0.0176 0.0003x1 + 0.0313x4 + 0.00001x12 + 2.2101x42 0.0060x1x4 y3 = 14.4378 29.4008y1 26.9829x4 + 14.5168y12 + 13.5454x42 + 28.6533y1x4 y4 = 0.2627 + 1.4747y2 + 1.6834x2 3.0804y22 0.4754x22 0.6473x2y2 x21 = 0.0398 + 1.4780y3 0.3739y4 22.3829y32 21.1699y42 + 43.4862y3y4 x13 (mole fraction of water in organic phase) y1 = 0.0048 + 0.0072x3 + 0.0612x4 + 0.00001x32 + 1.9339x42 + 0.0013x3x4 y2 = 0.0176 0.0003x2 + 0.0313x1 + 0.00001x22 + 2.2101x12 0.0060x2x1 y3 = 14.4378 29.4008y1 26.9829x4 + 14.5168y12 + 13.5454x42 + 28.6533y1x4 x13 = 0.2627 + 1.4747y2 + 1.6834y3 3.0804y22 0.4754y32 0.6473y2y3 x23 (mole fraction of alcohol in organic phase) y1 = 0.0048 + 0.0072x4 + 0.0612x3 + 0.00001x42 + 1.9339x32 + 0.0013x4x3 y2 = 0.0176 0.0003x1 + 0.0313x3 + 0.00001x12 + 2.2101x32 0.0060x1x3 y3 = 14.4378 29.4008x2 26.9829y1 + 14.5168x22 + 13.5454y12 + 28.6533x2y1 y4 = 0.2627 + 1.4747y2 + 1.6834x1 3.0804y22 0.4754x12 0.6473x1y2 x23 = 0.0398 + 1.4780y3 0.3739y4 22.3829y32 21.1699y42 + 43.4862y3y4
In the basic form of the GMDH algorithm, all the possibilities of two independent variables out of the total n input variables are taken to construct the regression polynomial in the form of eq 5 that provides the best fit with the dependent observations (yi, i = 1, 2, ..., M).38 Using the quadratic subexpression in the form of eq 5 for each row of M data triples, the following matrix equation can be readily obtained as Aa ¼ Y
ð7Þ
where a is the vector of unknown coefficients of the quadratic polynomial in eq 5 a ¼ fao , a1 , a2 , a3 , a4 , a5 g
ð8Þ
and
T Y ¼ y1 , y2 , y3 , :::, yM
ð9Þ
Table 4. Polynomial Equations of the GMDH Model for the System (Water + Acetic acid + 2EH) x11 (mole fraction of water in aqueous phase) Y1 = 0.6665 + 2.8629z2 2.1985z4 1.2511z22 8.2906z42 + 4.4163z2z4 Y2 = 0.00004 + 0.0060z1 + 0.00005z4 0.00001z12 + 0.000006z42 + 0.0147z1z4 Y3 = 0.8780 20.4823z2 + 17.8765y1 + 115.1454z22 + 77.4637y12 188.9360z2y1 Y4 = 0.1182 + 1.1226z3 + 1.8754y2 1.4610z32 0.9900y22 2.0631z3y2 x11 = 0.0041 + 0.5801y3 + 0.4115y4 178.5454y32 178.6082y42 + 357.1579y3y4 x21 (mole fraction of acid in aqueous phase) Y1 = 0.0005 + 0.0042z1 0.0280z2 0.000005z12 0.3670z22 0.0018z1z2 Y2 = 0.0400 + 1.0015z3 + 0.3367z4 + 0.1694z32 + 1.2172z42 0.4800z3z4 Y3 = 0.0367 + 0.8689y1 0.2970z4 + 0.3325y12 1.2173z42 1.5382y1z4 Y4 = 0.0002 0.1030z3 + 1.0972y2 + 11.5102z32 + 12.7323y22 24.1778z3y2 x21 = 0.00009 + 0.0393y3 + 0.9602y4 40.2470y32 42.2506y42 + 82.4972y3y4 x31 (mole fraction of 2EH in aqueous phase) Y1 = 10.6417 23.2880z2 16.5302z3 + 12.7808z22 + 6.4240z32 + 18.0832z2z3 Y2 = 0.0000007 0.0001z1 0.000001z4 + 0.0000004z12 0.0000001z42 0.0004z1z4 Y3 = 0.0007 + 0.9644y1 + 0.0185z4 10.0050y12 0.1412z42 + 7.0469y1z4 Y4 = 0.2469 50.2498y2 6.6085z4 + 2816.0600y22 + 43.8112z42 + 660.5264z4y2 x31 = 0.00005 + 1.3737y3 0.0909y4 4.0010y32 + 12.8576y42 41.7301y3y4 x13 (mole fraction of water in organic phase) Y1 = 1.4920 6.7541z4 26.7141z3 + 11.2604z42 + 125.6225z32 + 62.2677z3z4 Y2 = 10.0400 224.5310z2 169.0174z3 + 125.6225z22 + 74.6152z32 + 188.9773z2z3 Y3 = 0.00001 + 0.0019z1 0.00003z4 0.000002z12 0.000004z42 0.0114z1z4 Y4 = 0.3236 + 2.4857y1 + 5.0852z4 1.8345y12 22.6687z42 9.3191y1z4 Y5 = 0.0397 + 1.4501y2 + 0.0046y3 1.8499y22 1.9058y32 + 2.6883y2y3 x13 = 0.0004 + 17.9130y4 16.9172y5 38.5793y42 + 38.5890y52 0.0009y4y5 x23 (mole fraction of acid in organic phase) Y1 = 0.00001 + 0.0018z1 0.00008z4 + 0.000001z12 0.000009z42 0.0249z1z4 Y2 = 0.3605 + 0.9860z3 3.0302z4 1.5247z32 10.9551z42 + 4.3184z3z4 Y3 = 0.00001 + 0.0018z1 0.00008z4 + 0.000001z12 0.000009z42 0.0249z1z4 Y4 = 0.3085 0.0275y1 + 1.2221z2 0.2650y12 1.3901z22 + 0.9673y1z2 Y5 = 0.0056 + 0.6816y2 + 0.3783y3 + 3.4771y22 + 3.0340y32 6.6354y2y3 x23 = 0.0008 0.8891y4 + 1.8710y5 + 63.8928y42 + 59.4812y52 123.3386y4y5 x33 (mole fraction of 2EH in organic phase) Y1 = 0.0076 0.0069z1 + 0.4237z2 + 0.00004z12 + 5.5472z22 0.0209z1z2 Y2 = 85.7756 + 199.5904z3 + 138.7723z4 114.6674z32 57.8168z42 162.7486z3z4 Y3 = 0.0080 + 0.2309y1 + 0.8019y2 + 1.1985y12 + 1.5978y22 2.8250y2y1 Y4 = 0.8526 + 29.7443z4 + 5.7680z3 114.6674z42 9.7356z32 66.5861z3z4 x33 = 0.0029 + 1.8385y3 0.8294y4 + 67.8869y32 + 70.2118y42 138.1058y3y4 10161
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Figure 1. Developed structure of GMDH-type NN model for the ternary system (water + ethanol + 2EH) in aqueous phase: (a) x11 and (b) x21.
Figure 3. Developed structure of GMDH-type NN model for the ternary system (water + acetic acid + 2EH) in aqueous phase: (a) x11, (b) x21, and (C) x31.
Figure 2. Developed structure of GMDH-type NN model for the ternary system (water + ethanol + 2EH) in organic phase: (a) x13 and (b) x23.
Here, Y is the vector of the output values from observation. It can be readily seen that 2 3 1 x1p x1q x1p x1q x1p 2 x1q 2 6 6 1 x2p x2q x2p x2q x2p 2 x2q 2 7 7 7 ð10Þ A¼6 6l l l l l l 7 4 5 2 2 1 xMp xMq xMp xMq xMp xMq The least-squares technique from multiple regression analysis leads to the solution of the normal equations in the form of38 a ¼ ðAT AÞ1 AT Y ð11Þ
3. THE LLE PREDICTION USING THE GMDH-TYPE NEURAL NETWORK The feed-forward GMDH-type neural network for the ternary systems of (water + ethanol + 2EH) and (water + acetic
acid + 2EH) was constructed using previously reported experimental data sets.36,37 The experimental compositions of the mixtures together with the correlated or predicted LLE data, using the thermodynamic models,20 in the aqueous and organic phases at different temperatures are shown in Tables 1 and 2. For each ternary system, the data set consist of 36 points in four different temperatures. For both ternary systems, the data was divided into two parts: 70% (26 points) was used as training data, and 30% (10 points) was used as test data. The three feed fractions in the compositions and temperature were used as inputs of the GMDH-type network. The 3 mol fractions in the aqueous phase (x11, x21, x31) and 3 mol fractions in the organic phase (x13, x23, x33) were used as desired outputs of the neural network. To estimate the mole fractions in the aqueous and organic phases, using the GMDH-type network, six polynomial equations were obtained for both investigated systems (Tables 3 and 4). In this table, z1 is the temperature, and z2, z3, and z4 are the normalized feed fractions of water, the solute (ethanol or acetic acid), and the organic solvent (2EH), respectively. The proposed model was used to calculate the mole fractions (the output data) of the components in the aqueous and organic phases. The calculated values are also presented in Tables 1 and 2. The developed GMDH neural network was successfully used to obtain six models for calculation of the LLE for both ternary systems. The optimal structures of the developed neural network with two hidden layers are shown in Figures 1, 2, 3, and 4. For instance, for the ternary system of 10162
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Figure 6. Plot of the mole fractions against data set number in the organic phase [(a) x13, (b) x23] to illustrate the prediction of the experimental data for (water + ethanol + 2EH) using the GMDH model; (O) experimental points and (+) calculated points.
Figure 4. Developed structure of GMDH-type NN model for the ternary system (water + acetic acid + 2EH) in organic phase: (a) x13, (b) x23, and (c) x31.
water + ethanol + 2EH, ccbbadac and bbaacddd are corresponding genome representations for the mole fractions of water in the aqueous and organic phases, respectively. In which a, b, c, and d stand for temperature and feed fractions (water, ethanol and 2EH), respectively. For the water + acetic acid + 2EH ternary system, bbbdccad and dcddbcad are the corresponding genome representations for the mole fractions of water in the aqueous and organic phases, respectively. All input variables were accepted by the models. In other words, the GMDH-type NN provides an automated selection of essential input variables and builds polynomial equations for the LLE modeling. These polynomial equations show the quantitative relationship between input and output variables (Tables 3 and 4). Our proposed model behavior in prediction of the LLE of the investigated systems is demonstrated in Figures 5, 6, 7, and 8. The results of the developed models give a close agreement between observed and predicted values of the LLE data. Some statistical measures are given in Tables 5 and 6 to determine the accuracy of the models. These statistical values are based on R2 as an absolute fraction of variance, RMSE as root-mean squared error, MSE as mean squared error, and MAD as mean absolute deviation. These factors are defined as follows: 2
2 3 YiðmodelÞ YiðactualÞ 7 6 i¼0 7 R2 ¼ 1 6 6 7 2 M 4 5 YiðactualÞ
Figure 5. Plot of the mole fractions against data set number in the aqueous phase [(a) x11, (b) x21] to illustrate the prediction of the experimental data for (water + ethanol + 2EH) using the GMDH model: (O) experimental points and (+) calculated points.
M
∑
∑
ð12Þ
i¼1
10163
dx.doi.org/10.1021/ie101425w |Ind. Eng. Chem. Res. 2011, 50, 10158–10167
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Figure 7. Plot of the mole fractions against data set number in the organic phase to illustrate the prediction of the experimental data for (water + acetic acid + 2EH) using the GMDH model: (O) experimental points and (+) calculated points.
2
2 31=2 ∑ Y Y iðacutalÞ 7 6i ¼ 0 iðmodelÞ 7 RMSE ¼ 6 4 5 M
M
MSE ¼
∑
i¼0
M
YiðmodelÞ YiðactualÞ
ð13Þ
Figure 8. Plot of the distribution coefficient (D2) of acetic acid as a function of the mole fraction (x23) of acetic acid in the organic phase at different temperatures: () 298.2, (•) 303.2, (Δ) 308.2, and (O) 313.2 K.
Table 5. Model Statistics and Information for the Group Method of Data Handling Type Neural Network Model for Estimating LLE Data of the (Water + Ethanol + 2EH) System statistic
x11
training
0.99999
testing training testing
2
M YiðmodelÞ YiðactualÞ i¼0
MAD ¼
M
MSE
RMSE
MAD
0.00001
0.0025
0.0019
0.99999
0.00001
0.0027
0.0023
0.99974
0.00001
0.0023
0.0017
0.99952
0.00001
0.0023
0.0020
Aqueous-Rich Phase
ð14Þ x21
M
∑
R2
mole fraction
Organic-Rich Phase x13
ð15Þ
x23
After training with the GMDH algorithm, the output (the mole fractions) is generated. The output data of the models are given in Tables 1 and 2. Now we can compare the experimental data and those correlated or predicted by the thermodynamic
training
0.99798
0.00024
0.016
0.011
testing
0.99831
0.00012
0.011
0.0093
training
0.99895
0.00015
0.012
0.010
testing
0.99911
0.00008
0.0091
0.0066
models, which have been reported in our earlier publications,36,37 with the output of GMDH. 10164
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Table 6. Model Statistics and Information for the Group Method of Data Handling Type Neural Network Model for Predicting LLE Data of the (Water + Acetic acid + 2EH) system mole fraction
R2
statistic
Table 9. Experimental and Calculated Separation Factors (S) and Distribution Coefficients of Ethanol (D2) at Different Temperatures experimental
RMSE
MSE
UNIQUAC
GMDH
MAD D2
S
D2
0.00055
7.48
34.80
8.42
9.26
2.10
S
D2
S
45.56
8.30
39.02
9.66
2.37
9.79
Aqueous-Rich Phase x11
training testing
x21 x31
1.0000 0.9999
0.00052 0.00065
