Estimation of Thermal Conductivity of Ionic Liquids Using a Perceptron

Jun 15, 2012 - proposed network is able to correlate and predict the thermal ... mean relative deviation percent (MARD %) of 0.5% and mean square erro...
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Estimation of Thermal Conductivity of Ionic Liquids Using a Perceptron Neural Network Ali Zeinolabedini Hezave, Sona Raeissi,* and Mostafa Lashkarbolooki School of Chemical and Petroleum Engineering, Shiraz University, Mollasadra Avenue, Shiraz 71345, Iran ABSTRACT: On the basis of an artificial neural network (ANN), a model is proposed to predict the thermal conductivity of pure ionic liquids. A total of 209 data points from 21 different ionic liquids was used to train and test the proposed network. The optimum number of hidden layers was determined to be 1, with 13 neurons in the hidden layer and logarithmic−sigmoid and purelin functions as the transfer functions in the hidden and output layers, respectively. The results obtained reveal that the proposed network is able to correlate and predict the thermal conductivity of all of the pure ionic liquids with an overall absolute mean relative deviation percent (MARD %) of 0.5% and mean square error (MSE) of 1.2 × 10−6. The optimized network was also compared with literature correlations and a predictive group contribution method. The results indicate the rather good accuracy of the proposed neural network compared to the previously proposed literature methods.

1. INTRODUCTION Current world economy and increasing energy demands have led to investigations on alternative energies, optimization of present technologies, and the search for new and more environmentally friendly fluids. In the heat transfer area, conventional liquid coolants used at low and moderate temperatures exhibit very poor thermal conductivity and heat storage capacities.1 Therefore, there is a need for new and efficient heat transfer liquids. Recently, ionic liquids have proven to be suitable alternatives for many applications in industry and chemical manufacturing, even in the field of heat transfer and energy storage due to their suitable thermal conductivities.1 Ionic liquids (ILs) are novel solvents that belong to the class of molten salts which are liquid at room temperature. It is their unique properties, such as negligible vapor pressure, wide liquidus range, wide electrochemical window, wide thermal window, and high solvating capacity for organic, inorganic, and organometallic compounds, that has made them the focus of research and industrial interest.2 Since ILs are composed of large organic cations with various alkyl substituents and inorganic or organic anions, the possible number of ILs from chemical combinations is estimated to be more than a trillion, which not only provides a wide range of suitable alternatives to conventional solvents and working fluids but also allows their use in different areas of industries.2−7 Generally, to verify not only the feasibility of using a specific IL structure, e.g., as a solvent in catalytic reactions,8 entrainer in distillative separations,9 electrolyte in batteries,10 lubricant in difficult metal−metal wear contacts,11 or heat transfer fluid in energy technology,12 but also for a better technological design, knowledge of the physiochemical properties is crucial.1 Transport properties, particularly the thermal conductivities of aqueous electrolyte solutions, have to be calculated in many industrial applications such as the chemical industry, geochemistry, development and utilization of geothermal and ocean thermal energy, geology and mineralogy, desalination processes and hydrothermal synthesis. Although many physicochemical properties of ILs, including their equilibrium © 2012 American Chemical Society

and transport properties, have been studied extensively, a fundamental lack of reliable data still exists. In particular, only limited information on the thermal conductivity of ILs is available in literature.13 Estimation of the thermal conductivities of electrolyte solutions has attracted the attention of many researchers14 because measurement and experiments are not always easy and cheap. In this respect, proposing predictive methods to correlate the thermophysical properties of ionic liquids could be quite useful to the numerous researchers investigating these novel liquids. Previously, a number of researchers reported correlations and group contribution methods to predict the thermal conductivities of pure ionic liquids.15 However, these methods have their own disadvantages. Group contribution methods require knowledge of the exact structure of the ionic liquids and are, in some cases, complex and time consuming. However, most importantly, the group parameters of many different functional groups in the ionic liquid structures are not yet determined, so such methods are still very limited in their usage. Correlations are also not well generalized to predict the thermal conductivity of a wide range of ionic liquids. In addition, due to the complex ionic nature of such liquids, simple correlations have difficulty in predicting properties within an acceptable range of deviation. Because of these shortcomings, in the present work it is attempted to use a different approach which can predict unknown values from data observed at other known locations with any degree of complexity. Such methods include the Kriging method,1617 the radial basis function,18 artificial neural networks, etc. Artificial neural networks have gained popularity in the past decades as feasible tools in a variety of industries. An artificial neural network is a system based on the operation of biological neural networks, i.e., it is an emulation of the biological neural system. The advantages of artificial neural Received: Revised: Accepted: Published: 9886

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Among the different types of neural networks, it is the feedforward structures that have proven most useful in solving real problems.24 A typical feed-forward network has neurons arranged in a distinct layered topology. The input layer is not really neural at all: these units simply serve to introduce the values of the input variables. The hidden and output layer neurons are each connected to all of the units in the preceding layer. It is possible to define networks that are partially connected to only some units in the preceding layer; however, for most applications fully connected networks are more suitable. Generally, a multilayer neural network consists of a number of layers, namely, the input layer, hidden layer(s), and output layer (see Figure 1 for the case where one hidden layer is

networks can generally be considered as its ability to map any relation with any complexity and its self-learning ability makes it independent from reprogramming. Because of this, in the present study an artificial neural network (ANN) was proposed to correlate the thermal conductivity of pure ionic liquids with temperature, pressure, and the ionic liquid’s molecular weight as input parameters.

2. AVAILABLE THERMAL CONDUCTIVITY CORRELATIONS AND GROUP CONTRIBUTION METHODS Since the model proposed in this study will be compared to predictive methods in the literature, a brief review of published correlations and the group contribution method available in literature is provided below. Frez et al.20 reported a correlation to predict the thermal conductivity of four pure ionic liquids (1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, 1-butyl-3-methylimidazolium tetrafluoroborate, and 1-butyl-3-methyl imidazolium hexafluorophosphate) based on the thermal diffusivity (a), density (ρ), and heat capacity (cp) of an ionic liquid λ = aρcp

(1) 21

Also, Tomida et al. predicted the thermal conductivity of 1butyl-3-methylimidazolium tetrafluoroborate using the following equation22 with a maximum error of 0.6%. λ 0 = 0.177(W·m−1·K−1) − 2.5 × 10−5(W·m−1·K−2)T (2)

⎛ 11.4(MPa) + P ⎞ λ − λ0 = 2.30 × 10−2 ln⎜ ⎟ λ ⎝ 11.4(MPa) + 0.1 ⎠

Figure 1. Schematic of the feed-forward multilayer perceptron neural network (FFMLPNN) used in this study for prediction of thermal conductivity.

(3)

18

Furthermore, Fröba et al. proposed a generalized correlation to predict the thermal conductivity of ionic liquids based on density (g·cm−3) and molecular weight (g·mol−1)

considered). Data are fed to the input layer and then dispatched to the hidden layer, where the selected transfer function correlates the input to the desire output. According to Cybenko,25 a network that has only one hidden layer is able to approximate almost any type of nonlinear mapping. Too few neurons in the hidden layer prevent the network from being trained accurately enough. On the other hand, too many neurons cause the network to memorize the training data points without capturing the underlying relationship between the input and the output variables. This problem is usually called “overfitting”. Thus, following Cybenko,25 only one hidden layer was considered in this study. One of the most important stages using the ANN model is the training of the network to find the optimum network parameters, including the number of hidden layer(s), the number of neurons in the hidden layer, and the appropriate transfer function of the hidden and output layers. In this study, the training stage was carried out using more than two-thirds of the collected experimental data points and the optimal network architecture was determined. Two different network architectures were tested (cascade and feedforward structure) to select the most accurate scheme. As input information, it is necessary to use criteria that can distinguish between the different ionic liquids. In addition, the systems’ operational conditions are also necessary as they affect the numeric value of thermal conductivity. We have chosen molecular weight and melting point to discriminate the different substances and temperature and pressure as additional

λ ·MW ·ρ = 0.1130(g·cm−3·W·m−1·K−1)M W + 22.65(g 2·cm−3·W ·m−1·K−1·mol−1)

(4)

2

Gardas and Countinho proposed a group contribution method to predict the thermophysical properties of ionic liquids, including the thermal conductivities of imidazolium-, pyrrolidinium-, and phosphonium-based ionic liquids. A detailed description of the proposed group contribution method is provided elsewhere.2

3. METHODOLOGY 3.1. Neural Network Training and Testing. Usually the most important features concerning models obtained from computational tools are the flexibility to model multiple mechanisms of action, the capability to deal with highdimensional data, and the level of predictive accuracy.23 Neural networks are strongly able to correlate parameters with any possible complexity, which makes them good candidates to use as powerful predictive tools in the field of property estimation where the usual correlations are unable. An artificial neural network is developed with a systematic step-by-step procedure which optimizes a criterion commonly known as the learning rule. The input/output training data is fundamental for these networks as it conveys the information which is necessary to obtain the optimal operating parameters. 9887

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input data to determine the system’s condition. In other words, the functionality of the thermal conductivity was considered as follows λ = f (Tm , MW , T , P)

Table 1. Physicochemical Properties of the Ionic Liquids ionic liquid trihexyl(tetradecyl)phosphonium chloride 1-butyl-1-methylpyrrolidinium bis[(trifluoromethyl) sulfonyl]imide 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 1-ethyl-3-methylimidazolium tetrafluoroborate 1,2-dimethyl-3-propylimidazolium bis[(trifluoromethyl)sulfonyl]imide trihexyl(tetradecyl)phosphonium bis[(trifluoromethyl)sulfonyl]imide 1-hexyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium trifluoromethanesulfonate 1-hexyl-3-methylimidazolium hexafluorophosphate 1-octyl-3-methylimidazolium hexafluorophosphate 1-ethyl-3-methylimidazolium ethylsulfate 1-ethyl-3-methylimidazolium bis[(trifluoromethyl) sulfonyl]imide 1-hexyl-3-methylimidazolium bis[(trifluoromethyl) sulfonyl]imide 1-butyl-3-methylimidazolium bis[(trifluoromethyl) sulfonyl]imide 1-butyl-3-methyl imidazolium hexafluorophosphate 1-ethyl-3-methylimidazole acetate 1-ethyl-3-methylimidazole dicyanamide 1-ethyl-3-methylimidazolium ethylsulfate 1-ethyl-3-methylimidazolium octylsulfate [EMIM] [OcSO4] [P4444][Val]

(5)

After the network was trained and the parameters were obtained, a validation stage was performed to investigate the predictive capability of the proposed network. In this regard, less than one-third of the collected data, which were not used in the training stage, were used to validate the proposed network. The input data (T, P, MW, and Tm) of 21 ionic liquids (209 data points) were used, and the thermal conductivities were predicted at several temperatures and pressures. In this way, less than one-third of the total data were considered as the testing data in order to estimate the correlative capabilities of the model. It should be noted that the training and testing data sets were selected randomly in a way to cover the whole range of experimental conditions. In both stages of training and validation three statistical parameters, including mean absolute relative deviation % (MARD %, eq 6), mean square error (MSE, eq 7), and correlation coefficient (R2, eq 8) values, were utilized to compare the correlated thermal conductivity values to the experimental ones MARD % =

MSE =

1 N

1 N

N

⎛ λ exp − λ calcd i i exp λi ⎝

∑ ⎜⎜ i

R =

(6)

N

∑ (λiexp − λicalcd)2

(7)

i=1

N

2

⎞ ⎟⎟ × 100 ⎠

Tm (K)

519.31 422.41

217.15 267.15

503.53

271.15

197.98 419.37

288.15 288.15

764.01

306.10

254.08 226.03 288.29

191.15 192.15 289.55

312.24 340.29 236.29 319.32

212.15 233.15 236.29 261.15

447.42

266.00

419.37

269.15

284.18 170.21 177.21 236.29 320.45

212.15 253.15 252.15 236.29 264.15

375.59

298.92

Table 2. Error Analysis of Different Numbers of Neurons in the Hidden Layer

N

∑i (λiexp − λ ̅ )2 − ∑i (λiexp − λicalcd)2

λexp i

MW (g/ gmol)

N

∑i (λiexp − λ ̅ )2

(8)

error analysis

λcalcd i

where is the experimental thermal conductivity, is the estimated thermal conductivity, and λ̅ is the average experimental thermal conductivities of the pure ionic liquids. 3.2. Data Collection. To propose an accurate network using reliable experimental data points is vital. In this regard, 209 experimental thermal conductivity data points for 21 ionic liquids from various families were collected from previously published studies.12,15,26−29 Any data available in the literature which was obtained from theoretical methods, correlations, or extrapolations of any kind was not considered. Also, data for which the authors themselves indicated that the accuracy is not guaranteed for any reason (presence of impurities, instability of the fluid, or problems with the equipment) were not considered. The physicochemical properties of the ionic liquids used are given in Table 1.

hidden neuron 7 8 9 10 11 12 13 14

4. RESULTS AND DISCUSSION Following the methods described above, the results obtained are given in Table 2. This table reveals that 13 neurons in the hidden layer lead to good predictions of the thermal conductivities of ionic liquids, with overall average MARD %, MSE, and R2 values of 0.5%, 1.2 × 10−6, and 0.9983, respectively. The weights and biases of the best network configuration are given in Table 3. One can simply recalculate the thermal conductivities of the ionic liquids using these weights and biases.

15 16 17

MARD % train test train test train test train test train test train test train test train test train test train test train test

2.0 2.1 1.6 1.7 0.7 1.0 0.7 1.1 0.6 0.9 0.6 0.9 0.4 0.6 0.6 0.9 0.5 0.7 0.7 0.9 0.4 0.8

R2

MSE −5

1.5 × 10 1.6 × 10−5 9.6 × 10−6 1.1 × 10−5 2.4 × 10−6 3.6 × 10−6 2.6 × 10−6 4.7 × 10−6 1.7 × 10−6 3.0 × 10−6 1.7 × 10−6 3.3 × 10−6 1.0 × 10−6 1.8 × 10−6 1.6 × 10−6 3.6 × 10−6 1.1 × 10−6 2.1 × 10−6 2.1 × 10−6 2.7 × 10−6 9.6 × 10−7 2.6 × 10−6

0.9771 0.9785 0.9857 0.9854 0.9964 0.9951 0.9962 0.9935 0.9975 0.9958 0.9974 0.9955 0.9985 0.9975 0.9976 0.9951 0.9984 0.9972 0.9970 0.9962 0.9986 0.9965

In Figure 2, the results obtained from the training stage are shown. The solid line shows a hypothetical exact fit of the predicted values with the experimental ones, while the stars 9888

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Table 3. Weights and Biases of the Best Neural Network Architecture hidden layer

output layer

neuron

weights, MW

Tm

T

P

biases

weight, wjk

bias

1 2 3 4 5 6 7 8 9 10 11 12 13

0.0478 −0.0069 −0.0963 0.0042 0.0662 0.0141 0.4601 0.2220 0.0142 0.0246 0.0266 −0.0890 −0.2598

−0.0030 0.0231 0.0317 0.0984 1.0560 −0.0116 −1.2776 0.3084 −0.0102 −0.4200 −0.0217 0.1404 0.0947

−0.0203 −0.0037 −0.0496 −0.0002 0.0659 −0.0133 0.1609 0.2010 0.0007 0.0035 −0.0169 −0.0005 0.0040

−0.0001 −0.1544 0.0043 −0.2716 −2.7696 −0.0001 0.0106 −2.4690 0.0776 1.0893 0.1970 0.0313 0.5152

−7.4301 13.2769 10.9841 −2.0652 −6.5866 7.5152 −16.0084 −6.7543 −9.4868 −8.2576 −11.8083 −6.0824 6.3969

−0.0051 −0.0324 0.0092 −0.0444 −0.0014 −1.4065 0.0010 −0.0005 −0.0704 −0.0166 0.1534 0.0513 −0.0530

1.3809

Figure 2. Plot of experimental thermal conductivity data vs developed ANN predictions in the training stage.

Figure 4. Comparison between the predicted and the experimentally measured thermal conductivities of the ionic liquids: (a) training data set and (b) testing data set. Figure 3. Plot of experimental thermal conductivity data vs developed ANN predictions for the test data.

network is trained well. In Figure 3, the predicted values of thermal conductivities are shown for the data points which were not considered in the training stage. These predicted results also show good agreement with actual experimental thermal conductivities of the ionic liquids, indicating the good interpolative capability of the trained network.

show the real predicted values of thermal conductivities compared with experimental ones. The close proximity of the stars to the line of Figure 2 demonstrates that the proposed 9889

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Table 4. Detailed Error Analysis of Predicted Thermal Conductivity of Different Pure Ionic Liquids error analysis component IL 1-butyl-1-methylpyrrolidinium bis[(trifluoromethyl)sulfonyl] imide 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 1-ethyl-3-methylimidazolium tetrafluoroborate 1,2-dimethyl-3-propylimidazolium bis[(trifluoromethyl) sulfonyl]imide trihexyl(tetradecyl)phosphonium bis[(trifluoromethyl)sulfonyl] imide 1-hexyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidazolium trifluoromethanesulfonate 1-hexyl-3-methylimidazolium hexafluorophosphate 1-octyl-3-methylimidazolium hexafluorophosphate 1-ethyl-3-methylimidazolium ethyl sulfate 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl] imide 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl] imide 1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl] imide 1-butyl-3-methyl imidazolium hexafluorophosphate 1-ethyl-3-methylimidazole acetate 1-ethyl-3-methylimidazole dicyanamide 1-ethyl-3-methylimidazolium ethylsulfate 1-ethyl-3-methylimidazolium octylsulfate [EMIM][OcSO4] [P4444][Val] trihexyl(tetradecyl)phosphonium chloride overall

temp. (K) 293−323

pressure (kPa) 101.15

no. of data 4

R2

ref

MARD %

MSE

26

2.7

1.2 × 10−5 × × × ×

−5

−45.95

293−333 293−353 300−390 300−390

101.15 101.15 101.15 101.15

5 7 10 10

27 26 12 12

2.8 0.2 0.1 0.2

1.1 1.4 2.2 9.5

10 10−7 10−7 10−8

−44.76 0.9377 0.9816 0.9526

293−353

101.15

7

26

0.2

2.0 × 10−7

0.9223

294.9−335.1 293−353 293−353 293−353 294.1−335.2 293−353 295.1−335.2 293−353 293−353

101.15 101.15 101.15 101.15 100−20000 101.15 100−20000 101.15 101.15

7 10 7 7 9 7 9 7 7

27 12 26 27 28 27 28 26 26

0.2 0.2 1.7 1.7 0.2 0.3 0.2 0.2 0.3

1.1 1.5 6.6 5.6 1.2 1.8 1.9 2.1 2.5

10−7 10−7 10−6 10−6 10−7 10−7 10−7 10−7 10−7

0.9813 0.9556 −1.61 −1.87 0.9332 0.9173 0.8288 0.9265 0.3981

273.15−353.15 293−353

100 101.15

9 7

15 26

0.5 0.3

4.8 × 10−7 1.7 × 10−7

0.4592 0.8037

273.15−353.15 293−353

100 101.15

9 7

15 26

0.6 0.2

7.1 × 10−7 8.6 × 10−8

0.5135 0.9595

294.9−335.1 293−353 273.15−353.15 273.15−353.15 273.15−353.15 273.15−353.15 313.15−353.15 293−353 273.15−390

100−20000 101.15 100 100 100 100 101.15 101.15 100−20000

9 7 9 9 9 9 5 7 209

28 27 15 15 15 15 29 26

0.1 0.3 0.3 0.5 0.3 0.4 0.2 0.2 0.5

2.5 1.9 5.4 1.3 9.6 7.4 1.4 2.1 1.2

10−8 10−7 10−7 10−6 10−7 10−7 10−7 10−7 10−6

0.9823 0.7734 0.9896 0.9448 0.7585 0.9356 0.9319 0.9485 0.9983

A comparison between predicted and experimentally measured thermal conductivity values as a function of temperature is given in Figure 4. The dots show the estimated values of thermal conductivity using the ANN model, while the closest circles show the corresponding experimental points. A detailed error analysis for all of the 21 ionic liquids is also provided in Table 4. The proposed neural network was able to predict the thermal conductivities of the pure ionic liquids with a very low percentage of deviation (0.1−0.6%), except for the two ionic liquids 1-butyl-1-methylpyrrolidinium bis(trifluoromethyl sulfonyl)imide26,27 and 1-butyl-3-methyl imidazolium trifluoromethanesulfonate.26,27 The MARD % obtained for the ANN model compared to the differing values of thermal conductivity reported by Ge et al.26 and Nieto de Castro et al.27 for 1-butyl1-methylpyrrolidinium bis(trifluoromethyl sulfonyl)imide was 2.7% and 2.8%, respectively. In addition, the ANN model predicts the thermal conductivity of 1-butyl-3-methylimidazolium trifluoromethanesulfonate reported by Ge et al.26 and Nieto de Castro et al.27 with a MARD % value of about 1.7. Also, the obtained R2 values of these two ionic liquids were negative, which means that the predicted values by the proposed ANN model lay in between these two reported values of thermal conductivities (see Figure 5).

× × × × × × × × ×

× × × × × × × × ×

The higher percentages of errors between model and experimental values for these two specific ionic liquids may be related to the fact that for each of these two ionic liquids two sets of inconsistent experimental thermal conductivity data were reported in literature,26,27 which could be related to the sample purity or the technique of thermal conductivity measurements. The matching trend of the estimated and experimental thermal conductivities for 1-ethyl-3-methylimidazolium tetrafluoroborate, 1-butyl-3-methylimidazolium trifluoromethanesulfonate, 1-ethyl-3-methylimidazolium octylsulfate, 1-hexyl-3methylimidazolium tetrafluoroborate, and 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide show the capability of the proposed neural network not only to well predict the thermal conductivities of pure ionic liquids but also to accurately predict the trends of thermal conductivity variations with temperature (see Figure 6). Generally, the obtained results given in Table 4 reveal that the proposed network predicts the thermal conductivity not only with lower MARD % values but also within a narrower range of MARD % for all of the ionic liquids, except for 1-butyl-1-methylpyrrolidinium bis(trifluoromethyl sulfonyl) imide26,27 and 1-butyl-3-methylimidazolium trifluoromethanesulfonate.26,27 The results obtained by the proposed neural network are also compared to empirical correlations15,20 and the group 9890

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Furthermore, the proposed correlation by Fröba et al.15 is able to correlate thermal conductivity only as function of temperature, while the proposed network is able to predict thermal conductivity as a function of both temperature and pressure. Predicted thermal conductivities of ionic liquids using Frez et al.’s20 correlation are compared with results from the proposed neural network in Table 5. This comparison reveals that the ANN model is able to predict the thermal conductivity of the four investigated ionic liquids within values of MARD % less than 5.1%, while Frez et al. 20 predicted the thermal conductivities with MARD % values of 7.7−24.8%. In addition, the results show that the proposed ANN model which was optimized for 21 ionic liquids is more global than the proposed correlation by Frez et al.20 covering only four ionic liquids. Further, since the extrapolated results obtained for [BMIM][BF4], [BMIM][PF6], and [EMIM][NTF2],which were not considered in either the training or testing data subsets, matched closely with experimental data (see Table 5), the capability of the proposed neural network for predicting thermal conductivities of pure ILs is shown. Finally, the results obtained from the proposed neural network were also compared with the group contribution method by Gardas and Coutinho2 for those particular ionic liquids for which the group parameters were available. Table 6 shows that the proposed neural network is as accurate as the group contribution method. Although the overall average MARD % is 0.2% for the proposed ANN method and 1.2% for the group contribution method, since both of these overall averages are smaller than the experimental uncertainties, no conclusion can be made on the superiority of our proposed method. Therefore, the results show that, in general, the proposed neural network could be a potential tool to predict the thermal conductivity of pure ionic liquids for a wide range of temperatures and pressures with acceptable accuracy compared to previously proposed correlations and group contribution methods.

Figure 5. Comparison between estimated and experimental thermal conductivities for the two ionic liquids 1-butyl-1-methylpyrrolidinium bis(trifluoromethyl sulfonyl)imide (temperature range of 293−323 K)26,27 and 1-butyl-3-methylimidazolium trifluoromethanesulfonate26,27 (temperature range of 293−353 K).

5. CONCLUSIONS In this study, we developed a neuromorphic formulation for estimation of thermal conductivities of pure ionic liquids. The developed model can be used for a vast variety of ionic liquids over a wide range of temperatures and pressures. In this regard, 209 data points were collected from different literature sources for 21 ionic liquids. The collected data were divided and used for two functions: (1) to train the network and find the best network parameters, including the number of hidden layers, number of neurons in the hidden layer, and best transfer functions of hidden and output layers, and (2) to investigate the predictive and correlative capability of the trained network. Using this approach, the optimum parameters of the proposed network were found to be as follows: 1 hidden layer with 13 neurons and the logarithmic−sigmoid and purelin functions as the transfer functions in the hidden and output layers, respectively. The accuracy of the proposed network with the best network architecture to predict the thermal conductivities of pure ionic liquids was validated with overall MARD %, MSE, and R2 values of 0.5%, 1.2 × 10−6, and 0.9983, respectively. In addition, comparing the results of the proposed neural network with the correlations of Fröba et al.15 and Frez et al.20 showed that the ANN model is more accurate than both of

Figure 6. Comparison between the temperature trend of the ANN model and experimental data.

contribution method proposed by Gardas et al.2 Previously, Fröba et al.15 reported that using eq 4, the standard percentage deviation and mean absolute percentage deviation of thermal conductivity of all the ionic liquids in their study at a temperature of 293.15 K and atmospheric pressure are within the range of 4.6−5.5% while the MARD % using the proposed neural network is in the range of 0.1−2.8%. They also reported that for 1-ethyl-3-methylimidazolium ethyl sulfate a maximum relative deviation of 10.2% is found. The maximum MARD % obtained by the proposed network in this study was 2.8% for 1butyl-1-methylpyrrolidinium bis(trifluoromethyl sulfonyl)imide, which, as discussed previously, had an unusually large error compared to possibly unreliable experimental data. 9891

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Table 5. Comparison between the Predicted Thermal Conductivities (W·m−1·K−1) of Four Ionic Liquids at 298.15 K Using Frez et al.’s Correlation20 and the Proposed ANN Model ionic liquid

experimental thermal conductivitya

predicted thermal conductivity using Frez et al.20

Frez et al.20 ARD %

predicted thermal conductivity by the ANN model

0.18624

0.162

12.9

0.186

0

0.14527

0.109

24.8

0.145

0

0.12826

0.108

15.6

0.121

5.1

0.13026

0.120

7.7

0.130

0

[BMIM] [BF4]b [BMIM] [PF6]c [BMIM] [NTF2]d [EMIM] [NTF2]e

ANN model ARD %

a

These data points were obtained using interpolation between values of thermal conductivity at the two known temperatures. b1-Butyl-3methylimidazolium tetrafluoroborate. c1-Butyl-3-methyl imidazolium hexafluorophosphate. d1-Butyl-1-methylpyrrolidinium bis(trifluoromethyl sulfonyl)imide. e1-Ethyl-3-methylimidazolium bis(trifluoromethyl sulfonyl)imide.

Table 6. Comparison between the Accuracy of the Group Contribution Method2 and the Proposed Neural Network To Predict the Thermal Conductivity of Some Ionic Liquids error analysis component IL 1-butyl-3-methyl imidazolium hexafluorophosphate 1-hexyl-3-methylimidazolium hexafluorophosphate 1-butyl-3-methylimidazolium tetrafluoroborate 1-ethyl-3-methylimidazolium tetrafluoroborate 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide 1-ethyl-3-methylimidazolium ethyl sulfate

no. of data points

ref

MARD % obtained by Gardas and Coutinho’s group contribution2

MARD % obtained by the ANN method

294.9−335.1

3

27

0.1

0.3

294.1−335.2

3

27

0.2

0.3

293−353

10

15

2.5

0.2

300−390

10

15

3.1

0.1

293−353

7

26

1.6

0.3

293−353

7

26

1.4

0.3

293−353

7

26

0.8

0.2

293−353

7

26

0.1

0.2

temp. (K)

(2) Gardas, R. L.; Coutinho, J. A. P. Group Contribution Methods for the Prediction of Thermophysical and Transport Properties of Ionic Liquids. AIChE J. 2009, 55, 1274. (3) Wilkes, J. S. A Short History of Ionic Liquidsfrom Molten Salts to Neoteric Solvents. Green Chem. 2002, 4, 73. (4) Torimoto, T.; Tsuda, T.; Okazaki, K.; Kuwabata, S. New Frontiers in Materials Science Opened by Ionic Liquids. Adv. Mater. 2010, 22, 1196. (5) Armand, M.; Endres, F.; MacFarlane, D. R.; Ohno, H.; Scrosati, B. Ionic-Liquid Materials for the Electrochemical Challenges of the Future. Nat. Mater. 2009, 8, 621. (6) Wishart, J. F. Energy applications of ionic liquids. Energy Environ. Sci. 2009, 2, 956. (7) Plechkova, N. V.; Seddon, K. R. Applications of Ionic Liquids in the Chemical Industry. Chem. Soc. Rev. 2008, 37, 123. (8) Wasserscheid, P.; Keim, W. Ionic liquids-New “Solutions” for Transition Metal Catalysis. Angew. Chem., Int. Ed. 2000, 39, 3772. (9) Jork, C.; Kristen, C.; Pieraccini, D.; Stark, A.; Chiappe, C.; Beste, Y. A.; Arlt, W. Tailor-Made Ionic Liquids. J. Chem. Thermodyn. 2005, 37, 537. (10) Sakaebe, H.; Matsumoto, H.; Tatsumi, K. Application of Room Temperature Ionic Liquids to Li Batteries. Electrochim. Acta 2007, 53, 1048. (11) Jiménez, A.-E.; Bermúdez, M.-D. Ionic Liquids as Lubricants for Steel-Aluminum Contacts at Low and Elevated Temperature. Tribol. Lett. 2007, 26, 53. (12) Van Valkenburg, M. E.; Vaughn, R. L.; Williams, M.; Wilkes, J. S. Thermo- Chemistry of Ionic Liquid Heat-Transfer Fluids. Thermochim. Acta 2005, 425, 181.

these literature models. Another advantage of the proposed neural network is functionalizing thermal conductivity to both the pressure and the temperature of the system, while Fröba et al.’s15 correlation is only correlated to the temperature of the system. Furthermore, the results of the neural network were compared to the group contribution method of Gardas and Coutinho.2 This comparison has shown that the proposed network was as accurate as or better than the group contribution method. In addition, a group contribution method can be more time consuming and complex compared to the proposed neural network. In general, the results obtained in this study showed that the ANN model could be a promising method to predict the thermal conductivity, or possibly other physiochemical properties, of pure ionic liquids.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +98-711-6133707. Fax: +98-711-6474619. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Konov, A. Ionic Liquids: Applications and Perspectives; InTech, 2011. 9892

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