Application and Performance of Neural Networks in the Correlation of

Application and Performance of Neural Networks in the Correlation of Thermophysical Properties of Long-Chain n- ... Publication Date (Web): May 23, 20...
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Ind. Eng. Chem. Res. 2007, 46, 4717-4725

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CORRELATIONS Application and Performance of Neural Networks in the Correlation of Thermophysical Properties of Long-Chain n-Alkanes Naif A. Darwish* Chemical Engineering Program, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates

In this study a comprehensive performance evaluation of artificial neural networks (ANN) in addition to many correlations and methods in the literature for the correlation of critical temperature (Tc), critical pressure (Pc), and normal boiling points (Tb) of long-chain n-alkanes has been conducted. The predictive capability of a 1-3-1 feed-forward network is explored by training it with a partial set of experimental data versus carbon number for CH4 (C1) to C21H44 (C21). Prediction made by ANN is found to be very sensitive to the training algorithm. Compared to many correlations and models available in the open literature, ANN employing the Bayesian regularization algorithm gives the best predictive capability in terms of average absolute deviations and other statistical measures. For the three properties considered the ANN predicts a nonzero limiting (asymptotic) values for n-alkanes with a high carbon number (i.e., Tc∞, Pc∞, Tb∞); asymptotic values obtained are as follows: 1015 K for Tc∞, 2.25 bar for Pc∞, and 1070-1120 K (depending on the on the training algorithm) for Tb∞. 1. Introduction Thermophysical properties such as critical temperature, critical pressure, and normal boiling points are among the most important properties of pure materials. Besides the very sound physics associated with these properties, they play a prominent role as essential input parameters for many thermodynamic models, which are used to correlate and predict pure as well as mixture thermophysical properties such as pressure-volumetemperature (PVT) and phase equilibrium behavior. Models based on the corresponding state principles and cubic equations of state (EOS), in particular, require (in addition to reliable mixing rules) credible critical properties for good prediction of thermophysical properties of pure and mixtures of fluids. In fact, most of these thermodynamic models are mathematically highly nonlinear functionals in these parameters, i.e., highly sensitive to small uncertainties in such input parameters.1,2 Whenever possible, therefore, accurate critical thermophysical properties must be employed in such thermodynamic models. The most accurate properties usually come from experimental measurements. Unfortunately, however, it is not always feasible to experimentally measure these properties; heavy hydrocarbons, for example, are thermally unstable at high temperatures, making direct measurements of their critical properties a difficult task. In particular, n-alkanes heavier than n-decane (C10) are thermally degradable far below their critical temperatures.3 Yet, there are many applications involving long-chain n-alkanes.4 Examples of these applications are modeling phase and PVT behavior of systems containing waxes and heavy oils and characterization of heavy oil/gas components. There are also important applications in the petrochemical industry, supercritical extraction, and adsorption/desorption industrial processes. Moreover, owing to their well-known and simple structure, n-alkanes lend them* To whom correspondence should be addressed. Tel.: +971 2 5085455. Fax: +971 2 5085200. E-mail: [email protected].

selves as perfect prototypes for many theoretical studies.5 In all of these applications it is imperative to use the best correlative and predictive methods. Various group-contribution predictive methods have been proposed for the estimation of Tc, Pc, and Tb. Correlation is mostly done in terms of the carbon number (CN) or (equivalently) the molecular weight of the alkane.6 Some of these methods require the normal boiling point of the pure material to make a good prediction of the critical properties,6,9,10 which necessitates careful determination and estimation methods for the normal boiling points themselves. Other models, including group-contribution methods, require input about the structure of the compound.11 A good compilation of the variety of methods for the correlation and estimation of critical constants, with many worked-out examples and comparisons, is presented in the fifth edition of “the properties of gases and liquids”.12 Critical constants and acentric factors for long-chain alkanes have been also reviewed and evaluated for the prediction of vapor pressures using corresponding state methods by Kontogeorgis and Tassios.6 Other relevant reviews have been also reported by Nikitin7 and Nikitin et al.8 These methods have been classified in some of these studies6 into two groups: methods predicting no finite asymptotic values for Tc, Pc, and Tb13-17 and methods predicting asymptotic values for these parameters.18-21 Critical properties of n-alkanes have been reviewed by Ambrose and Tsonopoulos.22 Also, the critical properties for 11 homologous series of 118 organic compounds, with n-alkanes up to C18, were correlated with molecular surface areas by Mebane et al.23 A strong correlation between Tc/xPc and the surface area for the 11 homologous series including n-alkanes has been found. Artificial neural networks (ANN), owing to their special feature of learning and modeling, have found many applications in various areas of chemical engineering.24-27 Because of this special feature in learning and generalization, ANN methods

10.1021/ie061250c CCC: $37.00 © 2007 American Chemical Society Published on Web 05/23/2007

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have a high interpolative capability. ANN modeling, when applicable, is more cost effective than developing models from first principles. They are also easier to use than conventional methods. A standard feed-forward ANN with one hidden layer has been used by Cherqaoui et al.28 for predicting boiling points, melting points, critical temperatures, and molar volumes of alkanes up to C10. Arupjyoti and Iragavarapu29 made prediction for boiling points of n-alkanes up to C100 using an electrotopolgical molecular descriptor. In this work, application of ANN for correlating critical temperatures, critical pressures, and normal boiling points of long-chain n-alkanes will be investigated making use of the newly developed experimental data of n-alkanes up to C36.3 Toward this end, many back-propagation algorithms, as built in Matlab, are incorporated into computer Matlab programs to investigate the performance of different ANN’s in correlation and prediction of these properties. In all cases, the net is trained with a partial set of input data (Tc, Pc, and Tb versus the number of carbon atoms, NC, in the n-alkane). Moreover, the input and corresponding output training and validation data are scaled to the range from -1 to +1. The number of neurons used in the hidden layer is three, which is the same as the optimum number found by the regularization algorithm, which automatically computes its own optimum parameters. Carbon number as a correlating parameter for long-chain n-alkanes has been used by many investigators.27-32 The striking finding is that most of the investigated algorithms, although convergence is attained in a reasonable number of epochs (iterations), resulted in instable results in the sense that different predictions were obtained at different prediction trials. This reflects the high sensitivity of these algorithms to initial guesses of the network’s weight and bias parameters, which are randomly reinitialized each iteration. Once a proper training algorithm is figured out, however, the net makes prediction better than most available methods. 2. Artificial Neural Networks Approach ANNs may be described as a group of simple processing elements (called neurons) arranged in parallel layers that are fully interconnected by weighted connections. There are several types of ANNs mentioned in the literature33 such as feed-forward networks (or perceptron network), competitive networks (Hamming network), and recurrent networks (Hopfield network). The most famous type of ANN in use is the feed-forward networks, particularly, the multilayer perceptron network. In this type of network information flows in the forward direction only. ANNs interface independent variables through the neurons of an input layer and transmit output of dependent variables through the neurons of an output layer. The number of independent variables and dependent variables, therefore, dictates the number of neurons in the input and output layers, respectively. In between the input and the output layers there is at least one hidden layer that can have any number of neurons. According to the universal approximation theory34 an ANN with a single hidden layer with enough number of hidden neurons can map any input to any output to any degree of accuracy. Additionally, bias is an extra input added to neurons which always has a value of 1 and is treated like other weights. The reason for adding the bias term is that it permits adding to the sum a constant input and allows a representation of phenomena having thresholds.35 Neurons in the hidden and output layers calculate their inputs by performing a weighted sum of the outputs they receive from the previous layer. Their outputs, on the other hand, are calculated by transforming the inputs using a specific transfer function. The most widely used transfer functions are the S-shaped log-

Figure 1. Architecture of feed-forward neural network with a single hidden layer of S neurons.

sigmoid (logsig) transfer functions (eq 1), the S-shaped tansigmoid (tansig) transfer functions (eq 2), and the pure linear transfer function (purlin). The ‘logsig’ transfer function produces outputs in the range from 0 to 1, whereas the tansig function produces output in the range from -1 to +1. Outputs in the range from -∞ to +∞ can be produced from the linear transfer function

f(x) )

1 1 + e-x

(1)

f(x) )

ex - e-x ex + e-x

(2)

The architecture of the feed-forward ANN with a single hidden layer of S neurons is shown in Figure 1. In this figure the input vector P is represented by the solid vertical bar at the left. The dimensions of P are displayed below the variable as R × 1, indicating that the input is a single vector of R elements. These inputs go to the weights matrix W1 in the hidden layer, which is an S × R dimensional matrix. A constant 1 enters the neurons as an input and is multiplied by a bias b1 forming the vector b. This vector is summed with the weighted inputs (W1P) to form the net S × 1 input vector n1 as seen in eq 3

n1 ) W1P + b1

(3)

The output vector which is of S × 1 dimension, is the result of the transfer function being operated on n1 a1,

a1 ) logsig (W1p1 + b1)

(4)

The same procedure can be followed to the output layer. The net input to output layer is formed similarly as shown in eq 5, while a2 represents the output of the network (eq 6). This output could be a scalar or vector which is dependent on the case under study

n2 ) W2a1 + b2

(5)

a2 ) purelin (W2a1 + b2)

(6)

There are usually four steps involved in the application of ANN: (1) assembling the training data of input (independent) and output (dependent) variables, (2) deciding the network architecture, (3) training the network, and (4) validation, i.e., simulating the network response to new inputs. The training process is simply an optimization process which aims at finding the set of weights and biases associated with each layer that will minimize an error objective function that is related to the deviations of the ANN predictions from the target true (experimental) values. The error function is usually the sum of squared errors between targets and ANN outputs. ANN weights and biases in this process are updated in a systemic way by means

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of minimizing the error function using any of the well-known optimizing techniques. Several iterative techniques have been proposed for optimizing the weights. The most widely used is the back-propagation algorithm,36 which is based on steepest descent. In this method ANN performance is influenced by several variables such as the learning rate, momentum coefficient, number of hidden layers, and number of hidden neurons. The network performance can be judged by a combination of some parameters such as absolute errors and number of training cycles or epochs. Besides the network architecture (topology) and the transfer function the propagation rule mentioned above is one of the essential network features which has a direct effect on the network performance. In this work many back-propagation training algorithms are investigated in view of their performance in correlating critical constants and normal boiling points on long-chain n-alkanes. The training functions used here will be identified with the same acronym adopted by Matlab (e.g., trainbr to stand for the training function based on Bayesian regularization). For a list of these training functions, refer to the nomenclature section of this paper. 3. Results and Discussions 3.1. Critical Temperature, Tc. Figure 2a presents the predicted critical temperatures of C20-C36 n-alkanes obtained from 100 different trials when a 1-3-1 network is trained using data of Tc for methane (C1) to eicosane (C20). The network uses a sigmoid tan (tansig) transfer function in the hidden layer and a pure linear function (purelin) in the output layer. To avoid slow convergence a variable (adaptable) learning rate with momentum has been implemented.37 The mathematical training algorithm used is based on the gradient descent principles. Moreover, the input and corresponding output training data have been scaled to the range from -1 to +1. As shown in Figure 2a the predicted critical temperatures differ widely from one trial to another, depending on the initial values of weights and biases which are picked up randomly whenever the training code is invoked in Matlab. The variation of the predicted critical temperature for any specific n-alkane from one run to another is large and getting worse with increasing carbon number. For example, for C36 the predicted critical temperature from 100 trials coming from the above 1-3-1 network spans a range of 600-1100 K. This shows clearly the inability of neural networks with gradient descent algorithms to predict critical temperatures of heavy alkanes. Networks employing training based on conjugate gradients, quasi-Newton, and optimization algorithms have also been found to experience the same symptoms as those of the gradient descents but with different degrees of severity. Figure 2b shows results when the network uses the LevenbergMarquardt optimization-training algorithm as coded in Matlab (trainlm). Predicted critical temperatures are still scattering over a wide range, though narrower than before. The objective function employed in these algorithms is the mean of squared errors with a value of 1 for the performance ratio.37 When the objective function is modified to take into account the mean of squared weights with a nonzero value of the performance ratio, a robust (i.e., independent of the initial values of weight and bias) and regular (i.e., generalizes correctly with no over fit) solution results. This process is called regularization.37 Regularization is made more efficient by the automatic determination of its parameters like the performance ratio and the number of weights and biases needed for the optimum solution. This approach, which is based on Bayesian regularization,38 has the inherent merit of deciding the optimum size of

Figure 2. (a) Variation of critical temperatures of n-alkanes as predicted by 100 different trials using gradient descent back-propagation algorithms trained using C1-C20 n-alkanes (traingdx, 500 epochs). (b) Variation of critical temperatures of n-alkanes as predicted by 100 different trials using gradient descent back-propagation algorithms trained using C1-C20 (trainlm, 500 epochs).

the network, which is in our case a 1-3-1 network. Bayesian regularization is implemented in Matlab in the training function “trainbr”.37 Using this training function with data on critical temperatures for n-alkanes from C1 to C20 (methane to eicosane), the predicted critical temperatures for C20-C36 are shown in Figure 3. The algorithm underestimates the critical temperature for n-alkanes heavier than C25. Surprisingly, when the reciprocal of the critical temperature versus the reciprocal of carbon number are processed in the same network, a very good match with the experimental data is obtained as clear in Figure 3. More importantly, the network predicts an asymptotic value of the critical temperature (Tc∞) around 1060 K (not shown in the figure). To further refine the predictive capability of the network, the (asymptotic) critical temperature corresponding to C200 was sought by minimizing the squared deviation of the predicted

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Figure 3. Variation of the critical temperatures (Tc) of n-alkanes as predicted by 100 different trials using the regularization back-propagation algorithm trained by Tc data for C1-C20 (trainbr, 500 epochs). Table 1. Limiting Critical Temperature (Tc∞) and Critical Pressure (Pc∞) of Long-Chain n-Alkanes As Extrapolated by Different Correlations method or correlation neural networks (this work) ABC39 Nikitin3 Tsonopoulos,48 Gray et al.,40 Magoulas and Tassios41 Teja et al.30 Hu et al.42 Morgan and Kobayashi43 Kreglewski18 Bolotin and Shelomentsev44 Robinson et al.21,6 Tsonopoulos and Tan20

limiting critical temp., Tc (K)

limiting critical pressure, Pc (bar)

1015 981.8 1258.73 959.98

2.25 0.0 0.0 0.0

1143.8 1354.65 981.38 960.0 955.4 897.9 1072.72

8.4203 0.0 0.0 0.0 2.22 2.68 or 0.0519

critical temperature from the experimental data. The best Tc∞ that corresponds to C200 was found to be in the range of 10101020. A value of 1015 K is used and added to the training data set for C1-C20. The prediction made with this network is also shown in Figure 3, where a very good agreement with the experimental data is evident. Moreover, prediction made with the raw data and their reciprocals becomes identical (Figure 3). Table 1 presents limiting values of the critical temperatures and critical pressures of long-chain n-alkanes as predicted by ANN and other correlations available in the technical open literature. The closest to ANN prediction of Tc∞ are those due to Morgan and Kobayashi43 (981.38 K), the ABC model of Gao et al.39 (981.8 K), and Tsonopoulos and Tan20 (1072 K). Using the ANN approach the predicted critical temperatures for C1-C20 have shown shallow response to values of Tc∞ in the range of 1010-1020 K. The uncertainty, therefore, in the predicted limiting values of Tc∞ by the ANN approach is assumed to be (5 K. The highest limiting critical temperature is predicted by the Hu et al.42 model, which gives a value of Tc∞ ) 1354.65 K. This is outside the range of 900-1150 K, which brackets predictions from all other models (Table 1). Results obtained from ANN (employing the trainbr algorithm in a 1-3-1 networks and using tansig-purelin transfer functions) trained with data of C1-C20, in addition to the observed limiting

Figure 4. Comparison of critical temperatures of long-chain n-alkanes as predicted by ANN (trainbr, tansig-purelin) and other correlations.

value of 1015 K for C200, are compared with predictions of other models in Figure 4. For C1-C20 n-alkanes, all models perfectly agree with experimental data. However, for higher carbon number models begin to show some discrepancy. Predictions made by Teja et al.30 and Hu et al.42 show striking agreement between each other over the whole range of carbon number but not with the experimental data for n-alkanes heavier than C35. Predictions made by the ABC model,39 the ANN model of this work, and that of Morgan and Kobayashi,43 on the other hand, show good agreement with each other and with the experimental data. The Magoulas and Tassio model41 agrees with the latter three models up to C45, after which it underestimates the critical temperatures as shown in Figure 4. The Nakanishi et al.45 model overestimates Tc and highly deviates from experimental data. The worst prediction resulted from Marrero and Pardillo,46 which breaks down after C20. This model is not even able to predict the limiting behavior of critical temperatures. Three statistical measures for the deviations of the predicted critical temperatures (from the experimental data) for C15-C36 are shown in Table 2 for different models including ANN of this work. The least average absolute deviation (AAD) is given by the ANN approach (1.58 K), which agrees very well with those of Magoulas and Tassio41 (1.61 K) and the ABC model39 (1.67 K). It is worth noting that the network has been trained using all available data, i.e., data on C1-C36 n-alkanes. This makes comparison with models such as the ABC model more meaningful since usually correlations such as these make use of the whole set of data available in the literature. The maximum and minimum deviations obtained from ANN are +5.17 (for C28) and +0.27 K (for C26), respectively. Those obtained from the ABC model are 0.64 (for C28) and +0.50 K (for C19). In view of the ANN prediction, the reported experimental critical temperature of C28 looks like an outlier, i.e., as if it does not belong to the rest of the data. The same thing has been reported by Gao et al.39 It is worth mentioning that the critical temperature predicted by ANN of this work for C48 is in very good agreement with that predicted by molecular simulation of Siepmann et al.,47 which is 923.72 K. The distribution of deviations in the correlated Tc for C15C36 n-alkanes from five different models, including the ANN of this work, is shown in Figure 5. All models show high

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Figure 5. Deviations in predicted critical temperatures (∆Tc) for C15-C36 n-alkanes from ANN and other models.

Figure 6. Critical pressure (Pc) of n-alkanes as predicted by ANN with different training algorithms and transfer functions.

deviation in the predicted critical temperature for C28 from the experimental value reported by Nikitin.3 The Nikitin model itself gives a deviation of about 4.2 K for C28 as shown in Figure 5. This, again, supports a high experimental error in the reported experimental value of C28, as mentioned previously. Models of Maggoulas and Tassios41 and Nikitin3 underestimate the true (experimental) critical temperature as reflected by the negative deviations for most alkanes (Figure 5). The ANN of this work and the ABC model39 scatter evenly about the zero-deviation line with comparable deviations. 3.2. Critical Pressure, Pc. The architecture of the neural network used for the prediction of critical pressures is the same as that used previously for critical temperatures, i.e., 1-3-1 network. Also, input and output training sets of data have been scaled into the range from -1 to +1. The network has been trained using critical pressures of C2-C21 n-alkanes. The network predicts a nonzero limiting (asymptotic) value for the critical pressure Pc∞, which is observed for n-alkanes of 65 or more carbon number. The best value of Pc∞ has been sought by solving for the critical pressure of C100, which minimizes the sum of squared deviations from experimental data of critical pressures of long-chain n-alkanes (Σ(Pc,pred - Pc,exp)2) using the network. The value of Pc∞ thus obtained is 2.25 bar. Interest-

ingly, the value predicted by the neural network approach for Pc∞, which is 2.25 bar, is in very close agreement with that reported by Robinson et al.21 and Tsonopoulos and Tan20 (Table 1). Different values for Pc∞ have been reported in the open technical literature; many correlations assume a zero limiting critical pressure,3,18,39-43 while many others predict nonzero limiting critical pressures20,21,30 (Table 1). To help the network make better predictions this limiting value has been incorporated into the training data set. The susceptibility of the network prediction to different transfer functions and training algorithms is shown in Figure 6. Contrary to what has been observed in critical temperature predictions, the most robust predictions have been obtained from a network employing a tansig transfer function in both the hidden and the output layers. This is clear from Figure 6, which shows some deterioration in prediction when the output layer is modeled using a pure linear transfer function (purelin). Moreover, networks employing a purelin transfer function in the output layer, with training algorithms based on regularization principles, give unstable prediction as reflected by the different curves obtained from two different trials (Figure 6). Algorithms based on gradient descent principles, which were extremely unstable for critical temperature predictions, are shown to give

Table 2. Deviations of Predicted Critical Temperatures (K) of Long-Chain n-Alkanes with 15-36 Carbon Atoms As Predicted by Different Correlations method or correlation neural networks (this work) ABC39 Magoulas and Tassios41 Teja et al.30 Hu et al.42 Morgan and Kobayashi43 GC-Lydersen6,49 GC-Ambrose6,50 GC-Joback and Reid6,51 Recent GC-Constantinous and Gani (general method)6,52 GC-Constantinous and Gani (hydrocarbon version)6,53 Al-Hassan-16,54 Al-Hassan-212 Nikitin3 Marrero and Pardillo46

average absolute deviation (AAD) 1.92 2.01 2.01 7.10 5.39 3.25 24.64 23.11 7.42 20.43 6.18 5.88 45.93 11.20 2.53 55.56

max deviation (MXD) 5.17 6.24 5.87 20.21 15.67 10.56 48.39 128.31 20.65 111.99 9.56 9.24 184.86 54.81 4.24 274.27

min deviation (MND) 0.27 0.50 0.45 0.23 0.14 0.003 10.79 1.23 0.03 1.52 0.80 1.62 0.65 0.21 0.093 0.138

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results better than those based on the regularization concepts. This is intimately tied to the nature of input-output functionality of the training data sets. In this work, therefore, a 1-3-1 neural network employing tansig-tansig transfer functions with a training algorithm based on a variable-learning-rate gradient descent is used to make critical pressure predictions. The predictive capability of the neural network approach, in comparison to other models and correlations, is displayed in Figure 7. Except for Teja et al.30 and, to a lesser extent, GC Ambrose,6,50 all models shown in Figure 7 give comparable prediction of critical pressures of n-alkanes up to C40. Beyond C40, however, one set of models (i.e., Marrero and Pardillo,46 Magoulas and Tassio,41 the recent GC of Constantious and Gani,6,12,52,53 and Nikitin3) shows comparable extrapolative prediction of critical pressures for very heavy n-alkanes. Of these, the Nikitin model3 is based on experimental data of critical pressures up to C36, which imparts it with some credibility. The critical pressure of C100 as predicted by the Nikitin model3 is 0.46 bar, whereas that predicted by the ABC model39 is 0.042 bar. It is seen, therefore, that although both the Nikitin and ABC models assume a zero limiting value, the ABC model goes faster toward that zero. Therefore, the ABC model predicts lower critical pressures for high carbon number alkanes. ANN, on the other hand, predicts higher critical pressures than those predicted by the previous models. This is quite explainable in terms of the assumed value of Pc∞, which is 2.25 bar. In the absence of accurate experimental data, however, nothing can be said about the accuracy of these models. Except for the Teja et al. model,30 Figure 7 shows that the maximum difference in these models’ prediction of the critical pressures for say C100 is within 2.5 bar. In view of the new experimental data of Nikitin,3 it seems obvious that the Teja et al. model30 is totally inappropriate for n-alkanes of carbon number greater than 20. Deviations of the predicted critical pressures from experimental data for C15-C36 n-alkanes for some models are shown in Table 3. It is seen from this table that an average absolute deviation (AAD) less than 0.2 bar is only given by the ABC,39 ANN of this work, and Nikitin3 models. Again, in this case the network has been trained using all available data, i.e., data on C2-C36 n-alkanes. The maximum and minimum deviations obtained from the ANN computations are -0.46 bar (for C18) and + 0.04 K (for C30), respectively. Those obtained from the ABC model are -0.618 (for C18) and -0.002 bar (for C24). The distribution of deviations predicted by different models is shown in Figure 8. Teja et al.30 and Abmrose50 overestimate critical pressures with increasing deviations for alkanes with carbon number beyond 20. On the other hand, Joback and Reid51 and the ABC model39 underestimate critical pressures for n-alkanes beyond C20 but with deviations below 1 bar. Deviations from other models and ANN are randomly scattered around the reference zero line with no abnormalities. 3.3. Normal Boiling Point, Tb. For the normal boiling points (Tb) of long-chain n-paraffins it is found that all ANN algorithms, when converged, predict an asymptotic limiting value (Tb∞). However, as will be shortly seen, asymptotic values obtained were found to depend on the size of the training data set, training algorithm, and functional form of the treated raw data (e.g., processing Tb versus CN or Tb-1 versus CN-1). This does emphasize the extreme care to be exercised when employing ANN to predict thermophysical properties. Figure 9 presents predictions made by four variants of ANN in addition to four other methods reported in the literature.12 The four variants of ANN, which employ the trainbar algorithm with tansig-purelin

transfer functions (refer to nomenclature), differ in size and form of the raw training set of as follows. Case 1: Tb versus CN for CN ) 1-24. Case 2: Tb versus CN for CN ) 1-16. Case 3: 103/Tb versus 103/CN for CN ) 1-24. Case 4: 103/Tb versus 103/CN for CN ) 1-16. The first observation that can easily be made from Figure 9 is that all methods match experimental data up to CN ) 20. This is no surprise, however, because most reported methods in the literature utilize the available experimental data for C1-C20. For n-alkanes with carbon number greater than 20, the Joback and Reid model51 and Marrero and Pardillo model46 deviate from the rest of the models in their prediction. Moreover, these two models predict no asymptotic limiting values for the normal boiling points. In essence, they both predict infinity boiling point for n-alkanes with very large carbon number. It is also evident from Figure 9 that the four different cases of ANN give predictions which are below those of the ABC model for the carbon number range shown. Most importantly, the limiting asymptotic value (Tb∞) is dependent upon the size of the training data set and the functional form of the training data. This is made more explicit in Table 4, which shows, besides Tb∞ and CN∞ (value of CN at which Tb∞ is first attained), some statistical measures for deviations from the experimental data (for C1-C28) made by the ANN and some other models. The best results were obtained from a net trained with the full set of data of Tb versus CN for CN ) 1-24 (Case 1). The asymptotic value observed here is 897.2 K, obtained for n-alkanes with carbon number of 200 or greater. Results obtained from Case 1 are in excellent agreement with those of the ABC model up to CN ) 50. After that, each model cruises toward its own asymptotic value. When the same net is trained with a partial set of available data, i.e., Tb versus CN for CN ) 1-16 (Case 2), the net prediction deteriorates as reflected by the high values of average absolute deviation (AAD), maximum absolute deviation (MAD), and root-mean-square (rms) shown in Table 4. This deterioration is also shown in Figure 10, which displays deviations from the experimental measurements for the n-alkanes of CN ) 15-24. The value of Tb∞ predicted in Case 2 is 753.9 K, which is obtained for n-alkanes with CN ) 60 or more (see Table 4). A better agreement with the ABC model is obtained from the net in Case 3, where the net is trained with the full set of data as 103/Tb versus 103/CN for CN ) 1-24. The asymptotic limiting value here, as shown in Table 4, is 1120 K, which is obtained for n-alkanes of CN larger than 8000. This is relatively in good agreement with that used in the ABC mode (1078 K). Interesting results are also furnished by ANNCase 4, where the net is trained with a partial set of data as 103/Tb versus 103/CN for CN ) 1-16. Figure 9 shows that prediction obtained from this case is of the same quality as those given by Case 1, which uses the full data set of Tb versus CN. The limiting value (Tb∞) given by Case 4 is 1072 K, which is reached at carbon numbers larger than 8000. This value of Tb∞ is in very good agreement with that used by the ABC model (1078 K). It is seen from the values of CN∞ shown in Table 4 that ANN methods utilizing the raw input data as 103/Tb versus 103/CN give more realistic results since it is unlikely that asymptotic behavior will be attained as early as that encountered in Cases 1 and 2. This imparts confidence to the values of Tb∞ in the range of 1070-1120 K. It is worthwhile to mention at this point that the majority of models addressed in this paper, including the ANN methods, suffer a serious inconsistency; they predict asymptotic normal boiling points for heavy alkanes higher than the corresponding asymptotic critical temperatures. This reflects a black-box nature and a high degree of empiricism embedded inside these models.

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Figure 7. Comparison of critical pressures (Pc) of long-chain n-alkanes as predicted by different correlations (ANN prediction uses trainbar, tansigtansig, with and Pc∞ ) 2.25 for CN∞ ) 100)

Figure 9. Normal boiling points (Tb) of long-chain n-alkanes as predicted ANN and other correlations. Table 4. Asymptotic Values and Statistical Measures for the Prediction of Normal Boiling Points from Different Models and ANN Methods model

Figure 8. Comparison of deviations in critical pressures (∆Pc) as predicted by different correlations.

The capability of the different ANN models (cases) considered here in addition to the ABC model in representing the behavior of Tb versus CN over the range of CN of 15-24 is given by the deviation plot in Figure 10. A very good description of the experimental data is furnished by the ABC model, ANNCase 1, and ANN-Case 3, where the full data set has been utilized. When a partial set of the available data set is utilized

Tb, K

ABC39

1078

GC12 Marrero and Pardillo46 Joback12 ANN-Case1 ANN-Case2 ANN-Case3 ANN-Case4

897.2 753.9 1120 1072

CN

AAD

MXD

RMS

175 135 8000 8000

0.50 20.00 6.96 38.30 0.41 3.42 1.02 3.73

1.43 32.08 17.13 84.07 0.94 10.42 2.12 7.64

0.68 21.51 8.55 46.81 0.50 4.98 1.19 4.40

as in Cases 2 and 4, deviations up to 10 K are obtained. These results are also reflected in the AAD, MAD, and rms values displayed in Table 4. Before closing this discussion it is worth mentioning that when other training algorithms, like trainlm, trainoss, and traincgp (refer to nomenclature), are employed in the 1-3-1 neural network to correlate normal boiling points, the results obtained were unstable, i.e., different results are obtained each time the net is initiated. This means that the network output becomes dependent on the initial guesses of the weights and biases. Also, when the tansig-tansig-trainbr net is used, stable but poor prediction was obtained (Tb∞ ) 664.45 K, AAD ) 2.03 K, and MXD ) 6.34 K). These observations reflect the care that should be exercised when ANN methods are applied in the prediction of thermophysical properties of substances.

Table 3. Deviations of Predicted Critical Pressures of C15-C36 n-Alkanes As Predicted by Different Correlations and Methods method or correlation

average absolute deviation (AAD)

max absolute deviation (MXD)

minabsolute deviation (MND)

neural networks (this work) ABC39 Magoulas and Tassios41 Teja et al.30 Hu et al.42 Morgan and Kobayashi43 GC-Lydersen6,49 GC-Ambrose6,50 GC-Joback and Reid51 recent GC-Constantinous and Gani52 recent GC-Constantinous and Gani (HC version)53 Al-Hassan-154 Nikitin3 Marrero and Pardillo46

0.156 0.141 0.253 1.385 0.776 0.592 1.206 1.155 0.739 0.412 0.283 0.600 0.170 0.574

0.460 0.618 0.440 4.190 1.858 1.025 2.370 2.34 1.378 1.053 0.847 1.581 0.618 1.180

0.040 0.002 0.026 0.016 0.029 0.237 0.264 0.204 0.143 0.115 0.004 0.014 0.018 0.057

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Ind. Eng. Chem. Res., Vol. 46, No. 13, 2007

Figure 10. Comparison of deviations in normal boiling points (∆Tb) as predicted by different ANN cases and the ABC model 39.

4. Conclusions In this study a comprehensive performance evaluation of artificial neural networks (ANN) in addition to many correlations and methods in the literature for the correlation of critical temperatures and pressures and normal boiling points of longchain n-alkanes has been conducted. A 1-3-1 feed-forward, backpropagation ANN has been applied for the prediction of critical temperatures (Tc), critical pressures (Pc), and normal boiling points (Tb) of long-chain n-alkanes. Different training algorithms available in the neural networks toolbox in Matlab have been incorporated into computer programs for that purpose. Most of the investigated algorithms, although convergence is attained in a reasonable number of epochs (iterations), resulted in instable results in the sense that different predictions were obtained at different prediction trials. This reflects the high sensitivity of these algorithms to initial guesses of the network’s weight and bias parameters, which are randomly reinitialized each iteration. Once a proper training algorithm is figured out, however, the net makes prediction better than most available methods. In treating critical temperature, ANNs employing gradient descents, conjugate gradients, quasi-Newton, and optimization-based training algorithms show poor robustness. A training algorithm based on Bayesian regularization principles, however, gives the most robust (stable and regular) results. ANN employing the Bayesian regularization algorithm gives the best interpolative and extrapolative results.The limiting critical temperature (Tc∞) predicted by ANN is 1015 K. Regarding critical pressure correlation, algorithms based on gradient descent principles give the most robust correlation. ANN predicts a nonzero limiting (asymptotic) value for the critical pressure Pc∞ of 2.25 bar. As to the normal boiling points, different ANN algorithms have been investigated; all predict an asymptotic limiting value for the normal boiling point of heavy normal paraffin (Tb∞). The asymptotic values obtained, however, were found to depend on the training algorithm and functional (scaled) form of the treated raw data; when the reciprocal of the absolute boiling points versus reciprocal of the carbon numbers is treated, the Tb∞ obtained is in the range of 1070-1120 K, which is in good agreement with predictions made by other correlations. Nomenclature logsig ) log sigmoid transfer function purelin ) linear transfer function tansig ) hyperbolic tangent sigmoid transfer function trainbfg ) BFGS quasi-Newton back-propagation trainbr ) Bayesian regularization

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ReceiVed for reView September 26, 2006 ReVised manuscript receiVed April 8, 2007 Accepted April 16, 2007 IE061250C