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RESEARCH NOTES Neural Network Modeling of Adsorption Equilibria of Mixtures in Supercritical Fluids Sujit Kumar Jha and Giridhar Madras* Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India
A neural network model was used to predict the ternary adsorption equilibria of 2,6- and 2,7dimethylnaphthalene isomers dissolved in supercritical carbon dioxide on NaY-type zeolite. The neural network was trained using binary (pure solute dissolved in supercritical carbon dioxide onto NaY-type zeolite) and ternary adsorption equilibrium data. Despite a limited number of data points available for the training of the network, the model was capable of predicting the ternary adsorption equilibria using the binary adsorption equilibrium data very precisely. Introduction process1
Adsorption is a widely used separation that has been studied extensively at low (subatmospheric) and at high pressures.2,3 Supercritical fluids are attractive solvents for adsorption and desorption processes because they have excellent solvent power and masstransfer properties. Supercritical carbon dioxide (ScCO2) is the most widely used supercritical fluid, because it is inexpensive, nontoxic, nonflammable, and has a low critical temperature. There are numerous investigations on the regeneration of adsorbents4-6 and adsorptive separation7-9 using ScF. Several adsorption isotherms for pure solutes dissolved in ScF onto different solid matrixes are available and are mentioned elsewhere.10,11 However, there are very few investigations on the adsorption of mixture of solutes dissolved in ScF, primarily because of the difficulty associated with their measurement. Cross and Akgerman9 studied the mixture adsorption isotherms of pentachlorophenol and hexachlorobenzene dissolved in ScCO2 onto calcium montmorillonite. Multicomponent Langmuir equations,12 ignoring the excess adsorption of carbon dioxide, were used to fit the measured isotherms. Zhou et al.13 studied the partition equilibrium of various mixtures of polychlorinated biphenyls between St. Lawrence River sediments and two supercritical fluids (ScCO2 and ScCO2/5 mol % methanol). Iwai et al.3 studied the mixture adsorption isotherm of 2,6- and 2,7-dimethylnaphthalene (DMN) isomers onto NaY-type zeolite. The adsorption amounts of 2,6- and 2,7-DMN were correlated using an extended Radke-Prausnitz equation.14 Numerous studies15-17 have been conducted to predict the ternary adsorption equilibria of nonideal mixtures from single-component binary adsorption data. Adsorbate phase activity coefficients have been introduced to account for the deviation of adsorbate mixture from ideal solution. Costa et al.16 used Wilson and UNIQUAC equations for the prediction of ternary adsorption equilibria with data of binary systems. The prediction * Corresponding author. Tel: +91-80-2293 2321. Fax: +91080-360 0683. E-mail address:
[email protected].
was in excellent agreement with experimental values. Talu and Zwiebel17 proposed a thermodynamically consistent spreading-pressure-dependent (SPD) equation to evaluate adsorbate phase activity coefficients for nonideal systems. The SPD equation with parameters determined by fitting the single-component and binary adsorption data has been shown to give excellent prediction of the ternary equilibria. In the present study, a neural network model was developed to predict the ternary adsorption equilibria in ScF from the knowledge of binary adsorption equilibria. An artificial neural network (ANN) is a modeling tool and it can approximate complex relationships without detailed knowledge of the underlying process. It is typically a massively parallel interconnected network of artificial neurons, also called nodes, operating by curve fitting. However, it is adaptive, has better filtering capacity, and generally performs better than empirical models with noisy or incomplete data. Neural networks are used in various areas in bioprocessing and chemical engineering.18 Fullana et al.19 used neural network computing for modeling and simulation of supercritical fluid extractors. Fu and Zhu20 modeled the supercritical adsorption of methane on activated carbon using ANN. Carsky and Do21 and Yang et al.22 did ANN modeling of the relationship between single-component adsorption data and binary adsorption isotherms. However, the procedure cannot be extended to even the binary adsorption in ScF, because the adsorption equilibria of individual components on solid adsorbents are not known (e.g., to consider the naphthalene-ScCO2soil matrix system, information about the naphthalenesoil matrix is required). To the best of our knowledge, this is the first work that uses a neural network to predict ternary adsorption equilibria in supercritical fluids based on the knowledge of binary adsorption equilibria. Because the experimental determination of adsorption isotherms of mixed solutes in supercritical fluids is extremely difficult, methods to predict multicomponent adsorption equilibria based on data of binary equilibria assumes significance. Therefore, in the present study, the ternary adsorption equilibria in ScF has been
10.1021/ie049010p CCC: $30.25 © 2005 American Chemical Society Published on Web 07/23/2005
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Figure 1. (A) An i-j-k feedforward neural network. (B) Flow of information in a typical i-j-k feedforward neural network. Table 1. Input Data for the Network pressure C2 (x 106 mol/ C3 (x 106 mol/ q2′ (x 105 mol/ q3′ (x 105 mol/ (MPa) cm3) cm3) g of zeolite) g of zeolite) 9 9 9 9 9 9
1.22 1.35 1.96 2.16 2.73 3.42
0.283 0.978 0.257 1.39 0.622 0.727
0.940 1.08 1.71 1.90 2.55 3.70
2.90 8.08 2.65 10.4 5.69 6.44
12 12 12 12 12
2.29 3.23 4.74 2.26 3.20
0.243 0.332 0.585 1.53 2.19
1.07 1.54 2.55 1.05 1.53
1.59 2.13 3.58 7.87 10.2
19.8 19.8 19.8 19.8 19.8
3.23 4.31 6.41 3.00 3.99
0.482 0.637 1.03 2.72 3.76
0.844 1.15 1.94 0.814 1.08
1.55 2.02 3.14 7.15 9.14
14.8 14.8 14.8 14.8 14.8
2.72 3.79 5.46 2.60 3.54
0.341 0.462 0.728 2.00 2.81
0.970 1.36 2.17 0.923 1.26
1.84 2.44 3.68 8.40 10.7
modeled using a neural network model based on binary equilibria data. Neural Network Modeling ANN architecture is determined by the way in which nodes are mutually interconnected. Here, the most common type of architecture, the feedforward network, which is also called the back-propagation network (Figure 1A), has been used. The inputs of the network are p1, p2, ..., pi, the outputs of the network are a21, a22, ..., a2k, and the corresponding targets (i.e., experimental values of outputs) are d1, d2, ..., dk. Figure 1B shows the signal flow for a single set of data. A hyperbolic tangent hidden layer with j nodes and a linear output layer have been used. W1 and W2 are weight matrixes and b1 and b2 are bias vectors. Cybenko23 proved that an ANN with a single hidden layer could uniformly approximate any continuous function with arbitrary accuracy. The number of neurons in the hidden layer is adjusted according to the fitting error during the training. However, as this number is increased, the network starts to lose its generalization, i.e., it can recall the training set more precisely but the error on new data starts increasing. A detailed discus-
Figure 2. Comparison of the model prediction of solute loading on NaY-type zeolite (qmodel) with the experimental values that were used for training the network (qexp): (A) 2,6-DMN and (B) 2,7DMN. Solid squares (9) represent data points, and lines (s) represent the relationship qmodel ) qexp.
sion on the determination of the effective number of parameters (weights and biases) in the Bayesian framework has been presented by Foresee and Hagan.24 The weights and biases are initialized by generating random numbers. The network is trained until convergence is reached. The network training with different initial values of the number of nodes in the hidden layer showed the same effective number of parameters (i.e., 32). The same degree of accuracy in recalling the training data set during different training session (i.e., with different initial values of weights and biases) shows the robustness of the model. In addition, the time required in training the network is insignificant. Thus, the optimized values of weights and biased are not to be recorded and the network can always be trained before using it for the prediction. This helps to easily upgrade the model when new experimental data are available. In the present study, Bayesian regularization, in combination with Levenberg-Marquardt training,24 was used to avoid overfitting and to improve generalization profile of the network. Complex hypotheses are automatically self-penalizing under Bayes’ rule. In the Bayesian framework, the weights and biases are assumed to be random variables with specified distributions. The regularization parameters are related to the unknown variances associated with these distributions. Four inputssC2, C3, q2′, and q3′swere chosen for the neural network modeling of the ternary adsorption system, where 2,6- and 2,7-dimethylnaphthalene iso-
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Figure 3. Comparison of the model prediction of solute loading on NaY-type zeolite (qmodel) with the experimental values that were never used during the training of the network (qnewexp): (A) 2,6DMN and (B) 2,7-DMN. Solid squares (9) represent data points, and lines (s) represent the relationship qmodel ) qnewexp.
eters (weights and biases) should reach constant values when the network has converged. It was found that five neurons in the hidden layer give excellent fitting accuracy (Figure 2). The effective number of parameters was 32. Even if the number of neurons in the hidden layer was increased to 20, in which case, the total number of parameters becomes 142, the training result showed the same effective number of parameters (i.e., 32). Thus, it was ensured that there was no overfitting. The trained network produced outputs with a mean and standard deviation of zero and unity, respectively. Therefore, these outputs were converted back to the same units that were used for the original targets. The trained network was used to predict the ternary adsorption equilibria of 2,6- and 2,7-DMN dissolved in ScCO2 on NaY-type zeolite at 308.2 K, 14.8 MPa, and different DMN concentrations (C2 and C3). The network was simulated using binary adsorption equilibrium data. The outputs (i.e., equilibrium loading of solutes) were compared with the experimentally observed values (Figure 3). Ideally, all the points should fall on the straight line passing through the origin and having a slope of unity. Although the network was kept unaware of these data, the prediction is very precise. This validates the model. Hence, the model can be used to predict ternary adsorption equilibria of the system considered in the present study. However, it is only the first step toward the neural network modeling of mixture adsorption equilibria in supercritical fluids. As mentioned previously, the mixture adsorption equilibrium data are very few, and data for different systems at various temperatures, pressures, and solute concentrations are needed to explore the wider applicability of ANN modeling in this area. Literature Cited
mers dissolved in ScCO2 was adsorbed onto NaY-type zeolite. C2 and C3 are, respectively, the concentration of 2,6-DMN and 2,7-DMN in the mobile phase (ScCO2) of the ternary system; q2′ and q3′ are, respectively, the stationary phase loading of 2,6-DMN and 2.7-DMN in binary systems (i.e., pure solute-ScCO2-NaY-type zeolite) at the same temperature, total pressure, and concentration of the ternary system. The parameters q2 and q3, representing the stationary phase loading of 2,6DMN and 2,7-DMN, respectively, were taken as outputs. The Neural Network Toolbox (version 4) of MATLAB25 was used for simulation. Before training, it is useful to scale inputs and targets so that they always fall within a specified range. Here, inputs and targets were normalized so that they had a mean and standard deviation of zero and unity, respectively. Iwai et al.3 has tabulated the ternary and binary adsorption equilibrium data for the system 2,6- and/or 2,7-DMN dissolved in ScCO2 onto NaY-type zeolite at 308.2 K. C2, C3, q2, and q3 were taken from the ternary data, whereas q2′ and q3′ were obtained by interpolating the tabulated binary data for different concentrations. Table 1 contains the input data used for training and prediction. The network was trained using the adsorption equilibrium data at a temperature of 308.2 K, pressures of 9, 12, and 19.8 MPa, and different concentrations. The training was performed in batch mode. The network is trained until convergence is reached. The sum squared error, the sum squared weights, and the effective number of param-
(1) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, 1987. (2) Myers, A. L.; Monson, P. A. Adsorption in porous materials at high pressure: theory and experiment. Langmuir 2002, 18, 10261. (3) Iwai, Y.; Higuchi, M.; Nishioka, H.; Takahashi, Y.; Arai, Y. Adsorption of supercritical carbon dioxide + 2,6- and 2,7-dimethylnaphthalene isomers on NaY-type zeolite. Ind. Eng. Chem. Res. 2003, 42, 5261. (4) Tan, C. S.; Liou, D. C. Supercritical regeneration of activated carbon loaded with benzene and toluene. Ind. Eng. Chem. Res. 1989, 28, 1222. (5) Srinivasan, M. P.; Smith, J. M.; McCoy, B. J. Supercritical fluid desorption from activated carbon. Chem. Eng. Sci. 1990, 45, 1885. (6) Madras, G.; Erkey, C.; Akgerman, A. Supercritical fluid regeneration of activated carbon loaded with heavy molecular weight organics. Ind. Eng. Chem. Res. 1993, 32, 1163. (7) Lin, W. F.; Tan, C. S. Separation of m-xylene and ethylbenzene on silicate. Sep. Sci. Technol. 1991, 26, 1549. (8) Uchida, H.; Iwai, Y.; Amiya, M.; Arai, Y. Adsorption behaviors of 2,6- and 2,7-dimethylnaphthalenes in supercritical carbon dioxide using NaY-type zeolite. Ind. Eng. Chem. Res. 1997, 36, 424. (9) Cross, W. M., Jr.; Akgerman, A. Adsorptive separations using supercritical frontal analysis chromatography. AIChE J. 1998, 44, 1542. (10) Xiaoning, Y. Thermodynamic modeling of solute adsorption equilibrium from near critical carbon dioxide. J. Colloid Interface Sci. 2004, 273, 362. (11) Jha, S. K.; Madras, G. Modeling of adsorption equilibria in supercritical fluids. J. Supercrit. Fluids 2004, 32, 161-166. (12) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984.
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 7041 (13) Zhou, W.; Anitescu, G.; Tavlarides, L. L. Polychlorinated biphenyl partition equilibrium between St. Lawrence River sediments and supercritical fluids. J. Supercrit. Fluids 2004, 29, 32. (14) Radke, C. J.; Prausnitz, J. M. Adsorption of organic solutes from dilute aqueous solution of activated carbon. Ind. Eng. Chem. Fundam. 1972, 11, 445. (15) Minka, C.; Myers, A. L. Adsorption from ternary liquid mixtures on solids. AIChE J. 1973, 19, 453. (16) Costa, E.; Sotelo, J. L.; Calleja, G.; Marron, C. Adsorption of binary and ternary hydrocarbon mixtures on activated carbon: experimental determination and theoretical prediction of the ternary equilibrium data. AIChE J. 1981, 27, 5. (17) Talu, O.; Zwiebel, I. Multicomponent adsorption equilibria of nonideal mixtures. AIChE J. 1986, 32, 1263. (18) Baughman, D. R.; Liu, Y. A. Neural Networks in Bioprocessing and Chemical Engineering; Academic Press: San Diego, CA, 1995. (19) Fullana, M.; Trabelsi, F.; Recasens, F. Use of neural net computing for statistical and kinetic modeling and simulation of supercritical fluid extractors. Chem. Eng. Sci. 2000, 55, 79.
(20) Fu, G.; Zhu, H. Neural network modeling of supercritical adsorption of methane on activated carbon. Carbon 2003, 41, 2653. (21) Carsky, M.; Do, D. D. Neural network modeling of adsorption of binary vapour mixtures. Adsorption 1999, 5, 183. (22) Yang, M.; Hubble, J.; Lockett, A. D.; Rathbone, R. R. A neural network model for prediction of binary adsorption using single solute and limited binary solute adsorption data. Sep. Sci. Technol. 1996, 31, 1259. (23) Cybenko, G. Approximation by superpositions of a sigmoidal function. Math. Control Signal 1989, 2, 303. (24) Foresee, F. D.; Hagan, M. T. Gauss-Newton approximation to Bayesian regularization. In Proceedings of the 1997 International Joint Conference on Neural Networks, 1997; p 1930. (25) Demuth, H.; Beale, M. In Neural Network Toolbox User’s Guide; Mathworks: Natick, MA, 2000; pp 5-1-5-75.
Received for review October 13, 2004 Revised manuscript received July 4, 2005 Accepted July 13, 2005 IE049010P
