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Modeling the n‑Hexane Isomerization over Iron Promoted Pt/ WOx‑ZrO2 Catalysts Using Artificial Neural Networks Martha Leticia Hernández-Pichardo* and Ricardo Macías-Salinas ESIQIE, SEPI-Departamento de Ingeniería Química, Instituto Politécnico Nacional, Zacatenco, México, D.F. 07738 S Supporting Information *
ABSTRACT: An artificial neural network approach was presently used to model the hydroisomerization reaction of n-hexane over platinum supported on tungstated zirconia (Pt/WOx-ZrO2) catalysts doped with different amounts of iron. Four different multilayer feed-forward neural network arrangements were then devised and trained using previously reported experimental catalyst activity in terms of selectivity (S) and yield (Y) of 2,2dimethylbutane measured at several Fe/W weight ratios and surfactant/zirconia molar ratios (SMR). The performance of the four neural networks during the training process was acceptable despite the fact that a limited database was used for such a purpose; the catalyst synthesis variables chosen as neural network inputs (SMR, %W, and %Fe) played a very important role in successfully correlating the catalyst activity (in terms of S and Y of 2,2-dimethylbutane) in all cases. The predictive capabilities of the trained neural networks were further verified by computing some selectivities and yields of 2,2-dimethylbutane not considered in the training process. The agreement between predicted and observed catalyst activity data was highly acceptable thus demonstrating the abilities of the four neural networks (particularly, the 3−3−2−2 arrangement) in predicting suitable values of catalyst activity within the present “gray-box” modeling work.
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INTRODUCTION
efficient and rapid heterogeneous catalyst discovery and optimization.7−9 On the other hand, the design and optimization of catalysts using traditional methods based on first-principles (theoretical modeling) are cumbersome to apply since the complexity and the unpredictable diversity space that one needs to experimentally explore is quite large, particularly in heterogeneous systems. In this context, the use of an artificial neural network (ANN) approach seems to be a promising and yet little explored technique during the determination of optimum conditions of heterogeneous catalysts aimed for maximum performance. Moreover, the successful application of the ANN approach in the modeling of chemical reactions over heterogeneous catalysts has been already demonstrated and reported by various investigators. For example, Rodemerck et al.10 used an ANN in combination with a genetic algorithm during their discovery and optimization of new solid catalytic materials for the oxidative dehydrogenation of propane to propene. Their work effectively showed the combined application of a genetic algorithm for searching the optimal catalyst composition, and an ANN for modeling the entire parameter space in an attempt to identify compositional areas where high propene yields can be expected. Further, Song et al.11 also used an ANN to design an optimum Ni/Al2O3 catalyst for the production of hydrogen by the catalytic reforming of
The gradual increase of pollutant levels caused by volatile hydrocarbons is nowadays a matter of great concern, urging environmental regulations for the quality of motor fuels to become progressively more rigorous. In this context, heterogeneous catalytic isomerization of n-alkanes is becoming increasingly important in the quality improvement of such fuels by increasing the added value of the light naphtha streams thus obtaining a higher octane gasoline.1 The type of catalysts used for hydroisomerization reactions of light paraffins are bifunctional solids comprising mainly platinum for the metallic function and several acid solids for the acidic function such as chlorinated alumina, sulfated zirconia, or tungstated zirconia. One of the most studied reactions for the normal C5−C6 isomerization is the one using platinum supported on tungstated zirconia;2−6 in particular, the n-hexane isomerization over this type of catalyst mainly produces 2-methylpentane (2MP), 3-methylpentane (3-MP), 2,3-dimethylbutane (2,3DMB), and 2,2-dimethylbutane (2,2-DMB) as well as some cracking products,3 where the biramified products exhibit the higher octane number. In general, the high activity and selectivity of the aforementioned catalyst depend not only on the composition of its materials but also on its various synthesis methodologies thus making the development and testing of this type of catalysts, to some extent, an inefficient and timeconsuming process. Nevertheless, there exist some approaches to overcome this limitation such as the combinatorial and highthroughput experimentation (HTE) which has been widely recognized as a relatively new scientific technology allowing an © 2016 American Chemical Society
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May 11, 2016 June 17, 2016 June 21, 2016 June 21, 2016 DOI: 10.1021/acs.iecr.6b01821 Ind. Eng. Chem. Res. 2016, 55, 8883−8889
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each simulated neuron should be able to (1) receive signals from other neurons, (2) sum these signals, (3) transform this sum, usually using a suitable transfer function, and (4) transmit the result to other neurons. One of the transfer functions more extensively used during ANN development is the sigmoidal function which is a monotonic, continuously differentiable and bounded function:
crude ethanol by establishing inter-relationships between preparation methods, nickel loading, catalyst characteristics, and performance. Using three ANN topologies, the authors were finally able to find the optimum catalyst conditions that ensured the maximum hydrogen-production performance. In other related works, Günay and Yildirim12,13 analyzed the experimental data of the low-temperature selective CO oxidation over two catalysts (promoted Pt−Ce−Co/Al2O312 and Pt−Co/Al2O313) by means of an ANN-based approach. The authors concluded that the use of an ANN can be very helpful in improving the experimental work during catalyst design and if combined with proper statistical experimental design techniques, a highly successful model can be established even using a relatively small number of data points. Just recently, Panahi et al.14 presented an ANN-based modeling of the relationship between catalyst composition and catalytic performance during the selective catalytic reduction of the NO process using various combinations of transition metals (Mn, Fe, Co, and Cu) and catalyst supports (γ-Al2O3, ZSM5, and SAPO-34). Their ANN results revealed that the electronegativity and ionization energy chosen as descriptors of the transition metals had the largest impact on catalyst performance while the acidic property was the most suitable descriptor for the support. As mentioned earlier, HTE has convincingly shown to be useful in expediting the time during the development and testing of new catalytic materials whereas an ANN-based approach has proved to successfully model the relationship between catalysts composition and their reactivity within the large parameter space usually searched in combinatorial catalysis. Accordingly, it seems reasonable at this point to combine the qualities of both valuable tools in order to optimize the design of new catalytic systems in a more efficient and rapid manner. Consequently, the purpose of the present work was to assess and confirm the suitability of this hybrid HTE-ANN approach during the modeling of the hydroisomerization reaction of n-hexane over platinum supported on tungstated zirconia (Pt/WOx-ZrO2) catalysts doped with different iron contents. More specifically, using available experimental catalytic data for this reaction, previously measured by Hernández-Pichardo et al.3 through HTE, the aim of this study was to obtain a well trained ANN capable of adequately predicting the optimum value of iron content that ensures the highest possible yields of the 2,2-DMB isomer.
f (x ) =
1 1 + exp( −x)
(1)
Evidently, based on the complexity of the learning process, other transfer functions can be considered; e.g. Gaussian, hyperbolic tangent, hyperbolic secant, among others. Signals are transmitted between neurons through weighted connections. The corresponding weights are in fact the ANN’s adjustable parameters, which are continuously adjusted until the network “actually” learns. The number of neurons and the structure of the neuron connections constitute the architecture of the ANN and consequently the model generated by the network. Multilayer Feed-Forward Back-Propagation ANNs. Of the many possible network configurations, the feed-forward back-propagation type is so far the most popular because of its wide range of applicability and suitability, particularly in the field of process engineering. In this network configuration, information flows forward in the prediction mode whereas error corrections propagate backward in the learning mode. Such networks are usually organized into layers of neurons; a minimum of three layers is required. Information enters the network through the input layer which then distributes the input data to the first intermediate layer; one or more intermediate (hidden) layers lie between the input and the output layer. Both hidden and output layer neurons process all incoming signals by applying weights to them. Alternatively, a “bias” neuron connected to each neuron in the hidden and output layers can supply an invariant output. As aforementioned, all signals entering a neuron are weighted, combined and then processed through a transfer function which serves to normalize the neuron’s output signal strength between 0 and 1. During the learning (or training) process, the network response at the output layer is compared to a supplied set of known answers (training targets); the resulting deviations are then determined and backpropagated through the network in an attempt to better represent the supplied training data by adjusting the weights. Consequently, the network learning process relies on an iterative procedure in which dedicated training algorithms adjust the weights at each iteration in order to minimize the network’s response error. On the other hand, during the training of a network, its learning capabilities can be verified at any time by processing selected input data (test cases): a portion of the training data that has been set aside for such a purpose. This verification operation thus requires an external user supervision and terminates until the network properly “predicts” the test cases. Accordingly, the learning process of feed-forward backpropagation ANNs can be basically classified into supervised learning and unsupervised learning. Supervised learning is by all means the most popular method to train an ANN. In the case of unsupervised learning, ANNs are trained to govern themselves without any external supervision; they are practically based on self-organization.
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ARTIFICIAL NEURAL NETWORK (ANN) APPROACH ANNs can be considered a kind of artificial-intelligence technology that offers a very strong learning ability; they are well suited for complex problems where no prior fundamental understanding of the processes or phenomena is required by the network in order to solve them. ANNs can therefore handle problems involving data that are imprecise or noisy, particularly those that are highly nonlinear or with complicated relations between their various dependent and independent variables. ANN Learning Process. ANNs mimic somewhat the human learning process. In this context, if an ANN can learn correlative patterns between sets of input data and corresponding target values, then such an ANN should be able to properly predict outcomes from new input data. ANNs comprise a number of interconnected processing units called neurons (in analogy to the biological neurons). Each neuron is capable of receiving and sending signals; this process is simulated by means of dedicated computer algorithms. More specifically, 8884
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network layers neurons per layer transfer function iterations RMS error training RMS error test correlation R2 training correlation R2 test
ANN 1
ANN 2
ANN 3
ANN 4
3 3−3−2 sigmoid 67245 0.0675 0.0926 0.9248 0.9125
4 3−3−3−2 sigmoid 20675 0.0850 0.0813 0.8883 0.8733
4 3−3−2−2 hybrid 9150 0.0479 0.0635 0.9660 0.9441
5 3−3−2−2−2 sigmoid 98745 0.0636 0.0575 0.9391 0.9357
EXPERIMENTAL DATA USED FOR ANN TRAINING Presently, an ANN-based approach was applied to model the nhexane hydroisomerization reaction over platinum supported on tungstated zirconia (Pt/WOx-ZrO2) catalysts doped with different iron contents. This particular reaction was experimentally studied by Hernández-Pichardo et al.3 by using HTE techniques; details about the corresponding experimental equipment and procedure are given by the same authors. Here, only a brief description of the experimental procedure and results is presented. The catalysts were first synthesized by surfactant-assisted coprecipitation from zirconyl chloride ZrOCl2·xH2O (Aldrich, 98%), ammonium meta-tungstate (NH4)6W12O39·xH2O (Aldrich, 99.99%), iron nitrate Fe(NO3)3·9H2O (Aldrich, 99.95%), and cetyltrimethylammonium bromide (CTAB) (Aldrich, 99%) as surfactant. The synthesis variables comprised: tungsten content (10, 15, and 20 wt %W), iron content (0−2 wt %Fe), and surfactant content in terms of surfactant/zirconia molar ratio (SMR = 0, 1, and 2) thus yielding Fe/W ratios varying from 0.025 to 0.2. To summarize, zirconia, tungsten, and iron precursors were first mixed together at room temperature for 1 h. An aqueous surfactant solution was then added and the resulting mixture was coprecipitated with NH4OH. The resulting precipitates were then aged for 16 h at 80 °C to be eventually washed and dried for 24 h at 110 °C. Further, a calcination treatment was performed in static air for 4 h at 800 °C. Lastly, the platinum (0.3 wt %) was deposited by simple impregnation via the use of a H2PtCl6·6H2O aqueous solution, followed by drying process at 110 °C to be finally calcined at 400 °C in static air during 3 h. The n-hexane hydroisomerization reaction was conducted at 260 °C, 0.689 MPa, and 3.7 h−1 WHSV. The hydrogen and nhexane flows were adjusted to give a molar ratio H2/n-C6 = 1.47; the resulting mixture thus comprised 100 cm3 min−1 of H2 and 0.4 cm3 min−1 of n-hexane that was fed to the reactor by the use of a HPLC pump. The catalysts samples were loaded into the reactor heads for pretreatment reasons followed by a reduction in the hydrogen flow. The catalytic evaluation was performed in a combinatorial multi-channel fixed bed reactor (MCFBR) (Symyx) applying high-throughput testing techniques with a matrix arrangement of 6 × 8. This system comprised six reactor heads each one containing eight microreactors of 4 mm inner diameter and 47 mm length. The six reactor heads were connected independently to six chromatographs (Agilent, 6850 Series) equipped with a SPB-1 capillary column (Supelco) having a length of 100 m, 0.32 mm diameter, and 0.25 mm film thickness and a flame ionization detector (FID) for the analysis of the products. The catalysts samples of 100 mg diluted in 200 mg of inert silicon carbide were loaded in each well and fixed into the reactor heads.
The Supporting Information gives the experimental catalyst activities obtained by the authors3 in terms of selectivity (S) and yield (Y) of 2,2-DMB; the other isomer (2,3-DMB) produced by the reaction, reached the equilibrium conditions much faster than 2,2-DMB and according to the authors, the product to maximize was therefore the yield of 2,2-DMB.
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RESULTS AND DISCUSSION The Supporting Information gives all the necessary experimental information to carry out the training and test of the
Figure 1. Parity plot for the selectivity of 2,2-DMB.
various ANN architectures devised here; it lists, for each experiment and sample, the catalyst synthesis variables (SMR, %W, %Fe, and Fe/W) and the catalyst performance in terms of the selectivity (S) and yield (Y) of the most feasible isomer product (2,2-DMB) including the overall conversion of nhexane:
XA = 100
Y S
(2)
Selection of Input Descriptors. In general, proper selection of all input descriptors entering the input layer can make the difference between successful and unsuccessful ANN models. For the present isomerization reaction, however, the identification of the input descriptors was straightforward; the catalyst performance in this case only depends on the catalyst 8885
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Figure 2. Parity plot for the yield of 2,2-DMB.
synthesis variables. Accordingly, the surfactant molar ratio (SMR), the %W and the %Fe were chosen here as the input descriptors. The Fe/W ratio could be also selected as input descriptor instead of %W and %Fe, however, knowing that the maximum number of neurons used in each hidden layer should be of the same order as the number in the input layer,15,16 the resulting two-neuron input layer would yield a network with too few hidden neurons thus hindering the learning process. The choice of three input descriptors seems reasonable here, it does not allow for too many hidden neurons that may degrade the network’s predictive capability. Creation of ANN Architectures. All ANN architectures devised here have a three-neuron input layer comprising the input descriptors (SMR, %W, and %Fe) and a two-neuron output layer comprising the target data (selectivity and yield of 2,2-DMB). Furthermore, the build and simulation of the various ANNs developed in this work were carried out by means of the commercial software QNET 2000 by Vesta Services, Inc. Accordingly, four multilayer feed-forward backpropagated ANNs were constructed; Table 1 briefly gives the configuration of each ANN, namely, its number of layers, number of neurons per layer, and the transfer function for the hidden and output layers. For example, the simplest ANN architecture considered here was the 3−3−2 configuration containing a three-neuron input layer, a single three-neuron hidden layer, and a two-neuron output layer, whereas the most complex one (the 3−3−2−2−2 configuration) comprised five layers: one input, three hidden, and one output. Lastly, the sigmoidal function was chosen as the transfer function for the hidden and output layers for nearly all ANN architectures except for the 3−3−2−2 configuration for which a hybrid approach was used, namely a sigmoidal function to connect the two hidden layers and a hyperbolic secant function to connect the second hidden layer to the output layer. ANN Training and Testing. The aforementioned ANN architectures were trained and tested using the 36 × 3 input descriptors (SMR, %W, and %Fe) and the 36 × 2 target cases (S2,2‑DMB and Y2,2‑DMB) given in the Supporting Information.
Figure 3. Catalyst activity at %W = 10: (a) 2,2-DMB selectivity versus Fe/W, (b) 2,2-DMB yield versus Fe/W.
More specifically, 80% of the total available data were used for training purposes while the remaining 20% were set aside as test cases to verify possible overtraining of the network and to check the integrity of the neural model. All test cases were randomly selected in order to ensure that the test set was not a limited subset of the training cases. Prior the training process, all input and target data were automatically normalized by QNET 2000 between 0 and 1 since each neuron’s output signal falls between 0 and 1; such a normalization is required to improve training characteristics. The learning progress of the present ANNs was conveniently supervised via the use of QNET-2000’s monitoring capabilities that allow the visual readout of the error histories of both the training and test cases. Such errors are expressed by QNET 2000 in terms of the root-mean-square (RMS) errors between the target set and the network’s outcome. For example, the monitoring of the training set’s RMS error allows the user to determine the network learning progress as well as the general 8886
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Figure 4. Variation of conversion of n-hexane with Fe/W.
state of convergence (or divergence); the network reaches convergence as long as the RMS error approaches a minimum value. On the other hand, the monitoring of the test set’s RMS error allows the user to determine the overtraining status of the network, that is, the point at which the test error starts increasing after having reached its global minimum. Beyond this point, even though the training error continues to decrease, it is assumed that the network is memorizing rather than learning; therefore, a good ANN-based model should only be trained to the point of the test set’s global minimum error. Following the aforementioned training strategy, Table 1 summarizes the statistical results of the four trained ANNs considered in this work. These results include the number of iterations to complete the training, the RMS errors for both training and test sets, and the correlation R2 coefficients for both training and test sets. It can be readily seen from Table 1 that the best trained network is the one having the 3−3−2−2 configuration (ANN 3) which yielded the least number of iterations, the lowest RMS errors as well as the closest correlation coefficients to unity (the closer to 1, the more accurate the model estimations). As a matter of fact, although the network with more layers (the 3−3−2−2−2 configuration, ANN 4) produced the lowest possible test set’s RMS error (0.0576 against 0.0635 from ANN 3), it required a significant number of iterations to reach convergence (98 745 iterations, almost 10 times larger than those of ANN 3!); moreover, it yielded a correlation coefficient for the test set (0.9357) somewhat lower than that achieved by ANN 3 (0.9441). Performance of the Best Trained Network (ANN 3). As noted above, ANN 3 was the best neural model that properly learned to associate and recognize the 2,2-DMB selectivity and 2,2-DMB yield patterns (target data) with their corresponding catalyst synthesis variables (input data). To demonstrate this, a series of graphical comparisons are further presented and discussed. First of all, Figures 1 and 2 are parity plots showing the comparison between estimated data using ANN 3 and experimental data for the 2,2-DMB selectivity and the 2,2-DMB yield, respectively; both training points (filled circles) and test points (open diamonds) were considered. It can be seen that the two parity plots evidence comparable results with the majority of the points falling within the ±10% deviation. This
Figure 5. Catalyst activity at %W = 15: (a) 2,2-DMB selectivity versus Fe/W, (b) 2,2-DMB yield versus Fe/W.
substantiates the accuracy and quality of the neural model developed here. Further, Figure 3 depicts measured and ANN estimated selectivities and yields of 2,2-DMB versus the Fe/W ratio at 3 surfactant molar ratios (0, 1 and 2) when the tungsten content was set equal to 10%. As seen in this figure, the present neural model is able to represent well, at least qualitatively, the variation of experimental catalyst performance (selectivity and yield) with iron content at all SMR values. In Figure 3a, however, the experimental selectivity at SMR = 2 and Fe/W = 0.2 (which is in fact an ANN test point), is largely underpredicted by the neural model. It is likely that this particular point (and some others) contains a high experimental uncertainty; if so, the neural model may be actually predicting the right behavior. Another important parameter in assessing the present catalyst activity is the conversion of n-hexane which can be readily obtained by dividing the yield over the selectivity of 2,2-DMB. In this context, Figure 4 demonstrates the abilities of the present ANN approach in representing the experimental 8887
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Figure 7. Variation of 2,2-DMB selectivity with conversion of nhexane.
2,2-DMB isomer, fall within the range of Fe/W ratios from 0.05 to 0.075. Lastly, Figure 7 shows ANN estimated and experimental 2,2-DMB selectivities versus the corresponding conversion of n-hexane at three %W values (10, 15, and 20%). The highest selectivities (both ANN and experimental values) along with the highest conversions were mostly attained at %W values of 10 and 15 as depicted by Figure 7 (see upper right region at S > 5.2%mol and XA > 52%).
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CONCLUSIONS A stochastic approach based on the use of an artificial neural network (ANN) was applied to model the hydroisomerization reaction of n-hexane over platinum supported on tungstated zirconia (Pt/WOx-ZrO2) catalysts doped with different iron contents. The following conclusions can be drawn from this work: • The catalyst synthesis variables (SMR, %Fe, and %W) chosen as neural network inputs played a very important role in successfully correlating the catalyst activity in all cases. • Four different multilayer feed-forward ANN architectures were built and trained using previously measured catalyst activity (using the HTE technique) in terms of selectivity and yield of the 2,2-DMB isomer at several Fe/W weight ratios and SMR values. • The agreement between predicted and observed catalyst activity data was highly acceptable thus demonstrating the abilities of the four ANNs (particularly, the 3−3−2− 2 configuration which yielded a RMS error of 6.35%) in learning to associate and recognize the 2,2-DMB selectivity and 2,2-DMB yield patterns with their corresponding catalyst synthesis variables. • More importantly, the use of the present ANN-based approach served to properly establish the optimum values of iron content and %W that ensured the highest possible selectivity and yield values of the most preferred n-hexane isomer: 2,2-DMB. Such optimum values fell within the range of Fe/W ratios from 0.05 to 0.075. • Finally, this work confirmed the suitability of present combined HTE-ANN approach which may eventually
Figure 6. Catalyst activity at %W = 20: (a) 2,2-DMB selectivity versus Fe/W, (b) 2,2-DMB yield versus Fe/W.
conversions of n-hexane at three different SMR values and at a fixed 10%W value. As in Figure 3a, the neural model somewhat overpredicts the conversion for the only test point within this data subset (at %W = 10). Similar plots as those shown in Figure 3 were prepared to analyze the results of the other two data subsets (at 15 and 20% W). Accordingly, Figures 5 and 6 show that the neural model provided sufficiently accurate representations of the experimental catalyst activity at 15 and 20%W, respectively. In Figure 5, for example, the ANN model predicts reasonably well the experimental data for the 5 test points (open diamonds), particularly the experimental yield of 2,2-DMB (Figure 5b). In general, it was found that the 2,2-DMB yield rather than the 2,2-DMB selectivity was better represented by the present neural model as depicted by Figures 3b, 5b, and 6b. According to Figures 3, 5, and 6, both ANN estimations and measured catalyst activities establish that the optimum values of iron content, that ensure the highest possible S and Y values of the 8888
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(9) Turner, H. W.; Volpe, A. F., Jr.; Weinberg, W. H. HighThroughput Heterogeneous Catalyst Research. Surf. Sci. 2009, 603, 1763. (10) Rodemerck, U.; Baerns, M.; Holena, M.; Wolf, D. Application of a Genetic Algorithm and a Neural Network for the Discovery and Optimization of New Solid Catalytic Materials. Appl. Surf. Sci. 2004, 223, 168. (11) Song, S.; Akande, A. J.; Idem, R. O.; Mahinpey, N. InterRelationship Between Preparation Methods, Nickel Loading, Characteristics and Performance in the Reforming of Crude Ethanol Over Ni/Al2O3 Catalysts: A Neural Network Approach. Eng. Appl. Artif. Intel. 2007, 20, 261. (12) Günay, M. E.; Yildirim, R. Neural Network Aided Design of PtCo-Ce/Al2O3 Catalyst for Selective CO Oxidation in Hydrogen-Rich Streams. Chem. Eng. J. 2008, 140, 324. (13) Günay, M. E.; Yildirim, R. Analysis of Selective CO Oxidation Over Promoted Pt/Al2O3 Catalysts Using Modular Neural Networks: Combining Preparation and Operational Variables. Appl. Catal., A 2010, 377, 174. (14) Panahi, P. N.; Niaei, A.; Tseng, H.; Salari, D.; Mousavi, S. M. Modeling of Catalyst Composition-Activity Relationship of Supported Catalysts in NH3-NO-SCR Process Using Artificial Neural Network. Neural Comput. & Applic. 2015, 26, 1515. (15) Bhagat, P. An Introduction to Neural Nets. Chem. Eng. Progress 1990, No. August, 55. (16) Himmelblau, D. M. Accounts of Experiences in the Application of Artificial Neural Networks in Chemical Engineering. Ind. Eng. Chem. Res. 2008, 47, 5782.
serve as a very valuable tool to optimize the design of new catalytic systems in a more efficient and rapid manner.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b01821. Original experimental data used for ANN training and test purposes (PDF)
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AUTHOR INFORMATION
Corresponding Author
*On sabbatical leave at the Department of Physics and Astronomy, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, U.S.A. E-mail: mhernandezp@ ipn.mx. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors gratefully acknowledge the Instituto Politécnico Nacional and CONACyT for providing financial support for this work. M.L.H.-P. wishes to thank Dr. M. J. Yacaman and the International Center for Nanotechnology and Advanced Materials (ICNAM) for allowing her to carry out a Sabbatical Stay in the Department of Physics and Astronomy of the University of Texas at San Antonio.
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LIST OF SYMBOLS f(x) = ANN transfer function x = independent variable in transfer function Fe/W = iron to tungsten weight ratio
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REFERENCES
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