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GENERAL RESEARCH New Hybrid Neural Network Model for Prediction of Phase Equilibrium in a Two-Phase Extraction System Li Gao and Norman W. Loney* Department of Chemical Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102
A novel approach to modeling prediction of phase equilibrium is presented. The method, evolutionary polymorphic neural network (EPNN), is developed by the authors on the basis of artificial neural networks and evolutionary computing. The system poly(ethylene glycol) (PEG)/ potassium phosphate/water at pH ) 7 was selected to demonstrate the performance of the model. The results were favorable as compared to a traditional neural network modeling approach and the experimental data set. Seven distinct data sets of varying PEG molecular weights were used in this work. Of the seven, five were used for training, while the remaining two were employed as the test cases. Following the training, a networked symbolic equation system evolved, which, in addition to reproducing the data, can also be used to improve understanding of the phase diagram mechanism through the discovered parameters. 1. Introduction In macromolecules and biological materials processing industries, aqueous two-phase systems (ATPS) have been applied in the separation and purification of various substances. The primary advantages of ATPS handling of materials are biocompatible environments, economical operation and scale-up, and adjustable factors to manipulate target product partitioning.1 However, while there are many factors available to manipulate the partition, it is difficult to correlate and model the phase equilibrium and partition because of the highly complicated interactions between those factors.2 A number of thermodynamic models have been developed and applied to describe the mechanism of phase separation. Flory and Huggins theory3 was applied successfully in modeling the poly(ethylene glycol) (PEG)/ dextran systems.4 Later, models based on statistical thermodynamics, such as UNIQUAC or UNIFAC,5,6 were developed and applied to correlate and extrapolate phase diagrams with system parameters by fitting other similar phase equilibrium data.7 However, there is a common drawback in predicting a phase diagram using the above models. Generally speaking, a thermodynamic model requires a set of specific parameters which is only valid exclusively for a particular system. For example, the model of phase equilibrium with parameters measured from the Dextran 500/PEG 1000 system may not be able to forecast a phase equilibrium for a Dextran 500/PEG 3400 system. Therefore, to apply those thermodynamic models in laboratory or industrial process design, a huge parameter database for numerous systems is required. However, the special equipment to determine these parameters may not be available in every laboratory or factory.2 Since the mid-1990s, to overcome the limitations of traditional thermodynamic models, there have been a
number of efforts applying neural networks to the predictions of phase equilibrium data, such as the prediction of a PEG/potassium phosphate/water system.2 The advantages of these neural network approaches are stated as nonparametric models.8,9 Structurally, they are adjustable and adaptable for different phase systems without reevaluation of thermodynamic parameters. However, there are limitations to these approaches. The most cited one is the nonparametric nature or “blackbox” operation, which is a difficulty in establishing efficient structures of neural network and training methods.10 While the trained model gives great performance on the testing data sets, it does not provide information about the potential mechanism of the system. Described in this paper is an alternative model that can overcome the limitations of both traditional thermodynamic and neural network models. The proposed model does not require measurement of thermodynamic parameters for each specific system. In addition, it can produce empirical symbolic formulas for a set of systems. The produced formulas can be used to estimate the system mechanism. Furthermore, the polymorphic nature of the proposed model results in more efficient and flexible model structural determination. 2. Model Description Evolutionary polymorphic neural network (EPNN) is based on recent artificial intelligence (AI) techniques, such as the artificial neural network (ANN) and genetic programming (GP). However, it must be stipulated that GP is very different from genetic algorithms (GAs). Also, while the proposed model has features from GP, it is not a replication of GP. In the proposed model, GAs are not employed.
10.1021/ie010004s CCC: $22.00 © 2002 American Chemical Society Published on Web 12/08/2001
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Figure 2. Sample EPNN individual network.
above functional form stands for is
N0 ) N1 )
N2 (0.2345) X + 0.8562
N2 ) N2X + 0.1745 ln(N1)
Figure 1. Typical EPNN system composition.
In GP, each evolving individual is a computing formula, while, in GAs, each individual is a fixed-length string or vector of numbers. Generally speaking, the string representation in GAs encodes each target process parameter as a variable in a multiple-dimensional vector. In contrast, in GP, each individual represents a computing formula in various tree forms.11 Further discussions on the differences between GA and GP can be found elsewhere.11-14 The EPNN modeling system inherits two features of GP. One feature is the evolution of a randomly or purposely generated population of individuals. The other feature is the tree representation of symbolic formulas in each individual. Descriptions about how to represent a symbolic formula in a tree form and how to measure the tree can be found elsewhere.13-15 2.1. System Structure. A typical composition of a complete EPNN modeling system is shown in Figure 1. When Figure 1 is worked vertically downward, a complete EPNN system consists of necessary methods, such as evolutionary algorithms, tuning methods, and individual survivor strategies. The EPNN system also includes the target data structure: a population of EPNN individuals. The population is further divided into two subpopulations: a gene pool and a mutation pool. The gene pool holds candidate solutions while the mutation pool acts as an auxiliary during evolution. Each pool is a collection of EPNN individuals. Each EPNN individual is a highly interconnected network of symbolic formulas. Each of such an EPNN network is defined with four sets of symbols: a function set, an input variables set, an output variables set, and intermediate nodes (neurons). Each intermediate or output node is associated with a symbolic formula. Figure 2 shows an example of an EPNN network with two intermediate nodes and one output node. The corresponding functional form of the network is
N0 ) f0(N0, N1, N2, input) N1 ) f1(N2, input) N2 ) f2(N1, N2, input)
X + N0 N1 + N2
(1)
One of the many possible symbolic formulas that the
(2)
The function set here is {+, -, ×, ÷, and ln}. The input variable is X, and the output variable is N0. The intermediate nodes are N1 and N2. The recursive functions (N0 and N2) represent feedback linkage. The purpose of feedback linkage is to provide state variables and to model potential delay responses in the target system. Each individual in an EPNN population encodes a potential solution for the given problem by representing the potential solution with a system of empirical equations. In the two-phase extraction system, the size of an EPNN population is 100; the size of a mutation pool is 50. The function set consists of the operators {+, -, ×, ÷, ln, and exp}. There are four intermediate nodes and four output nodes. The inputs to the model are a normalized PEG molecular weight, total concentration of potassium phosphate, and total concentration of PEG. 2.2. Modeling Algorithms. Similar to an existing ANN and GP, EPNN must be trained prior to data prediction. A typical training algorithm for an EPNN system is shown in Figure 3. At the beginning of the training step, every individual network in the population is randomly generated. In each individual, the nodes are initialized with random constants but no linkage to each other. Upon detailed examination, the values of the initial random constants do not affect the final training result in given problems. This holds true in the discussed PEG system as well as in other experiments. There are three levels of mutation in the training method of an EPNN system: elementary, intermediate, and macro mutations. Elementary mutation is the lowest level of mutation. It deals with single-node mutation, such as the addition, change, swap, and deletion of a single constant or a single linkage. Figure 4 shows some typical operations of elementary mutation in tree form. Intermediate mutation deals with the swap, replacement, and deletion of a subtree of an EPNN formula, while macro mutation deals with the swap and replacement of whole intermediate nodes. Replacement means to replace the old nodes or constants by newly randomly generated ones. In addition to mutations, there are two levels of crossover. One is the swap of a subtree during intermediate mutation. An
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Figure 5. Typical crossover operation in tree representation.
Figure 3. Typical training algorithm in EPNN modeling.
Figure 4. Some typical operations during elementary mutation.
example of a crossover operation between two treerepresented formulas is shown in Figure 5. Such crossover occurs during intermediate mutations. The corresponding system of equations (eq 3) is changed to the system in eq 4. The other level of crossover is an
y1 ) ab + 3 a y2 ) + b 2 a y1 ) ab + 2 y2 ) 3 + b
(3)
(4)
individual level crossover, which deals with the exchange of one whole formula tree between different individual networks. An example of an individual level of crossover is shown in Figure 6. It can be seen from Figure 6 that two intermediate nodes were exchanged during the operation. The three mutation levels mentioned result in the polymorphic nature of an EPNN network. All of the above mutations have an equal chance to occur. In addition, the individual level crossover happens for every individual in the mutation pool before mutation. Two individuals are randomly selected during each such crossover. After crossover and mutation, fitter individuals in the mutation pool are then inserted into the gene pool according to their fitness (selective insertion in Figure 3). Then, the whole population in the gene pool will be sorted again. The mutation pool here acts as an auxiliary population and will be completely recreated each time after selection from the gene pool. The criterion of an individual survivor strategy is defined by a comprehensive fitness function. This function consists of two components: a specific life span (LP) for each individual and a combined minimum error (CME) function. The life span is used as a penalty function for each individual and is a variable during the training step (eq 5). It is a function of the diversity of the whole population. The more diverse the population is, the longer life span each individual will have. The life span is defined as
LPi,j ) K(Gj - Hi)/Dj
(5)
where K is a proportional constant, Hi is the generation number when the ith individual was created, Gj is the jth generation number, and Dj is the diversity value of the jth population. Diversity is evaluated by the standard deviation of the CME function of the gene pool. The CME function is defined as
CME ) (R1 × R2 × ... × Rn)1/n
(6)
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Figure 6. Typical crossover operation at the EPNN individual level.
where Ri is
Ri )
x
m
(uj - vj)2 ∑ j)1 m
i ) 1, ..., n
(7)
and where m is the number of data points in each case, u stands for experimental data, and v stands for model prediction. n is the number of outputs in the PEG system. For the studied aqueous two-phase extraction system, m is the number of data points in one data set, which is 8. The value of n is 4 for the top PEG, top salt, bottom PEG, and bottom salt concentrations. Once the desired fitness has been reached, the training algorithm is stopped. The fittest individual network is selected as the final solution for the system. The next step is testing the prediction from the fittest individual network for both interpolation and extrapolation. In the aqueous two-phase extraction system, all of the data from PEG 600, 1500, 3400, 8000, and 20 000 are used to train the model until a satisfactory prediction precision is reached. Following the training is the testing of the prediction of the model for both interpolation and extrapolation. PEG 1000 and PEG 400 cases are used in this step. Each data set consists of eight data points. A total of 40 data points were used for the EPNN system training in a single run. The remaining 16 data points were used to test the EPNN system performance in a different run. The number of coefficients in an EPNN network formula is changeable during training because of the polymorphic nature of EPNN networks. Therefore, unlike a traditional neural network, EPNN can adapt to different systems without overfitting the data. 2.3. Major Modeling Parameters. Prior to the application of an EPNN system, modeling parameters must be determined. The parameters that require the
most attention are the function set, depth of the tree and its limitations, and number of total nodes. However, the number of input variables and number of output variables are predetermined by each specific case. In addition, all of the input and output data should be normalized to a reasonable range. In the presented case, all of the data are normalized to the range [-1; 1] by rescaling. Other parameters, such as the mutation rate and crossover percentage, are fixed within the algorithm using typical values in the literature.11 They are not critical to the modeling results and will not be discussed here. To reproduce the model in parts or in total, two special considerations need to be addressed. The first one is the depth of a formula tree. Because each formula in an EPNN network is using a tree structure representation, the maximum depth of a tree needs to be restricted during evolution to avoid excessively large and complicated unstable functions. However, if the restricted maximum depth of a tree representation is too small, the EPNN cannot converge or will converge very slowly. In the presented aqueous two-phase extraction system, the maximum depth of any formula tree is limited to 100. This number is generated from three trials according to their fitness convergence speeds and the efficiency to converge to a global maximum fitness. The second consideration is the appropriate design of the function set. For the two-phase extraction system, the function set is {+, -, ×, ÷, ln, and exp}. In other cases, it could include sine, cosine, or other periodical functions as needed for periodical or periodical-like experimental data. Therefore, before the training of an EPNN system is started, the set of functions must be properly designed. In the aqueous two-phase extraction system, the maximum depth of any formula tree is restricted to 100. In summary, an EPNN system consists of a randomly or purposely generated population of individuals. Each individual in the EPNN system is a highly interconnected network of symbolic formulas. Each of such an
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EPNN network is defined with four sets of symbols: a function set, an input variables set, an output variables set, and intermediate nodes (neurons). Every distinct symbolic formula is associated with one intermediate or output node. In traditional neural network modeling, the linkage structure must be determined prior to training and application of the model. This can be accomplished either manually or by applying a number of trials which demands a significant amount of time, substantial effort, and experience.10,16,17 However, EPNN can form a linkage structure dynamically during the training process through evolution, which reduces the time and effort in structural determination. During evolutionary computing (i.e., GP), only one mutation is employed,11,12 while EPNN has three levels of mutations. An increase in the number of mutation levels results in more adaptable and efficient structures. In more recent efforts,18-20 GAs were applied together with ANNs to reduce the inefficiency of ANN structure determination. However, they still cannot overcome the blackbox nature of ANN. The proposed model differs from the two separate modeling methods as presented by ref 18, which uses ANN as a forward-modeling approach and GA as a reverse-modeling method. EPNN is also different from those methods, which use GA only as a way to refine ANN parameters.19-20 The proposed EPNN network differs from these approaches fundamentally, because it is a native combination of GP and ANN. Therefore, unlike most ANNs (or ANNs with GAs), EPNN does not work as a blackbox and can produce formulas during modeling. In the following section, the proposed EPNN approach is used to forecast the behavior of an ATPS. Also, further comparisons are made with a traditional neural network model, which is a feedforward back-propagation neural network with two hidden layers and four subnets.2
Figure 7. Trained EPNN network structure for a PEG system.
ing symbolic networked formulas of the fittest EPNN network are
d32 N0 ) 1.083d2 + 0.2875 N1 )
4. Results and Discussion During the simulation, 20 000 iterations (generations) were evolved before the training program was automatically stopped. Then, the fittest individual network was selected as the solution (Figure 7). The correspond-
exp(1.2042N5)
N2 ) f1(N1, N4, N5, N6, d3) N3 )
d3(1.4497) d2 + 0.64995
N4 ) 0.7292 N 5 ) N6 + N6 )
3. Data Preprocessing The phase equilibrium data of the PEG/potassium phosphate ATPS (pH ) 7) were obtained from experiments reported in the literature.2,21 The total weight proportions of PEG and potassium phosphate in the twophase system are two inputs to the EPNN system. In addition, the normalized PEG MW2 was utilized as a third input. During the training step of the EPNN model, all of the data from PEG 600, 1500, 3400, 8000, and 20 000 are used to train the model until satisfactory prediction precision is reached. Following the training is the testing of the prediction of the model for both interpolation and extrapolation. PEG 1000 and PEG 400 cases are used in this step. Each data set consists of eight data points. A total of 40 data points were used for the EPNN system training in a single run. The remaining 16 data points were used to test the EPNN system performance in a different run.
N7(0.1)
N7 ) exp
{
{
d3 - 0.3779 0.1544
}
exp{(N2 + d3)(1.8221) + ln(d3/d2)} d1 1.9536 N0(0.9135)
}
1 (N6 - 0.4109)(N2 - 0.6260) N6 exp N3
(
)
(8)
where N0, N1, N2, and N3 are the output nodes. f1 is
f1(N1, N4, N5, N6, d3) ) 0.1N4 ln(exp(f2) + (0.8875 - d3)(0.1341)) and f2 is
f2 ) (0.8124 exp{(N6 - 0.9816 exp(0.7405 + N1) + d3)} + 0.917)/(d3 - exp(N5) ln(N4)) From eq 8 and Figure 7, we can see that the nodes in the network are interconnected with each other by backlinks (feedbacks). Those back-links were formed automatically during the training step. The degree of complexity of an evolved network is related to the precision of the given data. As indicated in the experimental data,2,21 bottom PEG (N1) and top salt (N2) concentrations are minor components in their respective phases. This may result in a higher level of error or uncertainty because of the magnitude of these concentrations.2,21 Therefore, the EPNN network, in
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Figure 8. Top-phase PEG concentration study.
striving to maintain a constant error, forms a more complicated structure. In this work, the intermediate nodes are not germane to the interpretation of the results. Examination of eq 8 reveals some interesting and useful information about the PEG/potassium phosphate/ water system: i. The concentration of PEG in the top phase (N3) is approximately a function of the total PEG (d3) and total salt (d2) concentrations. ii. The concentration of salt in the bottom phase (N0) is approximately independent of the molecular weight of PEG (d1) and, thus, only depends on the total concentration of PEG (d3) and potassium phosphate (d2). iii. The concentration of the minor component in each phase (N1 and N2) is more complicated and interconnected than the major component concentration (N0 and N3) in their respective phases. Albertsson1 showed that, for a constant pH (pH ) 7) and a proper experimental salt concentration range, the PEG concentration in both phases remains constant with respect to the PEG molecular weight. According to Figure 8, the experimental results of Albertsson are approximately confirmed by the EPNN model. In particular, we do see that the PEG concentration in the top phase is dependent on the total PEG concentration. However, the top-phase PEG concentration can be more accurately predicted (Figure 9) by tuning the EPNN model with different initial system conditions. Now, the top-phase PEG concentration, N3, depends on all three variables:
N3 ) 1.196d2 + 1.149d3 + 2.0149d1d2 + 0.1142d1d3 - 0.1909 (9) During the tuning step, only one total node was used, which is the output node. Fifty generations are evolved to produce the improved approximation. The EPNN training results can be tuned with only minimum changes in the initial conditions. Therefore, unlike traditional neural networks, which demand a significant effort to change network structures before tuning,10 the network structure of EPNN is more flexible and adaptable. This is an advantage of the EPNN's polymorphic nature over traditional neural networks. As indicated in the experimental data,21 the product of d1d2 or d1d3 is really small and only second-ordercorrelated to N3, while d2 and d3 are both larger and
Figure 9. Tuned top-phase PEG predictions.
Figure 10. Phase diagram (training results).
first-order-correlated to N3. Therefore, according to eq 9, N3 is less sensitive to d1 (MW of PEG). The observed result of independence for N3 (top PEG concentration) still holds true in approximation. Figure 10 shows the training results after the tuning. The top left part of the figure shows the concentrations in the top phase, and the bottom right part shows the concentrations in the bottom phase. Figure 11 shows the associated training errors. Both Figures 10 and 11 show good agreement with the experimental data21 over the five training data sets. Figure 12 shows the prediction results. Similar to Figure 10, the top left part of the figure shows the concentrations in the top phase, and the bottom right part shows the concentrations in the bottom phase. Figure 13 shows the associated prediction errors. Again, both Figures 12 and 13 show good agreement with the experimental data21 over the two testing data sets. The standard deviations of EPNN prediction are 0.0114 w/w (PEG 400) for extrapolation and 0.003 02 w/w (PEG 1000) for interpolation, while the traditional neural network approach deviations are respectively 0.0118 w/w (PEG 400) and 0.0085 w/w (PEG 1000).2 In addition to producing better results, EPNN also evolved empirical formulas (eqs 8 and 9), which can be applied over the whole range of the studied system. The formulas produced by EPNN not only provide valuable information for further theoretical study but can also
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Figure 11. EPNN training errors.
of constants in EPNN networks will converge to the same or similar forms of functions in separate runs until the training process ends. Therefore, the EPNN system is not sensitive to differences in initial constant values of the EPNN population. However, if there exists a significant larger noise in one or more data sets in the whole data composition, the EPNN system will probably fail to converge to a satisfactory level of prediction on these data sets. Therefore, the precision of the training data is critical to successful and efficient modeling using EPNN. It should also be pointed out that although only the fittest individual is selected as the final solution for the target system, other candidate solutions with similar fitness are also archived by EPNN. Because of the stochastic nature of EPNN, multiple sets of suitable mathematical equations with different forms and similar fitness may happen, and the user can retrieve the sets of candidate solutions from the archive. In the discussed ATPS system, EPNN has archived sets of reasonable equations, though only the fittest was examined in this work. Further work will involve the exploration of the multiple sets of possible solutions during the evolution process. The EPNN method has also been applied in other systems, such as a published work applying EPNN in dynamic reaction system modeling and a model reduction for an existing bioremediation process.22 However, discussion of EPNN in other fields is beyond the scope of this paper. 5. Conclusions
Figure 12. Phase diagram (prediction results).
Figure 13. EPNN prediction errors.
be used as empirical equations for ATPS process optimization. Furthermore, the EPNN system uses only eight neurons while the traditional feedforward backpropagation neural network consists of about 40 neurons. Too many neurons can introduce an overfitting of data and yield a poor generalization of the PEG system.2 Because of the polymorphic and evolutionary nature of the EPNN system, the initially randomized values
EPNN is a novel approach and concept developed by the authors. EPNN is capable of modeling chemical, biochemical, and physical processes. This approach has its basis in modern AI, especially in neural networks and evolutionary computing. EPNN can perform networked symbolic regressions for input-output data, while providing information about both the structure and complexity of a process during its own evolution. EPNN networks with a relatively small number of neurons do overcome the limitations of both traditional thermodynamic and neural network models. In addition to the precision of prediction, EPNN can also produce meaningful formulas. These formulas can be used as empirical models for the given system and for further theoretical examination and process optimization. The EPNN system can also be used for data prediction tuning, in which case only a minimum number of initial system conditions needs to be adjusted before tuning. The network structure of EPNN is more flexible and adaptable than traditional neural networks. The EPNN system is not sensitive to differences in initial values of the EPNN population. However, precision of the training data is critical to successful and efficient modeling. Multiple candidate solutions with similar fitness are archived by EPNN. The archive can be used for further study. Literature Cited (1) Albertsson, P.-A. Partition of Cell Particles and Macromolecules, 2nd ed.; Wiley-Interscience: New York, 1971. (2) Kan, P.; Lee, C.-J. A neural network model for prediction of phase equilibria in aqueous two-phase extraction. Ind. Eng. Chem. Res. 1996, 35, 2015-2023.
Ind. Eng. Chem. Res., Vol. 41, No. 1, 2002 119 (3) Flory, P. J.; et al. Statistical thermodynamic of chain molecule liquids. i. an equation of state for normal and para (n hydrocarbons. J. Am. Chem. Soc. 1964, 86, 3507. (4) Gustaffsson, A.; et al. The nature of phase separation in aqueous two-phase systems. Polymer 1986, 27 (21), 1768. (5) Abrams, D.; Prausnitz, J. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 1975, 21 (1), 116. (6) Fredenslund, A.; Jones, R.; Prausnitz, J. Group-contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE J. 1975, 21 (6), 1086. (7) Gao, Y.-L.; Peng, Q.-H.; Li, Z.-C.; Li, Y.-G. Thermodynamics of ammonium sulfate-poly(ethylene glycol) aqueous two-phase systems. Part 2. Correlation and prediction using extended UNIFAC equation. Fluid Phase Equilib. 1991, 63 (1-2), 173-182. (8) Tsoukalas, L. H., Uhrig, R. E., Eds. Fuzzy and neural approaches in engineering; Wiley-Interscience: New York, 1997. (9) Caudill, M.; Butler, C. Understanding Neural Networks: Computer Explorations; MIT Press: Cambridge, MA, 1992; Vols. 1 and 2. (10) Doherty, S.; Gomm, J.; Williams, D. Experiment design considerations for non-linear system identification using neural networks. Comput. Chem. Eng. 1997, 21 (3), 327-346. (11) Koza, J. R. Genetic Programming: On the Programming of Computers By Means of Natural Selection; MIT Press: Cambridge, MA, 1992. (12) Fogel, D. Evolutionary Computation: Toward a New Philosophy of Machine Intelligence; IEEE Press: Piscataway, NJ, 1995. (13) McKay, B.; Willis, M.; Barton, G. Steady-state modeling of chemical process systems using genetic programming. Comput. Chem. Eng. 1997, 21 (9), 981-996.
(14) Greef, D.; Aldrich, C. Empirical modeling of chemical process systems with evolutionary programming. Comput. Chem. Eng. 1998, 22 (7-8), 995-1005. (15) Aho, A. V.; et al. Data Structures and Algorithms; AddisonWesley: Reading, MA, 1987. (16) Nelson, M. M.; Illingsworth, W. A Practical Guide to Neural Networks; Addison-Wesley: Reading, MA, 1990. (17) Bailey, D.; Thompson, D. How to develop neural networks. AI Expert 1990, 5, 38. (18) Ghosh, P.; Sundaram, A.; et al. Integrated product engineering: A hybrid evolutionary framework. Comput. Chem. Eng. 2000, 24, 685-691. (19) Zhao, W.; Chen, D.; Hu, S. Optimizing operating conditions based on ANN and modified GAs. Comput. Chem. Eng. 2000, 24, 61-65. (20) Gao, F.; Li, M.; et al. Genetic algorithms and evolutionary programming hybrid strategy for structure and weight learning for multilayer feedforward neural networks. Ind. Eng. Chem. Res. 1999, 38, 4330-4336. (21) Lei, X.; Diamond, A. D.; Hsu, J. T. Equilibrium phase behavior of the poly(ethylene glycol)/potassium phosphate/water two-phase system at 4 c. J. Chem. Eng. Data 1990, 35, 420-423. (22) Gao, L. Evolutionary polymorphic neural networks in chemical engineering modeling. Ph.D. Thesis, New Jersey Institute of Technology, Newark, NJ, 2001.
Received for review January 8, 2001 Revised manuscript received October 3, 2001 Accepted October 16, 2001 IE010004S
