New Predictive Methods for Estimating the Vaporization Enthalpies of

Centre Energe´tique et Proce´de´s, Ecole Nationale Supe´rieure des Mines de Paris, CEP/TEP. CNRS FRE 2861,. 35 Rue Saint Honore´, 77305 Fontaineb...
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Ind. Eng. Chem. Res. 2007, 46, 2665-2671

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New Predictive Methods for Estimating the Vaporization Enthalpies of Hydrocarbons and Petroleum Fractions Amir H. Mohammadi and Dominique Richon* Centre Energe´ tique et Proce´ de´ s, Ecole Nationale Supe´ rieure des Mines de Paris, CEP/TEP. CNRS FRE 2861, 35 Rue Saint Honore´ , 77305 Fontainebleau, France

Various models and correlations are available that can predict the vaporization enthalpy of hydrocarbons. The available methods normally have lower accuracy for predicting the vaporization enthalpy of heavy hydrocarbons and require further verification, because, during the development of the original predictive methods, experimental data describing the vaporization enthalpy of heavy hydrocarbons and petroleum fractions were not available. In this communication, after a quick review of the existing correlations reported in the literature, an empirical correlation is first proposed, which is capable of predicting the vaporization enthalpy of hydrocarbons, especially heavy hydrocarbons and petroleum fractions, from the specific gravity values and the normal boiling point temperatures. The capability of artificial neural networks (ANNs), as an alternative method, to predict the vaporization enthalpies of hydrocarbons and petroleum fractions is then demonstrated. Among the various neural networks reported in the literature, the feed-forward neural network method with a modified Levenberg-Marquardt algorithm is used. The method is trained using recent experimental data, especially for heavy hydrocarbons and petroleum fractions. Independent experimental data are used to validate and examine the reliability of this method. The results are also compared with the predictions of other predictive techniques. The predictions of this method are shown to be in better agreement with the experimental data reported in the literature, demonstrating the reliability of the ANN used in this work. 1. Introduction Accurate prediction of the vaporization enthalpy for hydrocarbons and petroleum fluids is of interest in the petroleum industry and can be involved in heat flux calculations, design and optimization of oil and gas production, transportation and processing facilities. They are also an indirect essential tool for the correlation and prediction of many physical phenomena, such as the solubility parameters of hydrocarbons. The existing predictive methods normally have been reported based on experimental data for pure and light hydrocarbons and, therefore, may not be accurate for heavy hydrocarbons and petroleum fluids. Prediction of the vaporization enthalpy via conventional thermodynamic models requires the use of many adjusted parameters. These models usually require considerable efforts to determine an appropriate relationship for fitting the experimental data. Therefore, there is still a need for simple, yet robust, predictive methods for quick estimation of the vaporization enthalpy. The objective of this work is to develop reliable predictive techniques for estimating the vaporization enthalpy of hydrocarbons and petroleum fluids. For this purpose, a quick review is first made on the existing correlations reported in the literature. An empirical correlation is then developed for predicting the vaporization enthalpy of hydrocarbons, especially heavy hydrocarbons and petroleum fractions, from the specific gravity values and the normal boiling point temperatures. The capability of artificial neural networks (ANNs) to estimate the vaporization enthalpy of hydrocarbons and petroleum fluids is also demonstrated. To our knowledge, this method has not been previously reported and can provide fast and accurate determination of the vaporization enthalpy. Among the various ANNs reported in the literature, the feed-forward (back-propagation) neural net* To whom correspondence should be addressed. Tel.: +(33) 1 64 69 49 65. Fax: +(33) 1 64 69 49 68. E-mail address: [email protected].

work (FNN) method1 with a modified Levenberg-Marquardt algorithm2-5 is used, which is known to be effective to represent the nonlinear relationships between variables in complex systems and can be regarded as a large regression method between input and output variables.1,6 To develop this method, reliable and more-recent literature data on the vaporization enthalpy of hydrocarbons and petroleum fractions are used. The developed method is then employed to predict independent experimental data (not used in the training and development of the ANN method) and the predictions are also compared with the results of other predictive tools. It is shown that the results of the ANN are in better agreement with the experimental data, demonstrating the reliability of this method to estimate the vaporization enthalpy of hydrocarbons and petroleum fluids. 2. Vaporization Enthalpy of Hydrocarbons The enthalpy of vaporization data are usually calculated at the normal boiling point temperature through an appropriate method and then are extended to the required temperature, using the well-known relation reported by Watson.7 The correlations for predicting the enthalpy of vaporization are grouped into two classes. The first class is comprised of equations that relate the enthalpy of vaporization at the normal boiling point temperature to the critical values and the normal boiling point.8 The second class consists of empirical correlations, which relate the enthalpy of vaporization at the normal boiling temperature to a few easily obtainable parameters, such as the normal boiling point temperature, the molecular weight, and the specific gravity. Because the critical constants of petroleum fractions may be unknown or inaccurately evaluated with empirical methods, the first class of correlations may suffer a significant loss of accuracy for petroleum fractions. The second class of correlations, with the required input parameters, which are known as the characterization properties of the petroleum fractions, are more practical in the petroleum industry.9,10

10.1021/ie0613927 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/08/2007

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Table 1. Second Class of Correlations with the Required Inputs for Predicting the Vaporization Enthalpy at the Normal Boiling Point Temperature of Hydrocarbons equationa

reference

) 88Tb ∆Hvap ) Tb(36.6 + 8.314 ln Tb) ∆Hvap

∆Hvap )

Trouton’s rule (as quoted in ref 10) Kistiakowsky (as quoted in ref 10)

Tb(58.20 + 5.94 ln M - {6.485[Tb - (263M)0.581]1.037}) M

∆Hvap ) 19.73809(1.8Tb)1.1347S0.0214

[

∆Hvap ) 4.1868Tb 9.08 + 4.36 ln Tb + 0.0068

[

()

∆Hvap ) 1081 + (S-0.01418Tb) 31.98 log10 Tb +

[

Vetere11

( )] )] ( )

Tb Tb2 + 0.0009 M M

(

Vetere9

22.12Tb-1.573 M

()

Tb Tb2 - 0.06662 + M M 3 Tb 7.833 × 10-5 + 19.334 ln S M

∆Hvap ) Tb 9.549 + 14.811 ln Tb + 12.346

Riazi and Daubert12

( )

Gopinathan and Saraf13

Fang et al.10

]

) enthalpy of vaporization at the normal boiling point temperature (given in units of J/g-mol); M ) molecular weight (given in units of g/g-mol); S ) specific gravity at 288.8/288.8 K; Tb ) normal boiling point temperature (given in Kelvin). a

∆Hvap

Table 1 summarizes the second class of correlations with the required inputs. As can be observed, a few equations have been reported for the second class. The majority of these correlations were developed based on the experimental data of the vaporization enthalpy for pure and light hydrocarbons, where direct calorimetric data of the vaporization enthalpy for petroleum fractions were not available at that time. Fang et al.10 have recently reported new experimental data for 58 petroleum fluids and have developed a new correlation showing that their method gives more-accurate results, particularly for heavy hydrocarbons. As can be seen in Table 1, the specific gravity (S), molecular weight (M), and normal boiling point temperature (Tb) can be used as inputs of empirical correlations to estimate the vaporization enthalpy of hydrocarbons. Because of the fact that experimental measurements of the specific gravity and normal boiling point temperature for hydrocarbons, especially petroleum fluids, are normally easier to obtain than molecular weight measurements, any predictive method that uses S and Tb as inputs to predict the vaporization enthalpy of hydrocarbons therefore can have more-practical applications. Therefore, we develop new predictive methods that are based on S and Tb only.

of neurons in the hidden layer(s); the presence of a few neurons produces a network with low precision and a higher number leads to overfitting and bad-quality interpolations and extrapolations. The use of techniques such as Bayesian regularization, together with a Levenberg-Marquardt algorithm, can help overcome this problem.14-16 In this work, the FNN method with a single hidden layer is used for computation of the vaporization enthalpy of hydrocarbon fluids (output neuron), as a function of the specific gravity and normal boiling point temperature of the hydrocarbons (input neurons). The bias is set to 1, to add a constant to the weighted sum for each neuron of the hidden layer.1,6 In this method, each neuron of the hidden layer performs two tasks: a weighted summation of its input and the application of the transfer function to this summation.1,6 The exponential sigmoid transfer function is used for this purpose. The neuron of the output layer simply performs a weighted summation of the outputs of the hidden neurons.1,6 The mathematical form for the vaporization enthalpy (∆Hvap) can be expressed by eq 1: n

∆Hvap )

3. Artificial Neural Network Artificial neural networks have large numbers of computational units that are connected in a massively parallel structure,14-16 and they do not require an explicit formulation of the mathematical or physical relationships of the handled problem.2 These methods are first subjected to a set of training data, consisting of input data together with the corresponding outputs. After a sufficient number of training iterations, the neural network learns the patterns in the data fed into it and creates an internal model, which it uses to make predictions for new inputs.17 The accuracy of model representation is dependent directly on the topology of the neural network.1 The most commonly used ANNs are the feed-forward neural networks, which are designed with one input layer, one output layer, and hidden layers. One of the main advantages of a FNN is that the connection pattern does not contain any cycles and the number of neurons in the input and output layers is equal to the number of inputs and outputs, respectively.1 The disadvantage of FNNs is the determination of the ideal number

wif(Vi) ∑ i)1

(1)

1 1 + e-Vi

(2)

with

f(Vi) )

Vi ) w1iS + w2iTb + w3i

(3)

where ∆Hvap is expressed in units of kJ/g-mol and the parameters w, f, V, and n denote weight, function, weighted sum of input to the hidden neuron i, and the number of neurons in the hidden layer, respectively. Subscript i represents the hidden layer. As can be seen, the inputs that represent the independent variables enter the neurons of the input layers and then the transfer function f(Vi) converts the inputs to outputs in the neurons. The parameters w1-3i (for i ) 1,..., n) in the summations, which are usually referred as the weights, are the fitting parameters of the ANN. These parameters can be found by applying a leastsquares regression procedure to a given set of experimental data. (These parameters are available on request.) The fitting

Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2667

Figure 1. Topology of the neural network method used to predict the vaporization enthalpy of hydrocarbons (bias, 1; dot (b) represent the neuron). Output neuron is the vaporization enthalpy; input neurons are the specific gravity and the normal boiling point temperature.

Figure 3. Results of the artifical neural network (ANN) method versus the experimental values (from Tables 2 and 3) for the vaporization enthalpy of hydrocarbons: (4) data for vaporization enthalpy of petroleum fractions reported in Table 2 used for training (and testing), (2) data for vaporization enthalpy of petroleum fractions reported in Table 2 used for validation, (O) data for vaporization enthalpy of pure hydrocarbons reported in Table 3 used for training (and testing), and (b) data for vaporization enthalpy of pure hydrocarbons reported in Table 3 used for validation.

Figure 2. Objective function (AAD%) versus the number of neurons in the hidden layer.

procedure, which is normally called the learning of the ANN, is performed using a modified Levenberg-Marquardt algorithm:2-5

wj ) wj-1 - [H h + µjhI ]-1∇J(wj-1)

(4)

where µ, J, H h , and hI are the step values of the LevenbergMarquardt algorithm,2-5 the Jacobian matrix of the first derivative of global error to weight, the Hessian matrix, and the identity matrix, respectively. The objective function corresponding to the enthalpy of vaporization is the sum of squares of the relative deviations between the experimental and calculated values. To develop the ANN, the features in the training set are first standardized with unit variance. This standardization is necessary to prevent nonuniform learning. After standardization, a randomly chosen sample is put into the network for feedforwarding. For each sample, the features are used as inputs, and these inputs are processed from the hidden layer to the output layer. After each sample is passed through the network, the output value is calculated. After calculating the output value of the network, this value is compared to the target value and the difference is used to determine the training error. This process is repeated for every sample in each training. After feedforwarding the entire training set, the weights of the network are adjusted to meet the minima of the objective function. As a first step, the general behavior of the network is evaluated

Figure 4. Difference between the experimental value and the ANN result versus the experimental value for the vaporization enthalpy of hydrocarbons: (4) data for vaporization enthalpy of petroleum fractions reported in Table 2 used for training (and testing), (2) data for vaporization enthalpy of petroleum fractions reported in Table 2 used for validation, (O) data for vaporization enthalpy of pure hydrocarbons reported in Table 3 used for training (and testing), and (b) data for vaporization enthalpy of pure hydrocarbons reported in Table 3 used for validation.

using all data to confirm that the network could be trained. This can also yield the network parameters. The parameters obtained from general training are applied to individual cycles of training and testing. In the general training and the testing, the optimization algorithm, adjusting the weights after feedforwarding the entire training set, is used as the learning algorithm. The number of neurons in the hidden layer can be varied to search for both the lowest value of the minimized objective function and generalizing capability of the ANN method for various conditions.14-16 The developed model is finally validated against validation data (or independent data (not used in training and testing)). 4. Results and Discussions As mentioned earlier, few correlations are available that relate the enthalpy of vaporization at the normal boiling point temperature to parameters such as the normal boiling point temperature, the molecular weight, and the specific gravity. It is therefore of interest to develop first a new empirical correlation to estimate the vaporization enthalpy of the hydro-

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Table 2. Comparison (AAD%) of Vaporization Enthalpies Predicted/Calculated by Various Correlations for Petroleum Fractionsa petroleum fraction

Tb (K)

S

M (g/g-mol)

∆Hvap (kJ/g-mol)

Troutonb

Kistiakowskyb

R-D12b

Vetere11b

Vetere9b

G-S13b

Fang et al.10b

new correlation

ANNc

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

355.5 357.2 377.2 377.5 380.5 381.1 385.4 402.7 406.6 407.7 411.0 427.4 429.1 431.0 431.3 433.3 433.6 434.9 437.1 437.4 439.8 443.0 445.7 446.2 452.0 455.4 457.2 457.3 461.1 467.6 477.7 478.3 481.5 490.7 502.4 503.7 506.8 509.2 526.6 530.1 530.9 531.7 542.4 556.7 556.8 557.4 559.3 577.7 582.1 583.3 595.0 601.2 607.0 610.5 624.4 629.6 634.3 646.8

0.7015 0.7201 0.7321 0.7325 0.7253 0.7416 0.7534 0.7525 0.7594 0.7497 0.7717 0.7695 0.7709 0.7722 0.778 0.7603 0.7739 0.7750 0.7761 0.7763 0.7777 0.7791 0.7805 0.7848 0.7833 0.7926 0.7820 0.7948 0.8143 0.8069 0.7959 0.8024 0.8136 0.8090 0.8526 0.8148 0.8155 0.8264 0.8293 0.8268 0.8390 0.8319 0.8365 0.8398 0.8483 0.8414 0.8482 0.8640 0.8558 0.8512 0.8602 0.8748 0.8700 0.8699 0.8814 0.8798 0.8786 0.8933

95 95 108 108 107 107 120 120 120 119 134 133 130 131 133 131 132 133 139 134 136 137 139 147 147 155 146 147 159 171 162 163 162 171 201 180 180 187 199 199 205 199 210 221 226 220 232 243 249 244 256 267 272 268 295 299 296 325

30.8 32.02 33.83 32.89 33.95 33.24 36.21 35.46 35.29 37.27 37.09 38.74 38.99 39.46 39.16 39.69 39.39 39.9 39.14 40.51 40.44 39.66 41.09 40.94 41.4 39.88 42.74 42.45 41.05 41.39 45.13 45.38 45.44 44.78 51.27 48.03 47.82 48.21 49.79 51.16 51.63 52.14 52.56 55.5 54.97 55.36 53.07 58.17 58.45 58.69 59.71 61.37 62.34 62.86 66.37 68.15 69.65 73.03

1.6 1.8 1.9 1 1.4 0.9 6.3 0.1 1.4 3.7 2.5 2.9 3.2 3.9 3.1 3.9 3.1 4.1 1.7 5 4.3 1.7 4.5 4.1 3.9 0.5 5.9 5.2 1.2 0.6 6.9 7.2 6.8 3.6 13.8 7.7 6.7 7 6.9 8.8 9.5 10.3 9.2 11.7 10.9 11.4 7.3 12.6 12.4 12.5 12.3 13.8 14.3 14.5 17.2 18.7 19.9 22.1

1.3 4.6 4.1 1.3 3.5 1.3 8.3 1.7 0.2 5.2 3.9 4 4.2 4.9 4 4.8 4.1 5 2.6 5.8 5.1 2.4 5.2 4.8 4.5 0 6.3 5.6 1.5 0.8 6.9 7.3 6.7 3.4 13.4 7.3 6.3 6.5 6.1 8 8.6 9.4 8.1 10.5 9.6 10.1 5.9 11.1 10.8 10.9 10.5 12 12.4 12.6 15.2 16.6 17.7 19.9

2.9 6 5.3 2.6 4.8 2.5 9.3 2.7 1.1 6.1 4.7 4.6 4.8 5.5 4.7 5.5 4.7 5.6 3.2 6.4 5.6 3 5.7 5.2 4.9 0.4 6.7 6 1.8 1.1 7.1 7.4 6.8 3.4 13.2 7.2 6.2 6.4 5.8 7.7 8.3 9.1 7.7 10 9.1 9.6 5.4 10.4 10.1 10.2 9.7 11.1 11.5 11.7 14.2 15.6 16.7 18.8

1.3 4.8 3.5 0.7 3.4 1.2 6.3 1.1 0.1 5.2 1.8 3.2 4 4.7 3.6 4.9 4 4.8 1.7 5.7 4.9 2.3 5 3.5 3.6 1.7 5.9 5.1 0.3 1.9 5.9 6.2 6 2.2 10.4 6 5.1 4.9 4.3 6.4 6.7 7.9 6.3 8.6 7.4 8.3 3.3 8.7 8.2 8.7 8.1 9.2 9.7 10.2 12 13.6 15 16.5

3.5 0.2 0.5 3.4 1.2 3.6 4.2 3.1 4.8 0.4 0.4 0.7 0.7 0 0.8 0.1 0.8 0.1 2.2 1 0.3 2.6 0.4 0.2 0.2 4.7 1.5 0.9 3.1 3.5 2.3 2.7 2 1.4 9.7 2.7 1.6 2.1 1.6 3.5 4.4 5 3.7 6.2 5.4 5.8 1.7 7 6.7 6.7 6.4 8 8.5 8.5 11.5 13 14.1 16.5

1.9 5.1 4.5 1.7 3.9 1.7 8.6 2 0.4 5.4 4.1 4 4.2 4.9 4.1 4.9 4.1 5 2.6 5.8 5.1 2.4 5.2 4.7 4.4 0.1 6.2 5.5 1.4 0.6 6.6 7 6.5 3 13.1 6.9 5.8 6.1 5.5 7.4 8.1 8.8 7.5 9.8 8.9 9.4 5.2 10.3 9.9 10 9.6 11 11.4 11.6 14.1 15.6 16.7 18.8

2.5 5.2 4.3 1.5 4.1 1.4 7.5 1.1 0.5 4.8 2.2 2.4 2.7 3.3 2.2 3.6 2.4 3.2 0.4 4 3.1 0.3 2.9 2 1.7 3.6 3.5 2.5 2.8 3.9 2.7 2.9 2 2 6.6 1.3 0.1 0.2 1.7 0.2 0.5 1.5 0.7 1 0.2 0.5 4.5 0.3 0.8 0.6 2 1.1 0.8 0.9 1 2.4 3.3 4.9

2.7 5.7 4.7 1.9 4.1 1.7 8.4 1.3 0.5 4.6 2.9 2.3 2.4 3.0 2.1 3.1 2.1 2.9 0.4 3.7 2.8 0.1 2.6 2.1 1.5 3.4 3.2 2.3 2.4 3.4 2.5 2.8 1.9 2.1 7.3 1.2 0.0 0.0 1.6 0.2 0.8 1.6 0.5 1.2 0.1 0.7 4.2 0.2 0.3 0.2 1.5 0.5 0.3 0.3 1.6 3.0 4.0 5.5

2.1 5.2 4.5 1.5* 3.8* 1.4* 9.1 0.7 1.0 4.1 2.7* 1.4* 1.5* 2.1 1.2 2.2 1.0 1.9 0.7 2.7* 1.7* 1.2* 1.5 0.9 0.4 4.5 2.3 1.0 3.9 4.7 2.0 1.9* 0.6* 2.7* 4.2* 1.1 0.0 0.8 1.2 1.1 0.3* 2.1* 0.3* 2.5* 0.3 1.9 3.6 0.8 0.0 0.7 1.7* 2.6* 1.7* 1.8* 1.6 0.1 0.9 0.8

6.9

6.8

7.0

5.6

3.8

6.5

2.3

2.2

2.0

AAD%

N AAD% ) (1/N)∑i)1 |(experimental value - predicted/calculated value)/(experimental value)| b Quoted in Fang et al.10 c Values marked by an asterisk represent validated data; other data in this

a

carbons, especially for heavy hydrocarbons and petroleum fractions. The following equation is suggested:

∆Hvap ) 10.6988 + 0.000511Tb0.8008S0.1520(ln Tb + Tb)

(5)

where ∆Hvap is given in units of kJ/g-mol. Experimental vaporization enthalpy data reported in Tables 2 and 3 for 58 petroleum fraction10 and 64 pure hydrocarbons,9,10,13,18 with wide ranges of specific gravities and normal boiling point temperatures, were used to develop the aforementioned equation and

× 100. Experimental data taken from Fang et al.10 column have been used for training (and testing).

adjust its parameters. Note that the experimental data reported in Tables 2 and 3 were also mentioned by Fang et al.10 As can be seen, the new correlation has a form similar to that of Gopinathan and Saraf’s13 and Riazi and Daubert’s12 equations, in which the normal boiling point temperature and specific gravity are used as inputs. The parameters of this equation were tuned using the experimental enthalpy of vaporization data for 58 petroleum fractions over a wide boiling point temperature range (from 231.1 K to 722.8 K), obtained from four different crude oils (one from Russia, one from Iran, and two from China)

Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2669 Table 3. Comparison of Enthalpies of Vaporization Calculated by the Fang et al. Correlation10 and This Work for Pure Hydrocarbonsa hydrocarbon

Tb (K)b

Sb

Mb (g/g-mol)

exp. ∆Hvap (kJ/ g-mol)b

Fang et al.10b

AD%b,c

new correlation

AD%

ANN

AD%d

reference for exp. datab

propane n-butane isobutane n-pentane isopentane 2,2-dimethyl propane n-hexane isohexane 3-methyl pentane 2,2-dimethyl butane 2,3-dimethyl butane n-heptane 2-methyl hexane 3-methyl hexane 3-ethyl pentane 2,2-dimethyl pentane 2,3-dimethyl pentane 2,4-dimethyl pentane 3,3-dimethyl pentane 2,2,3-trimethyl n-octane 2-methyl heptane 3-methyl heptane 4-methyl heptane 3-ethylhexane 2,2-dimethyl hexane 2,3-dimethyl hexane 2,4-dimethyl hexane 2,5-dimethyl hexane 3,3-dimethyl hexane 3,4-dimethyl hexane 3-ethyl-2-methyl pentane 3-ethyl-3-methyl pentane 2,2,3-trimethyl pentane 2,2,4-trimethyl pentane 2,3,3-trimethyl pentane 2,3,4-trimethyl pentane n-nonane n-decane n-dodecane n-octadecane n-nonadecane benzene toluene o-xylene m-xylene p-xylene cyclohexane methyl cyclohexane cyclopentane methyl cyclopentane ethyl cyclohexane cis-butene-2 ethyl benzene n-heneicosane n-docosane n-tricosane n-tetracosane n-pentacosane n-hexacosane n-heptacosane n-octacosane n-nonacosane n-triacontane

231.1 272.7 261.3 309.2 301.0 282.7 341.9 333.4 336.4 322.9 321.1 371.6 363.2 365.0 366.6 352.4 362.9 352.4 352.4 352.4 398.8 390.8 392.1 390.9 391.7 382.0 388.8 382.6 382.3 385.1 390.9 388.8 391.4 383.0 372.4 387.9 386.6 426.5 447.0 487.5 590.0 603.0 353.1 383.8 417.0 412.3 411.5 353.0 374.0 322.4 345.0 404.9 276.9 409.3 629.7 641.8 653.3 664.5 675.1 685.4 695.4 704.8 714.0 722.8

0.508 0.585 0.564 0.631 0.625 0.597 0.664 0.658 0.669 0.655 0.667 0.689 0.684 0.692 0.703 0.679 0.700 0.678 0.698 0.695 0.707 0.703 0.711 0.709 0.718 0.700 0.717 0.705 0.698 0.715 0.724 0.724 0.732 0.721 0.697 0.731 0.724 0.722 0.735 0.753 0.786 0.789 0.885 0.872 0.885 0.869 0.866 0.784 0.774 0.751 0.754 0.793 0.628 0.872 0.795 0.798 0.801 0.803 0.805 0.807 0.809 0.810 0.812 0.813

44.1 58.1 58.1 72.2 72.2 72.2 86.2 86.2 86.2 86.2 86.2 100.2 100.2 100.2 100.2 100.2 100.2 100.2 100.2 100.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 114.2 128.3 142.3 170.3 254.5 268.5 78.1 92.1 106.2 106.2 106.2 84.2 98.2 70.1 84.2 112.2 56.1 106.2 296.6 310.6 324.6 338.7 352.7 366.7 380.7 394.8 408.8 422.8

19.04 22.44 21.3 25.79 24.69 22.74 28.85 27.79 28.06 26.31 27.38 31.77 30.62 30.89 31.12 29.23 30.46 29.55 29.62 28.9 34.41 33.26 33.66 33.35 33.59 32.07 33.17 32.51 32.54 32.31 33.24 32.93 32.78 31.94 30.79 32.12 32.36 36.39 38.95 43.31 54.37 56.03 30.49 33.17 36.51 36.11 35.76 29.81 31.44 27.3 29.08 34.04 23.34 35.57 66.08 67.82 69.51 71.16 72.74 74.31 75.8 77.3 78.73 80.12

19.18 22.44 21.48 25.45 24.82 23.26 28.38 27.68 28.01 26.86 26.84 31.24 30.51 30.74 30.98 29.6 30.65 29.59 29.79 29.76 34.05 33.3 33.5 33.38 33.55 32.52 33.28 32.62 32.52 32.94 33.54 33.35 33.66 32.82 31.66 33.34 33.16 37.06 39.5 44.55 59.05 61.06 30.97 33.78 37.29 36.7 36.6 30.29 32.24 27.4 29.41 35.44 23.08 36.45 65.28 67.24 69.11 70.96 72.72 74.45 76.14 77.74 79.3 80.81

0.8 0 0.8 1.3 0.5 2.3 1.6 0.4 0.2 2.1 2 1.7 0.4 0.5 0.4 1.3 0.6 0.1 0.6 3 1 0.1 0.5 0.1 0.1 1.4 0.3 0.3 0.1 1.9 0.9 1.3 2.7 2.7 2.8 3.8 2.5 1.9 1.4 2.9 8.6 9 1.6 1.8 2.1 1.6 2.3 1.6 2.5 0.4 1.1 4.1 1.1 2.5 1.2 0.9 0.6 0.3 0 0.2 0.5 0.6 0.7 0.9

19.19 22.35 21.44 25.45 24.74 23.16 28.49 27.68 28.00 26.73 26.61 31.46 30.60 30.82 31.03 29.54 30.65 29.53 29.62 29.61 34.35 33.48 33.66 33.53 33.66 32.56 33.35 32.65 32.59 32.95 33.60 33.38 33.69 32.76 31.58 33.32 33.15 37.45 39.87 44.90 59.15 61.11 30.38 33.50 37.20 36.60 36.50 30.01 32.08 27.02 29.13 35.43 22.80 36.28 65.25 67.17 69.02 70.86 72.61 74.34 76.04 77.65 79.24 80.78

0.8 0.4 0.6 1.3 0.2 1.9 1.2 0.4 0.2 1.6 2.8 1.0 0.0 0.2 0.3 1.1 0.6 0.1 0.0 2.4 0.2 0.7 0.0 0.5 0.2 1.5 0.5 0.4 0.1 2.0 1.1 1.4 2.8 2.6 2.6 3.7 2.4 2.9 2.4 3.7 8.8 9.1 0.3 1.0 1.9 1.4 2.1 0.7 2.0 1.0 0.2 4.1 2.3 2.0 1.3 1.0 0.7 0.4 0.2 0.0 0.3 0.4 0.6 0.8

19.00 22.29 21.23 25.30 24.72 23.02 28.28 27.53 28.02 26.76 26.99 31.30 30.45 30.79 31.12 29.44 30.74 29.42 29.79 29.75 34.24 33.38 33.69 33.55 33.79 32.51 33.47 32.67 32.50 33.07 33.78 33.56 33.90 32.92 31.55 33.51 33.33 37.12 39.20 42.73 55.42 58.19 31.22 32.80 36.84 35.81 35.66 29.92 31.80 28.11 29.44 35.16 24.02 35.49 64.38 67.17 69.64 71.81 73.60 75.09 76.31 77.26 78.04 78.66

0.2 0.7 0.3 1.9* 0.1* 1.2* 2.0 0.9 0.1 1.7 1.4 1.5 0.6 0.3 0.0 0.7 0.9* 0.4* 0.6 2.9 0.5 0.4 0.1 0.6 0.6 1.4 0.9* 0.5* 0.1* 2.3 1.6 1.9 3.3 3.0 2.4 4.1* 2.9* 2.0 0.6 1.4 1.9 3.7* 2.3* 1.1 0.9 0.8 0.3 0.4 1.1 2.9 1.2 3.3 2.8* 0.2* 2.6 1.0 0.2 0.9 1.2 1.0 0.7 0.1* 0.9* 1.9

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AAD%

1.5

1.4

1.3

a

Experimental data for vaporization enthalpy of pure hydrocarbons reported in Gopinathan and Saraf13 have been taken from Reid et al.,8 Majer and Svoboda,19 and Perry and Chilton.20 (Information taken from Fang et al.10) b Quoted in Fang et al.10 c AD% ) |(experimental value - predicted/calculated value)/experimental value| × 100. d Values marked by an asterisk represent validated data; other data in this column have been used for training (and testing).

and 64 pure hydrocarbons (C3-C30), reported in Tables 2 and 3, respectively. These tables also show comparisons of the enthalpy of vaporization that has been predicted by different correlations. As can be observed, the newly developed correlation yields good results, especially for heavy hydrocarbons (average absolute deviation of AAD% ) 1.4) and petroleum

fractions (AAD% ) 2.2). As can be seen from Table 2, the AAD% values for Fang et al.’s10 correlation (AAD% ) 2.3) and this work (AAD% ) 2.2) are the best of the seven correlations. However, the new correlation requires four adjustable parameters and only two inputs (normal boiling point temperature and specific gravity). Note that other correlations

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Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007

Table 4. Number of Neurons, Hidden Layers, Parameters, and Data Used in This Method for Predicting the Vaporization Enthalpy of Hydrocarbons property number of neurons layer 1 layer 2 layer 3 number of hidden layers number of parameters number of data used for training (and testing) number of data used for validation type of function

value/comment 2 4 1 1 17 86 36 exponential sigmoid

give reasonable results for the lower and middle petroleum fractions and have much greater error for the heavier fractions. This may be due to the fact that these methods were essentially developed for pure hydrocarbons, which can be found in the literature and usually cover only a limited boiling point range. As mentioned earlier, one of the objectives of this work is to indicate the capability of ANN, as an alternative method, to estimate the vaporization enthalpy of hydrocarbons and petroleum fractions. The ANN method shown in Figure 1 and detailed in Table 4, with one hidden layer, was then used for the computation of the enthalpy of vaporization, as a function of the specific gravity and normal boiling point temperature. The experimental data shown in Tables 2 and 3 were used for training (and testing). The number of the hidden neurons was varied between 2 and 6, and the best value according to both the accuracy of the fit (minimum value of the objective function) and the predictive power of the neural network was observed to be 4. Figure 2 shows the objective function (AAD%) versus the number of neurons in the hidden layer. A preliminary study shows that the results should be acceptable enough as the input variables were well chosen (two input variables, i.e., S and Tb) and there were sufficient data to train the network and to avoid an overfitting problem. (The best way to avoid overfitting problem is to use sufficient training data.) Tables 2 and 3 show comparisons of the vaporization enthalpies predicted by various correlations and validated data from the ANN method. As can be observed, the newly developed ANN method yields good results, especially for heavy hydrocarbons (AAD% ) 1.3) and petroleum fractions (AAD% ) 2). The tables show that the AAD% values for the ANN and the new empirical correlation are the lowest of the all the predictive methods. Other correlations gave reasonable results for the lower and middle petroleum fractions but undesired values for the heavier fractions (except Fang et al.’s10 correlation). A comparison is finally made in Figures 3 and 4 between experimental values and the results coming from the ANN method for the vaporization enthalpy of petroleum fractions and hydrocarbons reported in Tables 2 and 3, respectively. As can be observed, the maximum difference between the experimental values and the model results is ∼2.5 kJ/g-mol, which demonstrates the ability of the ANN method developed in this work to estimate the vaporization enthalpy of hydrocarbons and petroleum fractions from their specific gravities and normal boiling point temperatures. 5. Conclusion A feed-forward artificial neural network method and an empirical correlation were developed to relate the vaporization enthalpy of hydrocarbons and petroleum fractions at the normal boiling temperature to the normal boiling point temperature and

the specific gravity values. The methods worked well for wide ranges of pure hydrocarbons and petroleum fractions and gave reasonable improvement to the existing methods, particularly for heavy petroleum fractions and heavy hydrocarbons. Nomenclature N AAD% ) average absolute deviation ((1/N)∑i)1 |(experimental value - predicted/calculated value)/experimental value| × 100) AD% ) absolute deviation (|(experimental value - predicted/ calculated value)/experimental value| × 100) ANN ) artificial neural network FNN ) feed-forward (back-propagation) neural network H h ) Hessian matrix hI ) identity matrix J ) Jacobian matrix of the first derivative of global error to weight M ) molecular weight N ) number of experimental data S ) specific gravity T ) temperature f ) function n ) number of neurons in hidden layer V ) weighted sum of input to hidden neuron w ) weight 1 ) bias

Greek Symbols ∆H ) vaporization enthalpy µ ) step values of Levenberg-Marquardt algorithm Subscripts b ) normal boiling point i ) index for hidden layer Superscript vap ) vaporization Literature Cited (1) Mohammadi, A. H.; Richon, D. Use of Artificial Neural Networks for Estimating Water Content of Natural Gases. Ind. Eng. Chem. Res. 2007, 46, 1431-1438. (2) Rivollet, F. Etude des proprie´te´s volume´triques (PVT) d’hydrocarbures le´gers (C1-C4), du dioxyde de carbone et de l’hydroge`ne sulfure´: Mesures par densime´trie a` tube vibrant et mode´lisation, Ph.D. thesis, Paris School of Mines, Paris, France, December 2005. (3) Wilamowski, B.; Iplikci, S.; Kayank, O.; Efe, M. O. In International Joint Conference on Neural Networks (IJCNN’01), Washington, DC, July 15-19, 2001; pp 1778-1782. (4) Marquardt, D. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 1963, 11, 431-441. (5) Levenberg, K. A Method for the Solution of Certain Problems in Least Squares. Q. Appl. Math. 1944, 2, 164-168. (6) Normandin, A.; Grandjean, B. P. A.; Thibault, J. PVT Data Analysis Using Neural Network Models. Ind. Eng. Chem. Res. 1993, 32, 970-975. (7) Smith, J. M.; van Ness, H. C. Introduction to Chemical Engineering Thermodynamics, Third Edition; McGraw-Hill: New York, 1985. (8) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th Edition; McGraw-Hill: New York, 1987. (9) Vetere, A. Methods to predict the vaporization enthalpies at the normal boiling temperature of pure compounds revisited. Fluid Phase Equilib. 1995, 106, 1-10. (10) Fang, W.; Lei, Q.; Lin, R. Enthalpies of vaporization of petroleum fractions from vapor pressure measurements and their correlation along with pure hydrocarbons. Fluid Phase Equilib. 2003, 205, 149-161. (11) Vetere, A. New correlations for predicting vaporization enthalpies of pure compounds. Chem. Eng. J. 1979, 17, 157-162. (12) Riazi, M. R.; Daubert, T. E. Simplify Property Predictions. Hydrocarbon Process. 1980, 59, 115-116.

Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2671 (13) Gopinathan, N.; Saraf, D. N. Predict heat of vaporization of crudes and pure components: Revised II. Fluid Phase Equilib. 2001, 179, 277284. (14) Chouai, A.; Laugier, S.; Richon, D. Modeling of thermodynamic properties using neural networks: Application to refrigerants. Fluid Phase Equilib. 2002, 199, 53-62. (15) Piazza, L.; Scalabrin, G.; Marchi, P.; Richon, D. Enhancement of the extended corresponding states techniques for thermodynamic modelling. I. Pure fluids. Int. J. Refrig. 2006, 29, 1182-1194. (16) Scalabrin, G.; Marchi, P.; Bettio, L.; Richon, D. Enhancement of the extended corresponding states techniques for thermodynamic modelling. II. Mixtures. Int. J. Refrig. 2006, 29, 1195-1207. (17) Elgibaly, A. A.; Elkamel, A. M. A new correlation for predicting hydrate formation conditions for various gas mixtures and inhibitors. Fluid Phase Equilib. 1998, 152, 23-42.

(18) Kudchadker, A. P.; Zwolinski, B. J. Vapor Pressure and Boiling Points of Normal Alkanes, C21 to C100. J. Chem. Eng. Data 1966, 11, 253255. (19) Majer, V.; Svoboda, M. R. Enthalpies of Vaporization of Organic Compounds; IUPAC Data Series No. 32; Blackwell Scientific Publications: Oxford, U.K., 1985. (20) Perry, R. H.; Chilton, C. H. Chemical Engineers Handbook, 5th Edition; McGraw-Hill: Tokyo, 1993.

ReceiVed for reView October 30, 2006 ReVised manuscript receiVed February 23, 2007 Accepted February 26, 2007 IE0613927