- Email: [email protected]
Determining phase diagrams of tetrahydrofuran+methane, carbon dioxide or nitrogen clathrate hydrates using an artificial neural network algorithm
Chemical Engineering Science 65 (2010) 6059–6063
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Note
Determining phase diagrams of tetrahydrofuran +methane, carbon dioxide or nitrogen clathrate hydrates using an artificial neural network algorithm Amir H. Mohammadi a,n, Jose´ F. Martı´nez-Lo´pez b, Dominique Richon a a b
´nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP—Centre E ´nica y Quı´mica Fı´sica, Facultad de Ciencias (Edificio D), Universidad de Zaragoza, c/. Pedro Cerbuna, 12, C.P. 50009 Zaragoza, Spain Departamento de Quı´mica Orga
a r t i c l e in fo
abstract
Article history: Received 18 November 2009 Received in revised form 27 June 2010 Accepted 20 July 2010 Available online 27 July 2010
In this communication, we have developed a feed-forward artificial neural network algorithm for estimating dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+ methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in the aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). In order to develop this algorithm, the most reliable experimental data reported in the literature on the dissociation pressures of the aforementioned binary hydrates have been used. Moreover, we report few experimental data on the dissociation pressures of the binary hydrates of tetrahydrofuran +carbon dioxide or nitrogen at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which were measured using an isochoric pressure-search method. The latter experimental data are used to verify the reliability of the corresponding experimental data reported in the literature. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Gas hydrate Tetrahydrofuran Methane Carbon dioxide Nitrogen Artificial neural network algorithm
1. Introduction It has been proven that tetrahydrofuran (THF) can form structure II of clathrate hydrates with or without the presence of a small molecule, like methane (Sloan and Koh, 2008). Adding tetrahydrofuran to water (with low-intermediate concentrations) normally leads to reduction of hydrate formation pressures of gases with respect to pure water (Sloan and Koh, 2008). This effect is called hydrate promotion effect, which is used in gas separation, storage and transportation processes using gas hydrate crystallization (Sloan and Koh, 2008). Hydrate phase equilibrium information is therefore required for designing the aforementioned processes. Sufficient phase equilibrium measurements for the clathrate hydrates of tetrahydrofuran+ gas have been reported for the tetrahydrofuran+ methane (Zhang et al., 2005; Seo et al., 2001; De Deugd et al., 2001; Mohammadi and Richon, 2009), tetrahydrofuran+ carbon dioxide (Sabil and Peters, 2007; Delahaye et al., 2006), and tetrahydrofuran+nitrogen (Seo et al., 2001) systems. Few thermodynamic models have been developed for clathrate hydrates containing THF. Strobel et al. (2009) have done a brief review on these thermodynamic models: De Deugd et al. (2001) used a model, in which the hydrate phase chemical potential is calculated using the van der Waals and Platteeuw (1959) theory.
n
Corresponding author. Tel.: +33 1 64 69 49 70; fax: + 33 1 64 69 49 68. E-mail address: [email protected] (A.H. Mohammadi).
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.07.013
Their model (De Deugd et al., 2001) is capable of accurately predicting the dissociation conditions for the binary clathrate hydrates of tetrahydrofuran+ methane as well as other watersoluble hydrate formers. Seo et al. (2001) presented a similar thermodynamic model, which sufficiently reproduces equilibrium conditions for tetrahydrofuran+ methane and tetrahydrofuran+ nitrogen hydrates. The thermodynamic model developed by Lee et al. (2006) for tetrahydrofuran+hydrogen hydrates and hydrocarbon+tetrahydrofuran hydrates is believed to be unrealistic (Strobel et al., 2009). Although these thermodynamic models reproduce hydrate equilibria for the specific systems, it is argued that the parameters used between the different models vary significantly (Strobel et al., 2009). Strobel et al. (2009) were able to predict the phase behavior of any of these given systems with a single set of consistent parameters within the framework of a single thermodynamic model. The aim of this communication is to develop an alternative tool based on the feed-forward neural network (FNN) algorithm to estimate dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+ methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in the aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). Reliable literature data along with few experimental data generated in this work based on isochoric pressure-search method (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Tohidi et al., 2000) were used to develop this algorithm. It is shown that this algorithm can be regarded a useful tool for modeling the aforementioned systems.
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A.H. Mohammadi et al. / Chemical Engineering Science 65 (2010) 6059–6063
2. Feed-forward neural network algorithm
3. Results and discussion
Feed-forward neural networks are the most frequently used type of artificial neural networks (ANN), which are designed with one input layer, one output layer and hidden layers (Rivollet, 2005; Wilamowski et al., 2001; Normandin et al., 1993; Chouai et al., 2002; Piazza et al., 2006; Scalabrin et al., 2006; Mohammadi and Richon, 2010b, 2008). The number of neurons in the input and output layers is equal to the number of inputs and outputs, respectively (Rivollet, 2005; Wilamowski et al., 2001; Normandin et al., 1993; Chouai et al., 2002; Piazza et al., 2006; Scalabrin et al., 2006; Mohammadi and Richon, 2010b, 2008). In the FNN model, the input layer of the network receives all the input data and introduces scaled data to the network (Mohammadi and Richon, 2010b, 2008). The data from the input neurons are propagated through the network via weighted interconnections (Mohammadi and Richon, 2010b, 2008). Every i neuron in a k layer is connected to every neuron in adjacent layers (Mohammadi and Richon, 2010b, 2008). The i neuron within the hidden k layer performs the following tasks: summation of the arriving weighted inputs and propagations of the resulting summation through an activation function, f, to the adjacent neurons of the next hidden layer or to the output neuron(s). In this work, the activation function is tangent sigmoid (Mohammadi and Richon, 2008)
The ANN algorithm shown in Fig. 1, with one hidden layer was used for the computation of the logarithm of hydrate dissociation pressure as a function of temperature and concentration of tetrahydrofuran in aqueous solution. In order to develop this algorithm, the experimental data shown in Figs. 2–4 were used. We also report few experimental data for the dissociation conditions of the tetrahydrofuran+carbon dioxide and tetrahydrofuran+ nitrogen binary clathrate hydrates at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which
f ðxÞ ¼
1ex 1 þex
x A ½1, þ1
and
f ðxÞ A ½1, þ 1
N k1 X
Input layer
ðwk1,j,i Ik1,j Þ þ bk,i
1
ð1Þ
where x stands for the parameter of the activation function. A bias term, b, is associated with each interconnection in order to introduce a supplementary degree of freedom. The expression of the weighted sum, S, to the ith neuron in the kth layer (kZ 2) is (Mohammadi and Richon 2010b, 2008)
Sk,i ¼
1
Hidden layer
Output layer
Fig. 1. Architecture of the neural network algorithm [1: bias; K: neuron; output neuron: logarithm of hydrate dissociation pressure (pressure in MPa); input neurons: temperature and concentration of tetrahydrofuran in aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction of THF in aqueous solution)].
ð2Þ
100
j¼1
Ok,i ¼
PNk1
PNk1
1e
1þ e
j ¼ 1
j ¼ 1
½ðwk1,j,i Ik1,j Þ þ bk,i ½ðwk1,j,i Ik1,j Þ þ bk,i
ð3Þ
The Levenberg–Marquardt algorithm (Marquardt, 1963; Levenberg, 1944) is used for optimization purposes. To develop the ANN, the data sets are generally subdivided into three groups corresponding to the following three steps: training, testing and validation (Mohammadi and Richon, 2010b, 2008). After partitioning the data sets, the training set is used to adjust the parameters. All synaptic weights and biases are first initialized randomly. The network is then trained; its synaptic weights are adjusted by minimizing the average root mean square error until it correctly emulates the input/ output mapping (Mohammadi and Richon, 2010b, 2008). The testing set is used during the adjustment of the network’s synaptic weights to evaluate the algorithms performance on the data not used for adjustment and to stop the adjustment if the error on the testing set increases. Finally, the validation set measures the generalization ability of the model after the fitting process (Mohammadi and Richon, 2010b, 2008).
p /MPa
where w is the weight parameter between each neuron–neuron interconnection and Ii ¼[Ii,1, y, Ii,Nk 1] represents the input vector. Using this feed-forward network with the tangent sigmoid activation function, the output, O, of the i neuron within the hidden k layer is (Mohammadi and Richon, 2008) 10
1 273 275 277 279 281 283 285 287 289 291 293 295 T /K Fig. 2. Experimental and predicted hydrate dissociation conditions for the nitrogen+tetrahydrofuran + water and nitrogen+water systems. Curves represent ANN results. Symbols represent experimental data. Nitrogen + water system: K: Marshall et al. (1964) (used for training and testing); m: Jhaveri and Robinson (1965) (used for training and testing); E: Mohammadi et al. (2003) (used for training and testing). Nitrogen +tetrahydrofuran + water system: : 0.011 mole fraction of THF in aqueous solution, this work (used for validation); J: 0.01 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); D: 0.02 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); }: 0.03 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); &: 0.05 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing).
A.H. Mohammadi et al. / Chemical Engineering Science 65 (2010) 6059–6063
10
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100
p/ MPa
p/ MPa
10
1
1
0.1 273
275
277
279
281
283
285
287
289
291
293
0.1 270
275
T/K Fig. 3. Experimental and predicted hydrate dissociation conditions for the carbon dioxide + tetrahydrofuran+ water and carbon dioxide + water systems. Curves represent ANN results. Symbols represent experimental data. Carbon dioxide + water system: m: Adisasmito et al. (1991) (used for training and testing); &: Mooijer-van den Heuvel et al. (2001) (Used for training and testing); *: Mohammadi et al. (2005) (used for training and testing). Carbon dioxide+ tetrahydrofuran + water system: +: 0.011 mole fraction of THF in aqueous solution, this work (used for validation); J: 0.012 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing); K: 0.0156 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); E: 0.0275 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); ’: 0.0299 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); D: 0.03 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing); }: 0.05 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing).
were measured using an isochoric pressure-search method (Mohammadi and Richon, 2010a, in press; Mohammadi et al., 2009; Tohidi et al., 2000) (Appendix). The latter experimental data were used for validation of the algorithm and also for studying the reliability of the corresponding experimental data reported in the literature. As can be seen in Figs. 2 and 3, our experimental data are in better agreement with the experimental data of Sabil and Peters (2007). In Fig. 3, some inconsistencies in terms of tetrahydrofuran concentration is observed in the experimental data of Delahaye et al. (2006) compared with the experimental data of Sabil and Peters (2007). As correctly pointed out by Strobel et al. (2009) the points for the 2.75 mole% data set of Delahaye et al. (2006) lie at temperatures greater than or equal to the temperatures of the 2.99 mole% data set of Delahaye et al. (2006) The experimental data of Sabil and Peters (2007), measured using highly qualified Cailletet equipment, were taken to be the most accurate. The reliability of the literature data for the tetrahydrofuran+ methane clathrate hydrate has already been studied by Mohammadi and Richon (2009). In all the figures, we have also shown the experimental data reported in the literature in the absence of tetrahydrofuran to show the hydrate promotion effects of tetrahydrofuran. As can be observed, the presence of the latter chemical in the aqueous solutions studied in this work, shifts hydrate dissociation conditions in the presence of pure water to low pressures/high temperatures. The model can correctly show this behavior. Furthermore, as the addition of very high concentrations of tetrahydrofuran in the aqueous solution diminishes its pressure-reducing effect while concentrations smaller than 0.056 mole fraction relative to water lower significantly the equilibrium pressure at a given temperature (stoichiometric concentration) (De Deugd et al., 2001), therefore the ANN algorithm was developed
280
285
290 T /K
295
300
305
310
Fig. 4. Experimental and predicted hydrate dissociation conditions for the methane+ tetrahydrofuran+ water and methane +water systems. Curves represent ANN results. Symbols represent experimental data. Methane + water system: K: Jhaveri and Robinson (1965) (used for training and testing); m: Adisasmito et al. (1991) (used for training and testing); ’: Mohammadi et al. (2005) (used for training and testing). Methane + tetrahydrofuran+ water system: + : 0.0048 mole fraction of THF in aqueous solution, Mohammadi and Richon (2009) (used for validation); : 0.0105 mole fraction of THF in aqueous solution, Mohammadi and Richon (2009) (used for validation); J: 0.0107 mole fraction of THF in aqueous solution, De Deugd et al. (2001) (used for training and testing); D: 0.03 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); }: 0.05 mole fraction of THF in aqueous solution, De Deugd et al. (2001) (used for training and testing); &: 0.056 mole fraction of THF in aqueous solution, Zhang et al. (2005) (used for training and testing).
for concentrations below/equal 0.056 mole fraction. As can be observed in Figs. 2–4, the ANN yields promising results and can be considered an alternative tool in addition to thermodynamic models for estimating hydrate stability zones of the systems containing tetrahydrofuran. 4. Conclusions In this work: 1. We successfully developed a feed-forward artificial neural network algorithm for estimating dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). 2. We reported few experimental data for the dissociation conditions of the binary clathrate hydrates of tetrahydrofuran+carbon dioxide or nitrogen at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which were measured using an isochoric pressure-search method (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Tohidi et al., 2000). 3. We showed that there are some inconsistencies among the experimental data reported in the literature (Sabil and Peters, 2007; Delahaye et al., 2006) for the clathrate hydrate systems containing tetrahydrofuran.
Acknowledgements The financial support of Agence Nationale de la Recherche (ANR) is gratefully acknowledged. Dr Jose´ F. Martı´nez-Lo´pez would like
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to thank CEP/TEP for the opportunity to work as a visiting researcher. The authors thank Dr. Timothy A. Strobel from Colorado School of Mines for fruitful discussion on the literature data. Mr. Ali Eslamimanesh and Mr. Chien-Bin Soo are appreciated for their help in revising this manuscript.
Appendix Table A.1 reports the purities and suppliers of the materials used in this work. The aqueous solution was prepared following the gravimetric method using an accurate analytical balance. Consequently, the uncertainty in mole fraction is estimated to be below 0.01. Briefly, the main part of the apparatus used for doing our measurements is a sapphire cylindrical vessel, which can withstand pressures up to 15 MPa. The volume of the vessel is 33.1 cm3. A stirrer was installed in the vessel to agitate the fluids and hydrate crystals inside it. The stirrer and all metallic parts of the apparatus (flanges, etc.) were made of stainless steel. Two platinum resistance thermometers (Pt100) inserted into the vessel were used to measure temperatures and check for their equality within temperature measurement uncertainty, which is estimated to be less than 0.1 K. This temperature uncertainty estimation comes from calibration against a 25 O reference platinum resistance thermometer. The pressure in the vessel was measured with two DRUCK pressure transducers (Druck, type PTX611 for pressure ranges up to (2.5 and 8) MPa, respectively). Pressure measurement uncertainties are estimated to be less than 5 kPa, as a result of calibration against a dead weight balance (Desgranges and Huot, model 520) (Mohammadi and Richon, 2010a; Mohammadi et al., 2009). The dissociation conditions were measured with an isochoric pressure search method (Mohammadi and Richon, 2010a). The vessel containing the aqueous solution (approximately 10% by volume of the vessel was filled with the aqueous solution) was immersed into the temperature-controlled bath, and the gas was supplied from a cylinder through a pressure-regulating valve into the vessel. Note that the vessel was evacuated before introducing any aqueous solution and gas. After obtaining temperature and pressure stability (far enough from the hydrate formation region), the valve in the line connecting the vessel and the cylinder was closed. Subsequently, temperature was slowly decreased to form the hydrate. Hydrate formation in the vessel was detected by pressure drop. The temperature was then increased with steps of 0.1 K. At every temperature step, the temperature was kept constant with sufficient time to achieve an equilibrium state in the vessel. In this way, a pressure-temperature diagram was obtained for each experimental run, from which we determined the hydrate dissociation point (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). If the temperature is increased in the hydrate-forming region, hydrate crystals partially dissociate, thereby substantially increasing the pressure. If the temperature is increased outside the hydrate region, only a small increase in the pressure is observed as a result of temperature increase (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). Consequently, the Table A.1 Purities and suppliers of materials.a Material
Supplier
Purity
Nitrogen Carbon dioxide Tetrahydrofuran
Air Liquide Air Liquide Aldrich
499.9 (mole %) 99.995 (mole %) 99.5 (%, GC)
a
Deionized water was used in all experiments.
Table A.2 Experimental dissociation data for the clathrate hydrates of tetrahydrofuran + nitrogen and tetrahydrofuran + carbon dioxide (concentration of tetrahydrofuran in aqueous solution is equal to 0.011 mole fraction). T (K)
p (MPa)
Clathrate hydrates of tetrahydrofuran+ nitrogen 281.4 3.31 283.6 4.44 285.2 5.61 287.3 7.48 289.4 10.09 Clathrate hydrates of tetrahydrofuran+ carbon 283.8 284.4 285.7 287.4 288.6 289.8
dioxide 1.33 1.48 1.81 2.61 3.35 4.41
point at which the slope of pressure-temperature data plots changes sharply is considered to be the point at which all hydrate crystals have dissociated and hence reported as the dissociation point (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). The experimental data are reported in Table A.2. References Adisasmito, S., Frank, R.J., Sloan, E.D., 1991. Hydrates of carbon dioxide and methane mixtures. J. Chem. Eng. Data 36, 68–71 (Quoted in Ref. Sloan and Koh, 2008). Chouai, A., Laugier, S., Richon, D., 2002. Modeling of thermodynamic properties using neural networks application to refrigerants. Fluid Phase Equilib. 199, 53–62. De Deugd, R.M., Jager, M.D., De Swaan Arons, J., 2001. Mixed hydrates of methane and water-soluble hydrocarbons modeling of empirical results. AIChE J. 47/3, 693–704. Delahaye, A., Fournaison, L., Marinhas, S., Chatti, I., Petitet, J.P., Dalmazzone, D., ¨ Furst, W., 2006. Effect of THF on equilibrium pressure and dissociation enthalpy of CO2 hydrates applied to secondary refrigeration. Ind. Eng. Chem. Res. 45, 391–397. Jhaveri, J., Robinson, D.B., 1965. Hydrates in the methane–nitrogen system. Can. J. Chem. Eng. 43, 75–78 (Quoted in Ref. Sloan and Koh, 2008). Lee, S., Yedlapalli, P., Lee, J.W., 2006. Excess Gibbs potential model for multicomponent hydrogen clathrates. J. Phys. Chem. B 110/51, 26122–26128. Levenberg, K.A., 1944. Method for the solution of certain problems in least squares. Q. Appl. Math. 2, 164–168. Marquardt, D., 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441. Marshall, D.R., Saito, S., Kobayashi, R., 1964. Hydrates at high pressures: Part I. Methane–water, argon–water, and nitrogen–water systems. AIChE J. 10/2, 202–205. Mohammadi, A.H., Richon, D., 2008. Estimating sulfur content of hydrogen sulfide at elevated temperatures and pressures using an artificial neural network algorithm. Ind. Eng. Chem. Res. 47, 8499–8504. Mohammadi, A.H., Richon, D., 2009. Phase equilibria of clathrate hydrates of tetrahydrofuran + hydrogen sulfide and tetrahydrofuran + methane. Ind. Eng. Chem. Res. 48, 7838–7841. Mohammadi, A.H., Richon, D, 2010a. Equilibrium data of methyl cyclohexane +hydrogen sulfide and methyl cyclohexane+ methane clathrate hydrates. J. Chem. Eng. Data, 55, 566–569. Mohammadi, A.H., Richon, D., 2010b. Hydrate phase equilibria for hydrogen+water and hydrogen+tetrahydrofuran+water systems: predictions of dissociation conditions using an artificial neural network algorithm. Chem. Eng. Sci. 65/10, 3352–3355. Mohammadi, A.H., Tohidi, B., Burgass, R.W., 2003. Equilibrium data and thermodynamic modeling of nitrogen, oxygen, and air clathrate hydrates. J. Chem. Eng. Data 48, 612–616. Mohammadi, A.H., Anderson, R., Tohidi, B., 2005. Carbon monoxide clathrate hydrates: equilibrium data and thermodynamic modeling. AIChE J. 51, 2825–2833 (Quoted in Ref. Sloan and Koh, 2008). Mohammadi, A.H., Belandria, V., Richon, D., 2009. Can toluene or xylene form clathrate hydrates? Ind. Eng. Chem. Res. 48 5916–5918. Mooijer-van den Heuvel, M.M., Witteman, R., Peters, C.J., 2001. Phase behaviour of gas hydrates of carbon dioxide in the presence of tetrahydropyran, cyclobutanone, cyclohexane and methylcyclohexane. Fluid Phase Equilib. 182, 97–110 (Quoted in Ref. Sloan and Koh, 2008).
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Normandin, A., Grandjean, B.P.A, Thibault, J.P.V.T., 1993. Data analysis using neural network models. Ind. Eng. Chem. Res. 32, 970–975. Ohmura, R., Takeya, S., Uchida, T., Ebinuma, T., 2004. Clathrate hydrate formed with methane and 2-propanol: confirmation of structure. II. Hydrate formation. Ind. Eng. Chem. Res. 43, 4964–4966. Piazza, L., Scalabrin, G., Marchi, P., Richon, D., 2006. Enhancement of the extended corresponding states techniques for thermodynamic modelling. I. Pure fluids. Int. J. Refrig. 29/7, 1182–1194. Rivollet, F., 2005. 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, France, December (in French). Sabil, K.M., Peters, C.J., 2007. Phase equilibrium data of mixed carbon dioxide and tetrahydrofuran clathrate hydrate in aqueous electrolyte solutions. In: Proceedings of the 11th International Conference on Properties and Phase Equilibria PPEPPD, Crete, Greece. Scalabrin, G., Marchi, P., Bettio, L., Richon, D., 2006. Enhancement of the extended corresponding states techniques for thermodynamic modelling. II. Mixtures. Int. J. Refrig. 29/7, 1195–1207.
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Seo, Y.T., Kang, S.P., Lee, H., 2001. Experimental determination and thermodynamic modeling of methane and nitrogen hydrates in the presence of THF, propylene oxide, 1,4-dioxane and acetone. Fluid Phase Equilib. 189, 99–110. Sloan, E.D., Koh, C.A., 2008. Clathrate Hydrates of Natural Gases, third ed CRC Press, Taylor & Francis Group, Boca Raton. Strobel, T.A., Koh, C.A., Sloan, E.D., 2009. Thermodynamic predictions of various tetrahydrofuran and hydrogen clathrate hydrates. Fluid Phase Equilib. 280, 61–67. Tohidi, B., Burgass, R.W., Danesh, A., Østergaard, K.K., Todd, A.C., 2000. Improving the accuracy of gas hydrate dissociation point measurements. Ann. N. Y. Acad. Sci. 912, 924–931. van der Waals, J.H., Platteeuw, J.C., 1959. In: Prigogine, I. (Ed.), Advances in Chemical Physics. Interscience, London, pp. 1–57. Wilamowski, B., Iplikci, S., Kayank, O., Efe, M.O., 2001. In: International Joint Conference on Neural Networks (IJCNN’01), Washington, DC, 15–19 July 2001, pp. 1778–1782. Zhang, Q., Chen, G.J., Huang, Q., Sun, C.Y., Guo, X.Q., Ma, Q.L., 2005. Hydrate formation conditions of a hydrogen +methane gas mixture in tetrahydrofuran+ water. J. Chem. Eng. Data 50, 234–236.
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Note
Determining phase diagrams of tetrahydrofuran +methane, carbon dioxide or nitrogen clathrate hydrates using an artificial neural network algorithm Amir H. Mohammadi a,n, Jose´ F. Martı´nez-Lo´pez b, Dominique Richon a a b
´nerge´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP—Centre E ´nica y Quı´mica Fı´sica, Facultad de Ciencias (Edificio D), Universidad de Zaragoza, c/. Pedro Cerbuna, 12, C.P. 50009 Zaragoza, Spain Departamento de Quı´mica Orga
a r t i c l e in fo
abstract
Article history: Received 18 November 2009 Received in revised form 27 June 2010 Accepted 20 July 2010 Available online 27 July 2010
In this communication, we have developed a feed-forward artificial neural network algorithm for estimating dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+ methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in the aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). In order to develop this algorithm, the most reliable experimental data reported in the literature on the dissociation pressures of the aforementioned binary hydrates have been used. Moreover, we report few experimental data on the dissociation pressures of the binary hydrates of tetrahydrofuran +carbon dioxide or nitrogen at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which were measured using an isochoric pressure-search method. The latter experimental data are used to verify the reliability of the corresponding experimental data reported in the literature. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Gas hydrate Tetrahydrofuran Methane Carbon dioxide Nitrogen Artificial neural network algorithm
1. Introduction It has been proven that tetrahydrofuran (THF) can form structure II of clathrate hydrates with or without the presence of a small molecule, like methane (Sloan and Koh, 2008). Adding tetrahydrofuran to water (with low-intermediate concentrations) normally leads to reduction of hydrate formation pressures of gases with respect to pure water (Sloan and Koh, 2008). This effect is called hydrate promotion effect, which is used in gas separation, storage and transportation processes using gas hydrate crystallization (Sloan and Koh, 2008). Hydrate phase equilibrium information is therefore required for designing the aforementioned processes. Sufficient phase equilibrium measurements for the clathrate hydrates of tetrahydrofuran+ gas have been reported for the tetrahydrofuran+ methane (Zhang et al., 2005; Seo et al., 2001; De Deugd et al., 2001; Mohammadi and Richon, 2009), tetrahydrofuran+ carbon dioxide (Sabil and Peters, 2007; Delahaye et al., 2006), and tetrahydrofuran+nitrogen (Seo et al., 2001) systems. Few thermodynamic models have been developed for clathrate hydrates containing THF. Strobel et al. (2009) have done a brief review on these thermodynamic models: De Deugd et al. (2001) used a model, in which the hydrate phase chemical potential is calculated using the van der Waals and Platteeuw (1959) theory.
n
Corresponding author. Tel.: +33 1 64 69 49 70; fax: + 33 1 64 69 49 68. E-mail address: [email protected] (A.H. Mohammadi).
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.07.013
Their model (De Deugd et al., 2001) is capable of accurately predicting the dissociation conditions for the binary clathrate hydrates of tetrahydrofuran+ methane as well as other watersoluble hydrate formers. Seo et al. (2001) presented a similar thermodynamic model, which sufficiently reproduces equilibrium conditions for tetrahydrofuran+ methane and tetrahydrofuran+ nitrogen hydrates. The thermodynamic model developed by Lee et al. (2006) for tetrahydrofuran+hydrogen hydrates and hydrocarbon+tetrahydrofuran hydrates is believed to be unrealistic (Strobel et al., 2009). Although these thermodynamic models reproduce hydrate equilibria for the specific systems, it is argued that the parameters used between the different models vary significantly (Strobel et al., 2009). Strobel et al. (2009) were able to predict the phase behavior of any of these given systems with a single set of consistent parameters within the framework of a single thermodynamic model. The aim of this communication is to develop an alternative tool based on the feed-forward neural network (FNN) algorithm to estimate dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+ methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in the aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). Reliable literature data along with few experimental data generated in this work based on isochoric pressure-search method (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Tohidi et al., 2000) were used to develop this algorithm. It is shown that this algorithm can be regarded a useful tool for modeling the aforementioned systems.
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A.H. Mohammadi et al. / Chemical Engineering Science 65 (2010) 6059–6063
2. Feed-forward neural network algorithm
3. Results and discussion
Feed-forward neural networks are the most frequently used type of artificial neural networks (ANN), which are designed with one input layer, one output layer and hidden layers (Rivollet, 2005; Wilamowski et al., 2001; Normandin et al., 1993; Chouai et al., 2002; Piazza et al., 2006; Scalabrin et al., 2006; Mohammadi and Richon, 2010b, 2008). The number of neurons in the input and output layers is equal to the number of inputs and outputs, respectively (Rivollet, 2005; Wilamowski et al., 2001; Normandin et al., 1993; Chouai et al., 2002; Piazza et al., 2006; Scalabrin et al., 2006; Mohammadi and Richon, 2010b, 2008). In the FNN model, the input layer of the network receives all the input data and introduces scaled data to the network (Mohammadi and Richon, 2010b, 2008). The data from the input neurons are propagated through the network via weighted interconnections (Mohammadi and Richon, 2010b, 2008). Every i neuron in a k layer is connected to every neuron in adjacent layers (Mohammadi and Richon, 2010b, 2008). The i neuron within the hidden k layer performs the following tasks: summation of the arriving weighted inputs and propagations of the resulting summation through an activation function, f, to the adjacent neurons of the next hidden layer or to the output neuron(s). In this work, the activation function is tangent sigmoid (Mohammadi and Richon, 2008)
The ANN algorithm shown in Fig. 1, with one hidden layer was used for the computation of the logarithm of hydrate dissociation pressure as a function of temperature and concentration of tetrahydrofuran in aqueous solution. In order to develop this algorithm, the experimental data shown in Figs. 2–4 were used. We also report few experimental data for the dissociation conditions of the tetrahydrofuran+carbon dioxide and tetrahydrofuran+ nitrogen binary clathrate hydrates at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which
f ðxÞ ¼
1ex 1 þex
x A ½1, þ1
and
f ðxÞ A ½1, þ 1
N k1 X
Input layer
ðwk1,j,i Ik1,j Þ þ bk,i
1
ð1Þ
where x stands for the parameter of the activation function. A bias term, b, is associated with each interconnection in order to introduce a supplementary degree of freedom. The expression of the weighted sum, S, to the ith neuron in the kth layer (kZ 2) is (Mohammadi and Richon 2010b, 2008)
Sk,i ¼
1
Hidden layer
Output layer
Fig. 1. Architecture of the neural network algorithm [1: bias; K: neuron; output neuron: logarithm of hydrate dissociation pressure (pressure in MPa); input neurons: temperature and concentration of tetrahydrofuran in aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction of THF in aqueous solution)].
ð2Þ
100
j¼1
Ok,i ¼
PNk1
PNk1
1e
1þ e
j ¼ 1
j ¼ 1
½ðwk1,j,i Ik1,j Þ þ bk,i ½ðwk1,j,i Ik1,j Þ þ bk,i
ð3Þ
The Levenberg–Marquardt algorithm (Marquardt, 1963; Levenberg, 1944) is used for optimization purposes. To develop the ANN, the data sets are generally subdivided into three groups corresponding to the following three steps: training, testing and validation (Mohammadi and Richon, 2010b, 2008). After partitioning the data sets, the training set is used to adjust the parameters. All synaptic weights and biases are first initialized randomly. The network is then trained; its synaptic weights are adjusted by minimizing the average root mean square error until it correctly emulates the input/ output mapping (Mohammadi and Richon, 2010b, 2008). The testing set is used during the adjustment of the network’s synaptic weights to evaluate the algorithms performance on the data not used for adjustment and to stop the adjustment if the error on the testing set increases. Finally, the validation set measures the generalization ability of the model after the fitting process (Mohammadi and Richon, 2010b, 2008).
p /MPa
where w is the weight parameter between each neuron–neuron interconnection and Ii ¼[Ii,1, y, Ii,Nk 1] represents the input vector. Using this feed-forward network with the tangent sigmoid activation function, the output, O, of the i neuron within the hidden k layer is (Mohammadi and Richon, 2008) 10
1 273 275 277 279 281 283 285 287 289 291 293 295 T /K Fig. 2. Experimental and predicted hydrate dissociation conditions for the nitrogen+tetrahydrofuran + water and nitrogen+water systems. Curves represent ANN results. Symbols represent experimental data. Nitrogen + water system: K: Marshall et al. (1964) (used for training and testing); m: Jhaveri and Robinson (1965) (used for training and testing); E: Mohammadi et al. (2003) (used for training and testing). Nitrogen +tetrahydrofuran + water system: : 0.011 mole fraction of THF in aqueous solution, this work (used for validation); J: 0.01 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); D: 0.02 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); }: 0.03 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); &: 0.05 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing).
A.H. Mohammadi et al. / Chemical Engineering Science 65 (2010) 6059–6063
10
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100
p/ MPa
p/ MPa
10
1
1
0.1 273
275
277
279
281
283
285
287
289
291
293
0.1 270
275
T/K Fig. 3. Experimental and predicted hydrate dissociation conditions for the carbon dioxide + tetrahydrofuran+ water and carbon dioxide + water systems. Curves represent ANN results. Symbols represent experimental data. Carbon dioxide + water system: m: Adisasmito et al. (1991) (used for training and testing); &: Mooijer-van den Heuvel et al. (2001) (Used for training and testing); *: Mohammadi et al. (2005) (used for training and testing). Carbon dioxide+ tetrahydrofuran + water system: +: 0.011 mole fraction of THF in aqueous solution, this work (used for validation); J: 0.012 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing); K: 0.0156 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); E: 0.0275 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); ’: 0.0299 mole fraction of THF in aqueous solution, Delahaye et al. (2006) (used for validation); D: 0.03 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing); }: 0.05 mole fraction of THF in aqueous solution, Sabil and Peters (2007) (used for training and testing).
were measured using an isochoric pressure-search method (Mohammadi and Richon, 2010a, in press; Mohammadi et al., 2009; Tohidi et al., 2000) (Appendix). The latter experimental data were used for validation of the algorithm and also for studying the reliability of the corresponding experimental data reported in the literature. As can be seen in Figs. 2 and 3, our experimental data are in better agreement with the experimental data of Sabil and Peters (2007). In Fig. 3, some inconsistencies in terms of tetrahydrofuran concentration is observed in the experimental data of Delahaye et al. (2006) compared with the experimental data of Sabil and Peters (2007). As correctly pointed out by Strobel et al. (2009) the points for the 2.75 mole% data set of Delahaye et al. (2006) lie at temperatures greater than or equal to the temperatures of the 2.99 mole% data set of Delahaye et al. (2006) The experimental data of Sabil and Peters (2007), measured using highly qualified Cailletet equipment, were taken to be the most accurate. The reliability of the literature data for the tetrahydrofuran+ methane clathrate hydrate has already been studied by Mohammadi and Richon (2009). In all the figures, we have also shown the experimental data reported in the literature in the absence of tetrahydrofuran to show the hydrate promotion effects of tetrahydrofuran. As can be observed, the presence of the latter chemical in the aqueous solutions studied in this work, shifts hydrate dissociation conditions in the presence of pure water to low pressures/high temperatures. The model can correctly show this behavior. Furthermore, as the addition of very high concentrations of tetrahydrofuran in the aqueous solution diminishes its pressure-reducing effect while concentrations smaller than 0.056 mole fraction relative to water lower significantly the equilibrium pressure at a given temperature (stoichiometric concentration) (De Deugd et al., 2001), therefore the ANN algorithm was developed
280
285
290 T /K
295
300
305
310
Fig. 4. Experimental and predicted hydrate dissociation conditions for the methane+ tetrahydrofuran+ water and methane +water systems. Curves represent ANN results. Symbols represent experimental data. Methane + water system: K: Jhaveri and Robinson (1965) (used for training and testing); m: Adisasmito et al. (1991) (used for training and testing); ’: Mohammadi et al. (2005) (used for training and testing). Methane + tetrahydrofuran+ water system: + : 0.0048 mole fraction of THF in aqueous solution, Mohammadi and Richon (2009) (used for validation); : 0.0105 mole fraction of THF in aqueous solution, Mohammadi and Richon (2009) (used for validation); J: 0.0107 mole fraction of THF in aqueous solution, De Deugd et al. (2001) (used for training and testing); D: 0.03 mole fraction of THF in aqueous solution, Seo et al. (2001) (used for training and testing); }: 0.05 mole fraction of THF in aqueous solution, De Deugd et al. (2001) (used for training and testing); &: 0.056 mole fraction of THF in aqueous solution, Zhang et al. (2005) (used for training and testing).
for concentrations below/equal 0.056 mole fraction. As can be observed in Figs. 2–4, the ANN yields promising results and can be considered an alternative tool in addition to thermodynamic models for estimating hydrate stability zones of the systems containing tetrahydrofuran. 4. Conclusions In this work: 1. We successfully developed a feed-forward artificial neural network algorithm for estimating dissociation pressures of the binary clathrate hydrates of tetrahydrofuran+methane, carbon dioxide or nitrogen as a function of temperature and concentration of tetrahydrofuran in aqueous solution below/equal its stoichiometric concentration (i.e., 0.056 mole fraction). 2. We reported few experimental data for the dissociation conditions of the binary clathrate hydrates of tetrahydrofuran+carbon dioxide or nitrogen at 0.011 mole fraction of tetrahydrofuran in aqueous solution, which were measured using an isochoric pressure-search method (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Tohidi et al., 2000). 3. We showed that there are some inconsistencies among the experimental data reported in the literature (Sabil and Peters, 2007; Delahaye et al., 2006) for the clathrate hydrate systems containing tetrahydrofuran.
Acknowledgements The financial support of Agence Nationale de la Recherche (ANR) is gratefully acknowledged. Dr Jose´ F. Martı´nez-Lo´pez would like
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to thank CEP/TEP for the opportunity to work as a visiting researcher. The authors thank Dr. Timothy A. Strobel from Colorado School of Mines for fruitful discussion on the literature data. Mr. Ali Eslamimanesh and Mr. Chien-Bin Soo are appreciated for their help in revising this manuscript.
Appendix Table A.1 reports the purities and suppliers of the materials used in this work. The aqueous solution was prepared following the gravimetric method using an accurate analytical balance. Consequently, the uncertainty in mole fraction is estimated to be below 0.01. Briefly, the main part of the apparatus used for doing our measurements is a sapphire cylindrical vessel, which can withstand pressures up to 15 MPa. The volume of the vessel is 33.1 cm3. A stirrer was installed in the vessel to agitate the fluids and hydrate crystals inside it. The stirrer and all metallic parts of the apparatus (flanges, etc.) were made of stainless steel. Two platinum resistance thermometers (Pt100) inserted into the vessel were used to measure temperatures and check for their equality within temperature measurement uncertainty, which is estimated to be less than 0.1 K. This temperature uncertainty estimation comes from calibration against a 25 O reference platinum resistance thermometer. The pressure in the vessel was measured with two DRUCK pressure transducers (Druck, type PTX611 for pressure ranges up to (2.5 and 8) MPa, respectively). Pressure measurement uncertainties are estimated to be less than 5 kPa, as a result of calibration against a dead weight balance (Desgranges and Huot, model 520) (Mohammadi and Richon, 2010a; Mohammadi et al., 2009). The dissociation conditions were measured with an isochoric pressure search method (Mohammadi and Richon, 2010a). The vessel containing the aqueous solution (approximately 10% by volume of the vessel was filled with the aqueous solution) was immersed into the temperature-controlled bath, and the gas was supplied from a cylinder through a pressure-regulating valve into the vessel. Note that the vessel was evacuated before introducing any aqueous solution and gas. After obtaining temperature and pressure stability (far enough from the hydrate formation region), the valve in the line connecting the vessel and the cylinder was closed. Subsequently, temperature was slowly decreased to form the hydrate. Hydrate formation in the vessel was detected by pressure drop. The temperature was then increased with steps of 0.1 K. At every temperature step, the temperature was kept constant with sufficient time to achieve an equilibrium state in the vessel. In this way, a pressure-temperature diagram was obtained for each experimental run, from which we determined the hydrate dissociation point (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). If the temperature is increased in the hydrate-forming region, hydrate crystals partially dissociate, thereby substantially increasing the pressure. If the temperature is increased outside the hydrate region, only a small increase in the pressure is observed as a result of temperature increase (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). Consequently, the Table A.1 Purities and suppliers of materials.a Material
Supplier
Purity
Nitrogen Carbon dioxide Tetrahydrofuran
Air Liquide Air Liquide Aldrich
499.9 (mole %) 99.995 (mole %) 99.5 (%, GC)
a
Deionized water was used in all experiments.
Table A.2 Experimental dissociation data for the clathrate hydrates of tetrahydrofuran + nitrogen and tetrahydrofuran + carbon dioxide (concentration of tetrahydrofuran in aqueous solution is equal to 0.011 mole fraction). T (K)
p (MPa)
Clathrate hydrates of tetrahydrofuran+ nitrogen 281.4 3.31 283.6 4.44 285.2 5.61 287.3 7.48 289.4 10.09 Clathrate hydrates of tetrahydrofuran+ carbon 283.8 284.4 285.7 287.4 288.6 289.8
dioxide 1.33 1.48 1.81 2.61 3.35 4.41
point at which the slope of pressure-temperature data plots changes sharply is considered to be the point at which all hydrate crystals have dissociated and hence reported as the dissociation point (Mohammadi and Richon, 2010a; Mohammadi et al., 2009; Ohmura et al., 2004). The experimental data are reported in Table A.2. References Adisasmito, S., Frank, R.J., Sloan, E.D., 1991. Hydrates of carbon dioxide and methane mixtures. J. Chem. Eng. Data 36, 68–71 (Quoted in Ref. Sloan and Koh, 2008). Chouai, A., Laugier, S., Richon, D., 2002. Modeling of thermodynamic properties using neural networks application to refrigerants. Fluid Phase Equilib. 199, 53–62. De Deugd, R.M., Jager, M.D., De Swaan Arons, J., 2001. Mixed hydrates of methane and water-soluble hydrocarbons modeling of empirical results. AIChE J. 47/3, 693–704. Delahaye, A., Fournaison, L., Marinhas, S., Chatti, I., Petitet, J.P., Dalmazzone, D., ¨ Furst, W., 2006. Effect of THF on equilibrium pressure and dissociation enthalpy of CO2 hydrates applied to secondary refrigeration. Ind. Eng. Chem. Res. 45, 391–397. Jhaveri, J., Robinson, D.B., 1965. Hydrates in the methane–nitrogen system. Can. J. Chem. Eng. 43, 75–78 (Quoted in Ref. Sloan and Koh, 2008). Lee, S., Yedlapalli, P., Lee, J.W., 2006. Excess Gibbs potential model for multicomponent hydrogen clathrates. J. Phys. Chem. B 110/51, 26122–26128. Levenberg, K.A., 1944. Method for the solution of certain problems in least squares. Q. Appl. Math. 2, 164–168. Marquardt, D., 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441. Marshall, D.R., Saito, S., Kobayashi, R., 1964. Hydrates at high pressures: Part I. Methane–water, argon–water, and nitrogen–water systems. AIChE J. 10/2, 202–205. Mohammadi, A.H., Richon, D., 2008. Estimating sulfur content of hydrogen sulfide at elevated temperatures and pressures using an artificial neural network algorithm. Ind. Eng. Chem. Res. 47, 8499–8504. Mohammadi, A.H., Richon, D., 2009. Phase equilibria of clathrate hydrates of tetrahydrofuran + hydrogen sulfide and tetrahydrofuran + methane. Ind. Eng. Chem. Res. 48, 7838–7841. Mohammadi, A.H., Richon, D, 2010a. Equilibrium data of methyl cyclohexane +hydrogen sulfide and methyl cyclohexane+ methane clathrate hydrates. J. Chem. Eng. Data, 55, 566–569. Mohammadi, A.H., Richon, D., 2010b. Hydrate phase equilibria for hydrogen+water and hydrogen+tetrahydrofuran+water systems: predictions of dissociation conditions using an artificial neural network algorithm. Chem. Eng. Sci. 65/10, 3352–3355. Mohammadi, A.H., Tohidi, B., Burgass, R.W., 2003. Equilibrium data and thermodynamic modeling of nitrogen, oxygen, and air clathrate hydrates. J. Chem. Eng. Data 48, 612–616. Mohammadi, A.H., Anderson, R., Tohidi, B., 2005. Carbon monoxide clathrate hydrates: equilibrium data and thermodynamic modeling. AIChE J. 51, 2825–2833 (Quoted in Ref. Sloan and Koh, 2008). Mohammadi, A.H., Belandria, V., Richon, D., 2009. Can toluene or xylene form clathrate hydrates? Ind. Eng. Chem. Res. 48 5916–5918. Mooijer-van den Heuvel, M.M., Witteman, R., Peters, C.J., 2001. Phase behaviour of gas hydrates of carbon dioxide in the presence of tetrahydropyran, cyclobutanone, cyclohexane and methylcyclohexane. Fluid Phase Equilib. 182, 97–110 (Quoted in Ref. Sloan and Koh, 2008).
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