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Modeling the Thermal Conductivity of Ionic Liquids and Ionanofluids Based on Group Method of Data Handling and Modified Maxwell Model Saeid Atashrouz, MEHRDAD MOZAFFARIAN, and Gholamreza Pazuki Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b00932 • Publication Date (Web): 12 Aug 2015 Downloaded from http://pubs.acs.org on August 18, 2015

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Industrial & Engineering Chemistry Research

Modeling the Thermal Conductivity of Ionic Liquids and Ionanofluids Based on Group Method of Data Handling and Modified Maxwell Model

Saeid Atashrouz1, Mehrdad Mozaffarian1,*, Gholamreza Pazuki1 1

Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Hafez 424, PO Box 15875-4413, Tehran, Iran

Abstract The objective of this study is to develop a model to determine the thermal conductivity of pure ionic liquids and Ionanofluids. In order to estimate the thermal conductivity of pure ionic liquids, a group method of data handling model is proposed based on 23 ionic liquids corresponding to 216 experimental data points. The average absolute relative deviation for all studied systems was 1.81%, which is a satisfactory degree of accuracy for the proposed model. Furthermore, the Maxwell model is modified to correlate the thermal conductivity of Ionanofluids as a function of temperature and volume fraction of nanoparticles. The average absolute relative deviation for this model is 0.61%. Additionally, Maxwell and modified geometric mean (mGM) models are used to evaluate the models that *

Corresponding Author: [email protected]

ACS Paragon Plus Environment


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predict the thermal conductivity of Ionanofluids. The results show that mGM is more accurate for prediction of thermal conductivity of Ionanofluids. Keywords: Ionic liquids, Ionanofluid, Thermal conductivity, Maxwell

1. Introduction In recent years, the interest in green chemical technology has led to the development of a new set of unique and highly tunable compounds called ionic liquids (ILs)1. ILs have attracted the attention of scientific communities due to their unique properties such as non-volatility, ease of recycling, high solvability capacity

for

polar/nonpolar

compounds,

and

high

thermal/chemical/

electrochemical stability2. These new class of liquids have numerous applications in biotechnology, chemical synthesis, catalytic reactions, membrane separation technology, lubricants, and electrolytes in batteries2-6. Additionally, ILs have recently become attractive heat transfer fluids (HTFs). This has led to some studies focused on measuring the thermal conductivity of ILs7-11. Other studies have investigated the viability of ILs as HTFs in solar energy collector systems7,12,13. A typical solar energy collector system uses HTFs to transfer energy from one source to another. The resulting thermal energy could be used in domestic applications or converted to power. Commercial HTFs such as Therminol VP-1


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(mixture of diphenyloxide and biphenyl) have a high vapor pressure which could lead to a phase change phenomenon12. In contrast, ILs have negligible vapor pressure and do not volatilize at atmospheric pressure, which enables them to maintain their liquid phase state within a wider temperature range. Another advantage of ILs is their high volumetric heat capacity compared to commercial HTFs12. Consequently, ILs can be an ideal substitution for regular commercial HTFs in solar energy collector systems. The thermal properties of ILs can be enhanced by adding nanoparticles to them. This will produce a new and innovative class of heat transfer fluids called Ionanofluids, which have higher thermal conductivity than ILs. Wang et al. have reported that when graphene sheets are added to an ionic liquid of 1-hexyl-3methylimidazolium tetrafluoroborate, thermal conductivity is enhanced by 18.6%4. Castro et al. have measured thermal conductivity of four ILs containing 1 weight percent of multiwalled carbon nanotubes (MWCNT) and have reported moderate thermal conductivity enhancements between 2 and 9% 10. Therefore, since ILs and Ionanofluids are better agents in various applications due to their special properties, it is beneficial to develop their thermal conductivity models. The objective of the present study is to model the thermal conductivity of ILs based on the group method of data handling polynomial neural network (GMDHPNN) systems. In this regard, 216 experimental data points for 23 ILs were



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considered for training and testing the model. Temperature, pressure, critical temperature, critical pressure and boiling temperature of ILs were selected as input variables to the model and thermal conductivity was designated as the output variable. Additionally, an artificial neural network (ANN) was developed for comparing GMDH-PNN with ANN systems. The thermal conductivity model of Ionanofluids was developed based on modifications to Maxwell model14. Furthermore, the reliability of Maxwell model and modified geometric mean model15 to predict the thermal conductivity of Ionanofluids was checked.

2. Methodology 2.1. Available methods for modeling the thermal conductivity of pure ILs Some studies that have attempted to estimate the thermal conductivity of pure ILs, have proposed correlative and predictive models such as simple correlations, ANNs, genetic algorithm, and group contribution method (GCM). Tomida et al. have proposed a simple relation for thermal conductivity of ILs11: M λ log  W  = 1.9596 − 0.004499M W  η 

(1)

Where MW is molar mass, λ is thermal conductivity and η is viscosity of IL. However, this model lacks sufficient universality, and is only applicable for a limited number of anions16,17.


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Froba et al. have proposed a correlation for 10 ILs at 293.15 K and atmospheric pressure8: λρ = A +

B MW

(2)

Where A and B are two constants and ρ is IL density. Wu et al. have tried to extend this model for other ILs and temperatures, but the results have not been satisfactory18 (mean absolute relative deviation was 8.15%). Shojaee et al. proposed the following simple formula for prediction of thermal conductivity as a function of temperature, pressure, melting point and molecular weight for 21 ILs16: a + b (M W / T )c + d (T m / P )e λ= T

(3)

A genetic algorithm was applied for adjusting parameters (a, b, c, d and e) of the model. They have reported mean absolute relative deviation (MARD) values of 5.22% and 10.76% for the train and test data sets respectively. Although the relatively high MARD of their model could be justified due to intrinsic simplicity of the model, the main disadvantage of the model is that the melting point of ILs is considered as one of the input parameters; even though the melting point values are not available for a wide range of ILs. Hezave et al. proposed an ANN model for modeling the thermal conductivity of 21 ILs and 209 data points17. They have assumed that the thermal conductivity of any IL is a function of its temperature,



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pressure, molecular weight and melting point. Their optimized model consists of 13 neurons in a hidden layer with a MARD of 0.5%. Although their model has excellent accuracy, it faces two criticisms. The first criticism is considering the melting point as the input parameter of the model (like the model by Shojaee et al.16), and the second is the complex structure of ANNs which makes its application difficult. An ANN provides a huge set of complex equations19 governing its nodes and layers which makes it a difficult approach for estimating thermal conductivity of ILs due to its highly complex structure. Gardas and Coutinho proposed a simple correlation based on GCM20:

λ = A λ − B λT

(4)

Where T is the temperature in Kelvin and A λ and B λ are adjustable parameters which can be obtained through a GCM method. The MARD for their model was 1.06% for 16 ILs and 107 data points. However, this model defines the anions as a group, which means that the results are not universal for different ILs. Consequently, the GCM values should be recalculated in order to apply this model for new anions17. Recently, Wu et al. proposed a GCM method to estimate the thermal conductivity of ILs18. The MARD for their model was 1.66% which makes this GCM model a far more reliable one. The model by Wu et al. is accurate enough, but in such GCM methods the whole data set is used to obtain values of the GCM model parameters.


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Therefore, the next step is to develop a model which is also capable of checking the validity of thermal conductivity predictions. In recent years, various researchers have introduced ANNs as predictive models that employ the strategy of training and testing. Based on this strategy, a part of experimental data called training data set is used for training and developing the ANN model, and then the obtained model is evaluated with the rest of experimental data called testing data set. Many studies have shown that ANNs are powerful and highly accurate nonlinear systems with the ability to model complex problems19,21-23. Generally, in order to connect input data to output data, the final mathematical model of ANNs is a matrix of weights and biases, which should be determined based on minimization of error of the model. In order to use an optimized ANN model, a mathematical software is needed to build the network based on the obtained weights and biases. Furthermore, an ANN model does not have a simple mathematical formula to relate input parameters to output parameters24. At the same time, it should be mentioned that simple mathematical models and relations are more favored in engineering calculations. In order to overcome the limitations of ANNs, we proposed in our previous studies an alternative approach called “group method of data handling polynomial neural



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network” (GMDH-PNN) system, which is a modified and hybrid version of “group method of data handling” (GMDH)24-26. The (GMDH-PNN) methodology devises a self-organizing neural network of simple polynomials and uses common minimization algorithms to find the most appropriate configuration24. We applied the GMDH-PNN model to estimate the viscosity of nanofluids24 and the results provided good accuracy (MARD=2.14%). The GMDH-PNN was also considered for modeling water activity in a binary system of water and poly ethylene glycol (PEG)25. Root of mean square error for this model was 0.0348. The GMDH-PNN was also used for modeling the surface tension of ionic liquids26, and the resulting MARD was 4.59%. These results indicate that the GMDH-PNN models have performed well for the aforementioned systems, and they are a good alternative to ANNs. Since the simplicity and practicality of applying the mathematical model was of greater importance in conducting this study, it was decided to select GMDH-PNN for this study even though ANN is more accurate. The GMDH and GMDH-PNN will be described in the following sections.

2.2. Group method of data handling The application of the once-devised Group Method of Data Handling (GMDH) has become appealing in recent years as it provides both simplicity and accuracy simultaneously27,28,29. The GMDH is founded on the backbone of natural selection


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of species theory. This theory states that the most perfect idiosyncrasies propagate among generations. The GMDH imitates an analogous premise to cross over independent input variables in order to originate a new generation of virtual ones. The GMDH algorithm follows four steps28 which will be described in the following sections. 2.2.1. Generation of observation matrix First, the GMDH algorithm generates a nonhomogeneous matrix of the following form: x 11

x 12

L

x 1M

y 2 x 21 = M M

x 22 M

L M

x 2M M

yN

x N 2 L x NM

y1

x N1

(5)

Where, M stands for the number of independent variables and N represents the number of observations. The left matrix holds the vector of observed outputs uur V y = ( y 1 , y 2 ,… , y N ) , while the right one represents the corresponding vector of

uur independent variables V x = ( x 1 , x 2 ,… , x M ) .

2.2.2. Generation of virtual variables



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Next, the algorithm replaces the independent variables by virtual ones. The new virtual variable (z) is defined by selecting only two independent variables at a time to construct a nodal quadratic polynomial as follows: z ij = a + b1x i + b 2 x j + c1x i2 + c 2 x 2j + dx i x j

i , j = 1, 2,…, M

(6)

In this regard, equation (5) reshapes to: z 11

z 12

L

z 1S

y 2 z 21 = M M

z 22 M

L M

z 2S M

yN

z N 2 L z NS

y1

z N1

M  ,S =    2 

M represents the number of independent variables, and

(7)

M  S =   2 

represents the

number of possible combinations of two at a time selection of independent variables.

2.2.3. Determination of coefficients The coefficients of equation (6) are calculated by utilizing the least squares fitting approach. The objective is to minimize the least square error (R2) for each column. The solution to the following system of equations determines the coefficients of the equations corresponding to the virtual matrix entries: Nt

R = ∑ ( y i − z ij ) 2 j

2

j = 1, 2,…,S

i =1


(8)

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∂ ( R j2 )

Nt

= −2∑ ( y i − z ij ) = 0

∂a ∂ ( R j2 ) ∂b1

Nt

= −2∑ ( y i − z ij ) x i = 0

(10)

i =1

∂ ( R j2 ) ∂c1

Nt

= −2∑ ( y i − z ij ) x i2 = 0

(11)

i =1

∂ ( R j2 ) ∂d

(9)

i =1

Nt

= −2∑( y i − z ij ) x ij = 0

(12)

i =1

Where Nt, is the number of training sets. An arbitrary ratio is chosen to split the observed data into two sets: the training set and the testing set.

2.2.4. Selection of the most accurate expressions The GMDH algorithm applies the testing data set to generated nodal expressions in each column of the virtual matrix in order to find the most accurate ones. The summation of errors for each column must meet the following criterion:

δ j2 =

N

∑

i =N t

2

 y i − z ij  < ε j = 1, 2,…, S +1

(13)

If a virtual variable meets the criterion, it will be saved; otherwise the algorithm will omit it. The deviation from actual data for each iteration is saved by the algorithm. As the algorithm iterates, it compares the errors and stops when the



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minimum value is obtained. The topology of the GMDH network comprises intermediate layers in which several nodes combine interaction of independent and virtual variables to generate nodal expressions by which the genome expression of the model is gradually derived.

2.2.5. Hybrid GMDH-type neural network The original GMDH approach has two disadvantages which reduce the ability of modeling or tracing a treatment in nonlinear systems. The first one is the consideration of only two independent variables at a time (equation 6), and the second is connecting virtual variables or independent variables only to the next layer. In this regard, two modifications are applied to construct a hybrid GMDHtype polynomial neural network (GMDH-PNN). The first modification allows more than two independent variables to combine. The second modification allows nodal crossover with different layers24. The virtual variables or independent variables are only connected to their next layer in the original GMDH. Consequently, considering more interactions between virtual or independent variables and also an increase in the terms of nodal expression will enable the hybrid GMDH-PNN to have more flexibility for modeling of complex and nonlinear systems. The grand multinomial correlation for hybrid GMDH-PNN has the following form24,25:


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M

M

M

y = a + ∑∑L ∑b ij Lk x in x nj L x kn i =1 j =1

n = 0,1,…, 2l

(14)

k =1

Where l is for the number of network layers.

3. Development of models 3.1. GMDH-PNN for estimation of thermal conductivity of pure ILs In order to develop the GMDH-PNN model, 216 experimental data points corresponding to 23 ILs were considered for training and testing the model with their characteristics tabulated in Table 1. Temperature, pressure, critical temperature, critical pressure and boiling temperature of ILs were used as the input variables of the model, and thermal conductivity was selected as the output variable. Critical temperature, critical pressure and boiling temperature of ILs were obtained using the GCM method of Valderrama et al.30-33: T b (K ) = 198.2 + ∑ n ∆T b

Tc (K ) =

(15)

Tb A + B ∑ n ∆Tc − (∑ n ∆Tc )2 

Pc (bar ) =

M

(16)

(17)

[C + ∑ n ∆P ]

2

c



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Where A , B and C are constants and n is the number of times that a group appears in a molecule. ∆Tb and ∆Tc are critical boiling temperature and critical temperature of each group respectively. In order to test the consistency of this method, the estimated density30 data or saturation pressure33 data was compared with the available experimental data. The successful consistency test of the method indicated that it can be considered as a standard method for calculating critical properties, normal boiling point and acentric factor of ILs. It should be noted that most ILs will decompose before reaching their critical state. Therefore experimental investigation of their critical properties is not practical. In this regard, our study considers the proposed method by Valderrama et al. as the standard for prediction of critical properties of ILs. However, it should be noted that using another method to estimate the critical properties of ILs may provide a different set of results. For instance, Shariati et al. proposed a method to estimate critical properties of ILs based on the vapor liquid equilibria data of binary mixtures containing ILs34. Table S1 in supporting information shows a comparison between the results of the models by Valderrama et al. and Shariati et al. which display considerable difference between the two critical properties data sets.


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In order to develop the GMDH-PNN, 50% of data set was randomly used for training the model and the remaining 50% was used for testing the model. The optimized model is tabulated in Table 2. In addition to the GMDH-PNN model, an ANN model was also considered for comparison with the GMDH-PNN model. In this regard, 60% of experimental data was randomly considered for training the ANN model and the remaining 40% was used for testing. In order to find an optimized ANN, a trial and error method was considered and different topologies of ANNs with various numbers of hidden neurons were checked. The MARD of training, validation and testing for each topology are reported in Table S2 (supporting information). The statistical error analysis demonstrated that the best configuration is a network with 9 hidden neurons. A schematic of optimized ANN model is presented in Figure 1.

3.2. Thermal conductivity model for Ionanofluids The most popular model for prediction of thermal conductivity of nanofluids is the Maxwell model14: 

λ Maxwell = λIL 1 + 

 3(α − 1)ϕns  (α + 2) − (α − 1)ϕ ns 


(18)


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Where α is the ratio of thermal conductivity of nanoparticle to the thermal conductivity of IL ( α = λns λIL ) and ϕ ns is the volume fraction of the nanostructure. The Maxwell model can be applied for systems with uniform dispersion of spherical particles with no particle interactions. The model predicts thermal conductivities of nanofluids 0 − 15% higher than that of a base fluid when ϕ ≤ 0 .0 5 . However, it has been demonstrated in so many cases that Maxwell

model tends to underestimate the thermal conductivity values when its predicted results are compared to the experimental data. In this regard, modification of the Maxwell model is the purpose of this phase of the study. The difference between the real data (experimental data) and the estimated data (by Maxwell model) is considered as a modified term:

λm = λreal − λMaxwell

(19)

Where, superscripts ‘m’, and ‘real’ correspond to modified property and real data respectively. Then thermal conductivity of Ionanofluid is obtained by the following relation: 

λ real = λIL 1 + 

 m 3(α − 1)ϕns +λ (α + 2) − (α − 1)ϕns 


(20)

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λIL can be elicited from experimental thermal conductivity of ILs or can be predicted by GMDH-PNN in the previous section. The model is then developed when the appropriate relations are replaced for λns ( α = λns λIL ) and λ m . Thermal conductivity of solids is dependent on size, and its bulk value is different compared to its nanostructure value due to entropy generation and wall effects. Some experimental studies and molecular simulations have focused on the size dependency of thermal conductivity of carbon based nanomaterials such as Graphene, singlewalled carbon nanotubes (SWCNT) and multiwalled carbon nanotubes (MWCNT). Those results have shown that the thermal conductivity of nanostructures is strongly dependent on the length of their particles35-38. Such results have also been observed in the case of semiconducting materials such as silicon39-41. Consequently, the effect of size dependency of nanostructures on their thermal conductivity should be factored in modeling. The only remaining term in equation (20) is λ m which is the modification factor and is of great importance for maximizing the performance of Maxwell model. This factor represents the non-ideality of thermal conductivity for an Ionanofluid system. We have considered this modification term as an excess property for the system (λ m = λ E ) . We had developed a modified two suffix Margules (MTSM) Gibbs free energy model in a previous study2, which was coupled with the Eyring’s relation42 to



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construct the model for the viscosity of ionic liquids-cosolvent mixtures. That study demonstrated that the MTSM model performed well if the ionic liquidscosolvent mixtures were considered to be non-ideal mixtures strictly for the purpose of modeling the viscosity. In this regard, we decided to apply this model as the non-ideality term of the model in equation (20). The MTSM model has the following relation: gE = α12 X 12 X 21 RT

(21)

Where α12 is an empirical constant and X ij is the local mole fraction of i around j . The physical properties of the system for nanofluids are reported as a function of volume fraction of nanoparticles. In this regard, local volume fractions can be considered in the model. We consider λ E as excess Gibbs free energy model for the case of thermal conductivity:

λ E = λ12φ12φ21

(22)

Where φij is the local volume fraction of component i around j and λ12 is an empirical constant with thermal conductivity characteristic similar to α12 . The relations between local volume fractions and bulk volume fractions are as follows2: φ12 ϕ1 = G φ22 ϕ2 12

(23)


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and φ21 ϕ2 = G φ11 ϕ1 21

(24)

Thus, the above equations can be rearranged to obtain the local volume fractions of the molecules in the cells: φ12 =

ϕ1G12 ϕ1G12 + ϕ2

(25)

ϕ2G 21 ϕ2G 21 + ϕ1

(26)

and φ21 =

Where:  τ G12 = exp  − 12  RT

  , τ12 = g 12 − g 22 

(27)

  , τ 21 = g 21 − g 11 

(28)

and  τ G21 = exp  − 21  RT

Where g ij is the energy between components i and j. Using equations (23) to (28), the model takes the following form: λ E = λ12

ϕ1ϕ2G12G 21 (ϕ1G12 + ϕ2 )(ϕ2G 21 + ϕ1 )

(29)



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Equation (29) is known as “modified two suffix Margules” (MTSM) model with three energy parameters namely λ12, τ12 and τ21 , which should be adjusted based on the experimental thermal conductivity data. Now, the final model can be obtained based on equations (20), (21) and (29): 

λ real = λIL 1 + 

Where

 3(α − 1)ϕ1 ϕ1ϕ 2G12G 21  + λ12 (α + 2) − (α − 1)ϕ1  (ϕ1G12 + ϕ2 )(ϕ2G 21 + ϕ1 )

(30)

ϕ1 and ϕ2 are volume fraction of nanostructure and IL respectively.

The obtained model is called Maxwell-MTSM in this study.

3.3. Statistical functions In order to evaluate the proposed models in this study, some statistical functions including MARD%, mean absolute deviation (MAD%), mean square deviation (MSD) and regression coefficient (R2) are considered as shown below: MA RD (%) =

MAD (%) =

MSD =

1 N

100 N λ calc − λ exp ∑ λ exp N i =1

(31)

100 N calc λ − λ exp ∑ N i =1 N

∑ (λ

calc

(32)

− λ exp )2

(33)

i =1

 N exp calc 2   ∑ (λ − λ )   R 2 = 1 −  i =1 N exp 2   (λ ) ∑   i =1

(34)


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4. Results and discussion 4.1. Pure ionic liquids Table S3 (in supporting information) shows statistical analysis for GMDH-PNN and ANN models with both models demonstrating a good degree of accuracy. One of the problems of neural network models is the “over fitting” issue, which happens when the model has a good accuracy for estimation of the train data set, but lacks accuracy for estimation of the test data. However, the proposed models in Table S3 have a good degree of accuracy for both the train and test data sets, and they have no “over fitting” issues. Figures 2 and 3 show estimated data against experimental thermal conductivity data for GMDH-PNN and ANN respectively. As can be observed, the data points are generally close to the diagonal line demonstrating a reasonable conformity between the experimental data and the results from the models. The detailed statistical analysis for all the studied systems is reported in Table S4 (in supporting information). In order to compare the performance of GMDH-PNN and ANN models with other models in literature, three models from Wu et al.18, Gardas/Coutinho20 and Shojaee et al.16 were picked and the results of MARD for these systems are tabulated in Table 3. As can be seen, the ANN and the GMDH-PNN have higher accuracy than the three selected models, with ANN being the most accurate one.



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Page 22 of 46

However, as mentioned before, ANN models have a complex mathematical structure, whereas GMDH-PNNs have an optimal complex model that is not necessarily too simple and not too complex. Furthermore, by using GMDHPNN, an explicit analytical model representing relevant relationships between input and output variables is obtained, and it provides good accuracy. This subject has also been discussed by Reyhani et al.27. GMDH-PNN has a good degree of accuracy for 23 ILs mentioned in Table 1. Although only a limited number of experimental investigations have been conducted on thermal conductivity of ILs7-11, it would be intriguing to attempt predicting the thermal conductivity of new ILs. To investigate the possibility of thermal

conductivity

prediction

for

GMDH-PNN,

1-butyl-1-methyl-

pyrrolidinium dicyanamide ([C4mpyr][dca]) which had not been considered for training the GMHD-PNN model, was selected for estimation of thermal conductivity with the results shown in Figure 4. As can be observed, the results of GMDH-PNN model are in general agreement with the experimental data. It should be noted that experimental errors such as uncertainty of instruments accuracy for measurement of thermal conductivity are not negligible. Experimental data of Franca et al.43 shown in Figure 4 have been reported with an uncertainty equal to ±0.01 which could be one of the reasons for the difference between predicted and experimental data. For more information on this issue,


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the results of GMDH-PNN model for [EMIM][dca] are presented in Figure 5. Of all the studies in the literature on thermal conductivity of ILs, only two studies by Froba et al.8 and Franca et al.43 have focused on measuring the thermal conductivity of [EMIM][dca] as a function of temperature. We have used experimental data of Froba et al. for thermal conductivity of [EMIM][dca] in order to train the proposed GMDH-PNN and as can be seen in Figure 5, there is an excellent agreement between the estimated results and the experimental data by Froba et al. The experimental data reported by Franca et al. for [EMIM][dca] has a significant deviation with the data reported by Froba et al. We are not certain which one of these two studies has the more accurate set of reported data. These differences could possibly be due to purity of the samples or the method for measuring thermal conductivity. Despite this issue, we aimed to report the thermal conductivity of ILs for a wide range of systems. Although it is expected that the predictions by GMDH-PNN will not be very accurate, it can still be useful as a predictive tool to estimate thermal conductivity of ILs when no experimental data is available. In this regard, inputs of the model containing boiling temperature, critical temperature, critical pressure for various ILs were calculated based on the group contribution method by Valderrama et al.30-33. The thermal conductivity of 1000 ILs at 298.15 K and 101.32 kPa are reported in an Excel file in the supporting information



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 59 60

section. It should be emphasized again that these values are only an estimate of thermal conductivity of ILs, and it will be normal to expect that the reported data would not be very accurate. Too few ionic liquids have so far been considered for experimental measurement of thermal conductivity. However, it is expected that more future studies will take on this challenge due to the unique and special properties of ILs and their potential for a variety of applications such as heat transfer processes.

4.2. Ionanofluids Few experimental studies have been conducted and reported on Ionanofluids. Seven systems reported in Table 44,13,43 were selected to validate the ability of Maxwell-MTSM model to predict thermal conductivity of Ionanofluid. The experimental investigations for thermal conductivity of Ionanofluids have in most cases focused on MWCNT 4,10,43,44. However, other nanostructures such as Al2O3 and Graphene have also been considered in some of the studies4,13. As mentioned before, thermal conductivity dependency of nanoparticles on their size is a necessary factor to be considered in modeling. Therefore, thermal conductivity values of nanoparticles with their corresponding size were used from reported data in literature35,36,45. Originally, the Maxwell-MTSM model has three energy parameters namely, λ12,

τ12 and τ21 . Adjustable parameters of the Maxwell-MTSM model are high in their


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present form. Therefore, it was assumed that interaction parameters between IL and nanostructure are the same and equal toτ :

τ12 =τ21 =τ

(35)

The experimental data of thermal conductivity for Ionanofluids has usually demonstrated a variation with respect to temperature. In this regard, λ12 is considered to be temperature dependent: λ12 = λ12( I )T + λ12( II )

(36)

Now the Maxwell-MTSM model has two energy parameters ( τ and λ12) which should be determined based on available experimental thermal conductivity data for Ionanofluids. In this regard, the following objective function (OF) was considered: OF =

1 N

N

∑ i =1

(λ calc − λ exp ) 2

(37)

λ exp

Values of energy parameters are tabulated in Table 4. Furthermore, MARD for all Ionanofluids are reported in this table, which show good performance of the model for correlation of thermal conductivity. Figure 6 shows the results of Maxwell-MTSM model for three Ionanofluids containing [EMIM][dca]MWCNT, [BMIM][dca]-MWCNT and [C4mpyr][dca]-MWCNT. It can be seen that the model is capable to correlate thermal conductivity data with variations of temperature. Furthermore, estimated thermal conductivity for [HMIM][BF4]-



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Page 26 of 46

MWCNT and [HMIM][BF4]-Graphene systems derived from the model is shown in Figure 7. The results of Table 4 and Figures 6 and 7 confirm that Maxwell-MTSM has a good accuracy for modeling the thermal conductivity of Ionanofluids. However, the main shortcoming of this model is the lack of available experimental data for obtaining energy parameters. Consequently, it is necessary to develop predictive models in order to obtain the thermal conductivity of new ILs. To the best of our knowledge, there is no predictive model available yet for thermal conductivity of Ionanofluids due to their novelty. However, various physical models have been proposed in literature for nanofluids. In the present study, Maxwell model14 and modified geometric mean (mGM) model15 were considered for prediction of thermal conductivity. The mGM has the following relation for an Ionanofluid:

λIonano

λ  = λIL  ns   λIL 

ϕ ns

(38)

Where λIonano is the thermal conductivity of Ionanofluid. The results of predicting the thermal conductivity of Ionanofluids are tabulated in Table 5. The mGM model displays better accuracy than Maxwell model, whereas the accuracy of this model is not satisfactory for the two systems containing [HMIM][BF4]MWCNT and [HMIM][BF4]-Graphene. However, the MARD of the other


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systems are below 5.7% which is a good performance for prediction without using adjustable parameters. Figure 8 shows predicted results for [C4mpyr][dca]MWCNT. As can be observed, predicted results of the models are lower than their experimental value. However, mGM results are much closer to experimental data, and consequently it is better suited for prediction of thermal conductivity of Ionanofluid systems.

Conclusion The present study has demonstrated that GMDH-PNN delivers good results for estimation of thermal conductivity of ILs with a MARD of 1.81%. Both neural network models have their advantages. GMDH-PNN has an optimal complex model that is neither too simple nor too complex compared to an ANN model. The ANN model was more accurate than the GMDH-PNN. However, since the accuracy of GMDH-PNN is reasonable in this study (MARD=1.81%), it is still better to use GMDH-PNN due to its simpler mathematical structure. The study also shows that the Maxwell-MTSM model can estimate thermal conductivity data for Ionanofluid with a high degree of accuracy (MARD=0.61%). However, this model is dependent on experimental data for obtaining energy parameters. This means that predictive models are also of great importance when experimental measurement is not conducted or available for an Ionanofluid. The



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mGM model had a better accuracy than Maxwell model with a MARD of 5.60%. In this regard, mGM is proposed to be the choice for prediction of thermal conductivity of Ionanofluids.

Supporting Information comparison between Shariati et al. and Valderrama et al. models for estimation of critical properties and acentric factor of ILs; The performance analysis of different ANN topologies; statistical analysis of train and test data for GMDH-PNN and ANN models; comparison of accuracy of GMDH-PNN and ANN models for all of ILs; prediction of thermal conductivity of ILs. Prediction of thermal conductivity for 1000 ILs (Excel file). This material is available free of charge via the Internet at http://pubs.acs.org.

References (1) Salgado, J.; Regueira, T.; Lugo, L.; Vijande, J.; Fernandez, J.; Garcia, J. Density and viscosity of three (2,2,2-trifluoroethanol + 1-butyl-3-methylimidazolium) ionic liquid binary systems. J. Chem. Thermodyn. 2014, 70, 101. (2) Atashrouz, S.; Zarghampour, M.; Abdolrahimi, S.; Pazuki, G. R.; Nasernejad, B. Estimation of the Viscosity of Ionic Liquids Containing Binary Mixtures Based on the Eyring’s Theory and a Modified Gibbs Energy Model. J. Chem. Eng. Data 2014, 59, 3691. (3) Fatehi, M.; Raeissi, S.; Mowla, D. Estimation of viscosity of binary mixtures of ionic liquids and solvents using an artificial neural network based on the structure groups of the ionic liquid. Fluid Phase Equilibr. 2014, 364, 88.


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(4) Wang, F.; Han, L.; Zhang, Z.; Fang, X.; Shi, J.; Ma, W. Surfactant free ionic liquid-based nanofluids with remarkable thermal conductivity enhancement at very low loading of graphene. Nanoscale Res. Lett. 2012, 7, 314. (5) Welton, T. Room-temperature ionic liquids. Solvents for synthesis and catalysis. Chem. Rev. 1999, 99, 2071.

(6) Plechkova, N.; Seddon, K. Application of ionic liquids in the chemical industry. Chem. Soc. Rev. 2008, 37, 123.

(7) Valkenburg, M.; Vaughn, R.; Williams, M.; Wilkes, J.; Thermochemistry of ionic liquid heat-transfer fluids. Thermochim. Acta 2005, 425, 181. (8) Froba, A. P.; Rausch, M. H.; Krzeminski, K.; Assenbaum, D.; Wasserscheid, P.; Leipertz, A. Thermal Conductivity of Ionic Liquids: Measurement and Prediction. Int. J. Thermophys. 2010, 31, 2059. (9) Ge, R.; Hardacre, C.; Nancarrow, P.; Rooney, D. W. Thermal Conductivities of Ionic Liquids over the Temperature Range from 293 K to 353 K. J. Chem. Eng. Data 2007, 52, 1819. (10) Castro, C. A.; Lourenco, M. J. V.; Ribeiro, A. P. C.; Langa, E.; Vieira, S. I. C.; Thermal Properties of Ionic Liquids and IoNanofluids of Imidazolium and Pyrrolidinium Liquids. J. Chem. Eng. Data 2010, 55, 653.

(11) Tomida, D.; Kenmochi, S.; Tsukada, T.; Qiao, K.; Yokoyama, C. Thermal Conductivities of [bmim][PF6], [hmim][PF6], and [omim][PF6] from 294 to 335K at Pressures up to 20MPa. Int. J. Thermophys. 2007, 28, 1147.

(12) Bridges, N. J.; Visser, A. E.; Fox, E. B.; Potential of Nanoparticle-Enhanced Ionic Liquids (NEILs) as Advanced Heat-Transfer Fluids. Energ. Fuel. 2011, 25, 4862. (13) Paul, T. C.; Morshed, A. M.; Khan, J. A. Nanoparticle enhanced Ionic liquids (NEILs) as working fluid for the next generation solar collector. Procedia Engineering 2013, 56, 631. (14) Maxwell, J. C. A Treatise on Electricity and Magnetism, 3rd ed., Vol. II. London: Oxford University Press, 1892. (15) Warrier, P.; Yuan, Y.; Beck, M. P.; Teja, A. S. Heat Transfer in Nanoparticle Suspensions: Modeling the Thermal Conductivity of Nanofluids, AIChE J. 2010, 56, 3243.



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(16) Shojaee, S.; Farzam, S.; Hezave, A. Z.; Lashkarbolooki, M.; Ayatollahi, S.; A new correlation for estimating thermal conductivity of pure ionic liquid, Fluid Phase Equilibr. 2013, 354,199. (17) Hezave, A. Z.; Raeissi, S.; Lashkarbolooki, M. Estimation of Thermal Conductivity of Ionic Liquids Using a Perceptron Neural Network. Ind. Eng. Chem. Res. 2012, 51, 9886. (18) Wu, K.; Zhao, C.; He, C.; Development of a group contribution method for determination of thermal conductivity of ionic liquids. Fluid Phase Equilibr. 2013, 339, 10. (19) Atashrouz, S.; Mirshekar, H. Phase equilibrium modeling for binary systems containing CO2 using artificial neural networks. Bulg. Chem. Commun. 2014, 46, 104. (20) 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. (21) Rohani, A.; Pazuki, G. R.; Najafabadi, H. A.; S. Seyfi, Vossoughi, M. Comparison between the artificial neural network system and SAFT equation in obtaining vapor pressure and liquid density of pure alcohols. Expert Syst. Appl. 2011, 38, 1738. (22) Yousefi, F.; Karimi, H.; Mohammad Mehdi Papari, Modeling viscosity of nanofluids using diffusional neural networks, J. Mol. Liq. 2012, 175, 85. (23) Alvarez, Marleny, V.; Saldana, D. A. Thermodynamic prediction of vapor–liquid equilibrium of supercritical CO2 or CHF3 + ionic liquids. J. Supercrit. Fluid 2012, 66, 29. (24) Atashrouz, S.; Pazuki, G. R.; Alimoradi, Y. Estimation of the viscosity of nine nanofluids using a hybrid GMDH-type neural network system. Fluid Phase Equilibr. 2014, 372, 43. (25) Atashrouz, S.; Pazuki, G. R.; Kakhki, S. A GMDH-type neural network for prediction of water activity in glycol and Poly (ethylene glycol) solutions. J. Mol. Liq. 2015, 202, 95. (26) Atashrouz, S.; Amini, E.; Pazuki, G. R. Modeling of surface tension for ionic liquids using Group Method of Data Handling, Ionics, 2015, 21, 1595. (27) Reyhani, S. Z.; Ghanadzadeh, H.; Puigjaner, L.; Recances, F. Estimation of LiquidLiquid Equilibrium for a Quaternary System Using the GMDH Algorithm, Ind. Eng. Chem. Res. 2009, 48, 2129.

(28) Ivakhnenko, A. G. Group method of data handling –A rival of the method of stochastic approximation. Soviet Automatic Control, 1966, 13, 43. (29) Farlow, S. J. The GMDH algorithm of Ivakhneko. American Statistician 1981, 35, 210.


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(30) Valderrama, J. O.; Robles, P. A. Critical Properties, Normal Boiling Temperatures, and Acentric Factors of Fifty Ionic Liquids. Ind. Eng. Chem. Res. 2007, 46, 1338. (31) Valderrama, J. O.; Sanga, W.; Lazzus, J. A. Critical Properties, Normal Boiling Temperature, and Acentric Factor of Another 200 Ionic Liquids, Ind. Eng. Chem. Res. 2008, 47, 1318. (32) Valderrama, J. O.; Rojas, R. E. Critical Properties of Ionic Liquids. Revisited, Ind. Eng. Chem. Res. 2009, 48, 6890.

(33) Valderrama, J. O.; Forero, L. A.; Rojas, R. E. Critical Properties and Normal Boiling Temperature of Ionic Liquids. Update and a New Consistency Test. Ind. Eng. Chem. Res. 2012, 51, 7838. (34) Shariati, A.; Ashrafmansouri, S.; Osbuei, M.; Hooshdaran, B. Critical properties and acentric factors of ionic liquids. Korean J. Chem. Eng. 2013, 30, 187. (35) Marconnet, A. M.; Panzer, M. A.; Goodson, K. E. Thermal conduction phenomena in carbon nanotubes and related nanostructured materials. Rev. Mod. Phys. 2013, 85, 1295. (36) Cao, A.; Qu, J. Size dependent thermal conductivity of single-walled carbon nanotubes, J. Appl. Phys. 2012, 112, 013503.

(37) Salaway, R. N.; Zhigilei, L. V. Molecular dynamics simulations of thermal conductivity of carbon nanotubes: Resolving the effects of computational parameters. Int. J. Heat Mass Tran. 2014, 70, 954. (38) Xu, X.; Pereira, L.; Wang, Y.; Wu, J.; Zhang, K.; Zhao, X.; Bae, S.; Bui, C. T.; Xie, R.; Thong, Hong, B. H.; Loh, K. P.; Donadio, D.; Li, B.; ozyilmaz, B. Length dependent thermal conductivity in suspended single-layer graphene. Nature, 2014, 5, 1. (39) Asheghi, Leung, Y. K.; Wong, S. S.; Goodson, K. E. Phonon-boundary scattering in thin silicon layers. Appl. Phys. Lett. 1997, 71, 1798. (40) Cahill, D. G.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. Nanoscale thermal transport. Appl. Phys. Rev. 2003, 93, 793. (41) Liu, W.; Asheghi, M. Phonon–boundary scattering in ultrathin single-crystal silicon layers. Appl. Phys. Lett. 2004, 84, 3819. (42) Eyring, H. Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Chem. Phys. 1936, 4, 283.



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(43) Franca, J. M. P.; Reis, F.; Vieira, S. I. C.; Lourenço, M. J. V.; Santos, F. J. V.; de castro, C. A. N.; Padua, A. A. H.; Thermophysical properties of ionic liquid dicyanamide (DCA) nanosystems, J Chem. Thermodyn. 2014, 79, 248. (44) Ferreira, A. G. M.; Simoes, P. N.; Ferreira, A. F.; Fonseca, M. A.; Oliveira, M. S. A.; Trino, A. S. M. Transport and thermal properties of quaternary phosphonium ionic liquids and IoNanofluids. J. Chem. Thermodyn. 2013, 64, 80. (45) Timofeeva, E. V.; Gavrilov, A. N.; McCloskey, J. M.; Tolmachev, Y. V.; Sprunt, S.; Lopatina, L. M.; Selinger, J. V.; Thermal conductivity and particle agglomeration in alumina nanofluids: Experiment and theory. Phys. Rev. E 2007, 76, 061203.

Abbreviations: [BBIM][NTf2]

1,3-dibutylimidazolium bis(trifluoromethylsulfonyl)imide

[BMIM][BF4]

1-butyl-3-methylimidazolium tetrafluoroborate

[BMIM][CF3SO3]

1-butyl-3-methylimidazolium trifluoromethanesulfonate

[BMIM][NTf2]

1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

[BMIM][PF6]

1-butyl-3-methylimidazolium hexafluorophosphate


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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 59 60


[(C6H13)3P(C14H29)]Cl

trihexyl(tetradecyl)phosphonium chloride

[C4mpyrr][(CF3SO2)2N]

1-butyl-1-methylpyrrolidinium bis{(trifluoromethyl)sulfonyl}imide

[C4mpyrr][NTf2]

1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide

[DMIM][NTf2]

1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

[DMPI]Im

1,2-dimethyl-3- propylimidazoliumbis(trifluorosulfonyl)imide

[EMIM][BF4]

1-ethyl-3-methylimidazolium tetrafluoroborate

[EMIM][C(CN)3]

1-ethyl-3-methylimidazolium tricyanomethide

[EMIM][Et(SO4)]

1-ethyl-3-methylimidazolium ethylsulfate

[EMIM][dca]

1-ethyl-3-methylimidazolium dicyanamide

[EMIM][NTf2]

1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

[EMIM][OAc]

1-ethyl-3-methylimidazolium acetate

[EMIM][OcSO4]

1-ethyl-3-methylimidazolium octylsulfate

[HMIM][BF4]

1-hexyl- 3-methylimidazolium tetrafluoroborate

[HMIM][NTf2]

1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

[HMIM][PF6]

1-hexyl-3-methylimidazolium hexafluorophosphate

[OMA][NTf2 ]

methyltrioctylammoniumbis(trifluoromethylsulfonyl)imide

[OMIM][NTf2]

1-octyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

[OMIM][PF6]

1-octyl-3-methylimidazolium hexafluorophosphate

Table 1: selected ILs for development of GMDH-PNN model. Temperature (K)

Pressure (kPa)

No. of data

Ref.

[EMIM][OAc]

273.15-353.15

101.32

9

8

[EMIM][dca]

273.15-353.15

101.32

9

8

[EMIM][C(CN)3]

273.15-353.15

101.32

9

8

Component IL



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Page 34 of 46

[EMIM][Et(SO4)]

273.15-353.15

101.32

16

8,9

[EMIM][OcSO4]

273.15-353.15

101.32

9

8

[BBIM][NTf2]

273.15-353.15

101.32

9

8

[OMA][NTf2]

273.15-353.15

101.32

9

8

[C4mpyrr][NTf2]

293-353

101.32

4

9

[(C6H13)3P(C14H29)]Cl

293-353

101.32

7

9

[HMIM][BF4]

293-353

100

7

10

[BMIM][CF3SO3]

293-353

100

14

9,10

[C4mpyrr][(CF3SO2)2N]

293-333

100

5

10

[BMIM][PF6]

293-353

100-20000

16

10,11

[HMIM][PF6]

293-353

100-20000

16

10,11

[OMIM][PF6]

293-333

100-20000

9

11

[EMIM][BF4]

298-388

101.32

6

7

[BMIM][BF4]

298-353

101.32

4

7

[DMPI]Im

273-373

101.32

5

7

[EMIM][NTf2]

273.15-353.15

101.32

16

8,9

[BMIM][NTf2]

293-353

101.32

7

9

[HMIM][NTf2]

273.15-353.15

101.32

16

8,9

[OMIM][NTf2]

293-353

101.32

7

9

[DMIM][NTf2]

293-353

101.32

7

9

Total

273-388

100-20000

216

Table 2. Nodal expressions for optimized GMDH-PNN.


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Layer 1 N 1 = 0.345585 + 7.7429e −7T b 2 − 1.32486e −16T b 5 − 0.000694805T c + 0.000319797 Pc 2 − 4.72555e −6 Pc 3

Layer 2 N 2 = −1.22482 − 1.39911e −7T 2 + 553.073N 12 − 6989.57 N 13 + 32944.8N 14 − 54621.2N 15

Layer 3 N 3 = −0.67169 + 0.00698842P − 7.4106e −7 P 2 + 4.91727e −20 P 5 − 9.44764e −9T c 2 + 0.896453N 2

Layer 4 N 4 = 0.0793127 + 3.65062e −5Tb − 8.52304e −12Tc 3 + 160.893N 14 − 850.361N 15 + 71.0263N 34

Output layer

λIL = 0.00695985 − 4.13528e −11Tb 3 + 1.81389e −11Tc 3 − 0.00167387Pc + 2.13897e −05Pc 2 + 1.10704N 4 Note: P: pressure (kPa), T: temperature (K), Tb: boiling temperature (K), Tc: critical temperature (K), Pc: critical pressure (MPa)

Table 3. Comparison of GMDH-PNN model with other available models in literature.

System [EMIM][OAc] [EMIM][dca]

GMDH-PNN 1.26 0.56

MARD% ANN Wu 0.29 0.40 -


Gardas -

Shojaee 6.54 5.34


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 59 60

[EMIM][C(CN)3] [EMIM][Et(SO4)] [EMIM][OcSO4] [BBIM][NTf2] [OMA][NTf2 ] [C4mpyrr][NTf2] [(C6H13)3P(C14H29)]Cl [HMIM][BF4] [BMIM][CF3SO3] [C4mpyrr][(CF3SO2)2N] [BMIM][PF6] [HMIM][PF6] [OMIM][PF6] [EMIM][BF4] [BMIM][BF4] [DMPI]Im [EMIM][NTf2] [BMIM][NTf2] [HMIM][NTf2] [OMIM][NTf2] [DMIM][NTf2] Average

0.88 1.44 2.53 6.76 0.99 0.44 1.12 1.43 1.56 0.91 0.90 1.60 0.88 1.41 1.01 3.52 3.90 3.41 1.85 1.28 1.48 1.79

0.31 1.27 0.65 0.59 0.61 3.35 0.15 0.25 1.78 1.98 0.33 1.09 0.23 0.37 0.29 0.33 3.74 0.53 1.72 0.93 0.41 0.94

0.74 1.20 0.67 4.50 0.77 1.20 1.73 1.55 1.51 3.02 5.42 4.08 0.65 1.92 3.01 5.16 2.32

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1.34 0.19 0.24 23.09 1.83 0.98 4.32 1.20 3.07 13.15 3.99 1.75 3.33 1.09 0.78 4.02

10.13 14.71 6.04 22.11 0.76 2.865 2.77 2.61 4.28 4.78 9.12 7.55 12.8 3.28 3.65 5.07 6.91

Table 4. Energy parameters and MARD% of Maxwell-MTSM model for Ionanofluids.

Ionanofluid

λ12I

λ12II

τ

MARD%

Ref.

[EMIM][dca]-MWCNT

0.00073

-0.09926

-0.18131

0.68

43

[BMIM][dca]-MWCNT

0.00034

-0.14648

-0.08648

0.59

43

[C4mpyr][dca]-MWCNT

0.00018

-0.12460

-0.02505

0.64

43


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[HMIM][BF4]- MWCNT

3.46158

0.03322

-882.59937

1.48

4

[HMIM][BF4]-Graphene

-0.00043

-0.20431

0.19912

0.59

4

[BMIM][NTf2]-Al2O3

0.00716

0.00103

-0.06232

0.18

13

[C4mpyrr][NTf2]-Al2O3

0.00568

0.00210

0.00283

0.16

13

Average

0.61

Table 5. Comparison between Maxwell and mGM models for prediction of thermal conductivity of Ionanofluids. MARD% Ionanofluid Volume fraction (%) Maxwell mGM [EMIM][dca]-MWCNT

0.3

3.30

2.31

0.6

6.54

2.93

0.29

6.27

4.38

0.57

8.83

5.19

0.27

5.39

3.61

0.55

7.92

4.35

0.018

5.45

5.03

0.037

14.12

12.15

0.018

5.14

5.03

0.037

12.36

12.14

[BMIM][NTf2]-Al2O3

0.36

5.36

4.45

[BMIM][mpyrr]-Al2O3

0.35

5.66

5.67

7.19

5.60

[BMIM][dca]-MWCNT

[C4mpyr][dca]-MWCNT [HMIM][BF4]- MWCNT [HMIM][BF4]-Graphene

Average



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Figure 1: Schematic of optimized ANN model with 9 neurons.


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Figure 2: plot of estimated data of GMDH-PNN vs. experimental thermal conductivity data for all of ILs.



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Figure 3: plot of estimated data of ANN vs. experimental thermal conductivity data for all of ILs.


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Figure 4: Comparison between predicted results of GMDH-PNN and experimental data of Franca et al.43 for [C4mpyr][dca] ionic liquid.



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Figure 5: Comparison between estimated results of GMDH-PNN and experimental data of Froba et al.8 and Franca et al.43 for [EMIM][dca] ionic liquid.


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Figure 6: Comparison between correlated results of Maxwell-MTSM and experimental data43 of Ionanofluids.



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Figure 7: Comparison between correlated results of Maxwell-MTSM and experimental data4 of Ionanofluids.


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Figure 8: Prediction of thermal conductivity of [C4mpyr][dca]-MWCNT system43 using Maxwell and mGM models.



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Graphical abstract:


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Modeling the Thermal Conductivity of Ionic Liquids ... - ACS Publications

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