Fluid Phase Equilibria 515 (2020) 112573
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Thermal conductivity of ionic liquids under pressure Francisco Yebra , Jacobo Troncoso *, Luis Romaní Universidad de Vigo, Departamento de Física Aplicada, Edificio Manuel Martínez Risco, Campus As Lagoas, 32004, Ourense, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 December 2019 Received in revised form 28 February 2020 Accepted 19 March 2020 Available online 23 March 2020
Thermal conductivity of eight ionic liquids has been measured in the temperature and pressure intervals (288.15e313.15) K and (0.1e65) MPa, selected to have a common ion. Values within the interval (0.12 e0.2) W$m1$K1 are found for these compounds. Thermal conductivity decreases with temperature and increases with pressure for all ionic liquids. The experimental results are compared with literature data, obtaining satisfactory agreement. The thermal conductivity data are discussed by taking into account the chemical nature of the compounds. The Bridgman equation and a heuristic modification of the Enskog theory are applied to quantitatively analyze the experimental data. This theoretical analysis is extended to other compounds to get a wider picture and thus, elucidate the main factors that could affect the thermal conductivity of ionic liquids. Finally, the prediction capability of these two theoretical approaches is evaluated using the reported thermal conductivity data. © 2020 Elsevier B.V. All rights reserved.
Keywords: Thermal conductivity Ionic liquids Pressure
1. Introduction Heat transfer constitutes a very important topic both from a fundamental and applied point of view. Three mechanisms have been classically described for transmission of heat: convection, radiation, and conduction. This latter does not involve mass transfer nor radiative processes and it needs a medium to be carried out. The magnitude that quantifies the ability of such medium for heat conduction is the thermal conductivity l. This quantity is found to vary up to five magnitude orders for the known materials. Diamond is the one with the highest l, around 1000 W m1 K1. Metals show also very large thermal conductivity, most of them well above 10 W m1 K1; for instance copper show l values around 300 W m1 K1. Non-metallic compounds show quite smaller l, both in solid and liquid state, although solids usually present larger l values. Thermal conductivities in the interval (0.1e10) W$m1$K1 are found for these materials. Finally, gases show the lowest values; most of them below 0.05 W m1 K1. Ionic liquids are materials that have received great attention in last years. They are characterized by being ionic compounds in liquid state at low temperatures: they are often defined as those melting below 100 C. Their ionic nature makes them to present very interesting properties, perhaps the most notable being their very low vapour pressure, a fact that makes them potential
* Corresponding author. E-mail address:
[email protected] (J. Troncoso). https://doi.org/10.1016/j.fluid.2020.112573 0378-3812/© 2020 Elsevier B.V. All rights reserved.
environmental-friendly substitutes to traditional organic solvents. A lot of work has been carried out to determine their physical properties [1], and, as a result of this great effort, there is a huge amount of available data for a large set of these compounds. However, although the thermophysical characterization is quite complete for some properties, as for instance, density, there is a lack of knowledge for other very important physical properties for such compounds. Thermal conductivity is among these lessstudied physical quantities. At atmospheric pressure, and around room temperature, l was experimentally determined for some ionic liquids [2e7], but temperature and, especially, pressure behaviour is not well known; in fact, it was measured against pressure (up to 20 MPa) only for eight ionic liquids [8e11]. Besides the interest that the knowledge of thermal conductivity presents, this quantity gives important information about microscopic mechanisms underlying in heat conduction. This quantity has been related with molecular-level parameters as lattice constants or mean free path, of great interest for microscopic characterization of the material. In the frame of the liquid state, a thermal conductivity first-principles theory was only successfully applied to very simple compounds as argon, nitrogen, or carbon dioxide [12]. As a result, semi empirical theories are often used for understanding and predicting thermal conductivity of liquids. In this work, thermal conductivity within the temperature and pressure intervals (283.15e313.15) K and (0.1e65) MPa is reported for eight ionic liquids with a common cation or anion. 1-Butyl-3methylimidazolium and bis-(trifluoromethylsulfonyl)imide are
2
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
selected as common ions. Experimental results are compared with available literature data. Two semi empirical theories are used for quantitatively analyzing the experimental data. This analysis is extended to molecular compounds, measured in a previous work [13], to check whether ionic liquids show significant differences in the frame of these theories. Finally, two prediction methodologies, based on these theories, are proposed and evaluated using the reported experimental data.
2. Experimental Table 1 shows the used chemicals, short name and purity. Ionic liquids were heated at 333 K under low pressure (1 Pa) for 48 h before thermal conductivity measurements, to remove volatile compounds. Water content for all chemicals was found to be below 0.0005 in mass fraction. Density and refractive index have been measured and compared with available literature data for the studied ionic liquids. A DMA 5000 densimeter from Anton Paar and a Mettler Toledo RE50 refractometer were used to this aim. Experimental setup is described in detail in Ref. [13]. The liquid under study is placed inside a cylindrical cell, which is attached through a connector to the pressure controller. It is formed by a pressure generator (model 50-6-15 from HiP manufacturer) driven by a DC-motor. The cell is immersed in a thermostatic bath (Lauda ProLine RP845) for controlling its temperature. The experimental procedure is based on the transient hot wire technique [14,15]. A thin platinum wire (Sigma-Aldrich, purity>0.999, 50 mm diameter, 70 mm length) is placed in the cavity disposed along the central axis, welded in one end to a stainless steel wire which works as support structure and electrical connection. The other end is soldered to two copper 0.2 mm diameter wires that are used to insert the current efixed to 107 mA for all experimentse and measure the voltage drop. For each thermal conductivity measurement, around 110 voltage data per second during 4e5 s are acquired. The transient hot wire technique is based on the transitory heating of an infinite one-dimensional wire immersed in an infinite three dimensional medium. The temperature increase is proportional to the measured voltage, which follows a logarithmic law against time t, which can be expressed as:
DV ¼ mLnðtÞ þ n
(1)
calibration of the system with a liquid with known thermal conductivity; n-hexane was used to this aim [16]. The thermal conductivity l for the unknown liquid is calculated from:
l ¼ lh
AðT; pÞ m
3. Results and discussion 3.1. Experimental results Table 2 shows density and refractive index data; good agreement with literature is found for all ionic liquids. Experimental thermal conductivity data are reported in Tables S1eS8 of the Supplementary Material. Thermal conductivity l was fitted against temperature T and pressure p to the next equation:
l ¼ a0 þ a1 T þ b1 p þ b2 p2
(4)
Fitting coefficients ai and bi and standard deviations of the fits s are given in Table 3. Figs. 1 and 2 show the experimental results as well as curve fits for all studied liquids except [Bpyr][NTf2]. Thermal conductivity for this ionic liquid could be determined only at few temperatures and pressures, some of them in the metastable liquid phase. This compound freezes above ambient temperature (the supplier reports a melting temperature at atmospheric pressure around 298 K), and therefore, it can crystallize, making impossible transmitting the pressure to the sample.
Table 2 Density and refractive index of the liquids of this work (r and nD) at 298.15 K and p ¼ 0.1 MPa compared with literature data (rRef and nD.Ref).a.
(2)
where A(T,p) is a function that does not depend on the liquid that fills the cell. Therefore, thermal conductivity can be obtained by
(3)
where lh denotes hexane thermal conductivity, and mh and m are the parameters of Eq. (1) obtained in the calibration and measurement experiments, respectively. Three experiments were performed for each experimental point; the reported thermal conductivity is the mean value of these three replicates. The uncertainty of the method is around 4e5%, rigorously estimated in a previous work [13] following the procedures recommended in Ref. [17].
where DV is the voltage variation, and m and n are parameters that depend on physical properties of the cell and the medium in which the wire is immersed. The thermal conductivity is directly related to m:
l¼
mh m
Liquid
r/g$cm3
rRef/g$cm3
nD
nD.Ref
[Bmim][MeSO4] [Bmim][TfO] [Bmim][SbF6] [Bmim][N(CN)2] [S222][NTf2] [Bmpip][NTf2] [Bmpyrr][NTf2] [Bpyr][NTf2]
1.2084 1.2978 1.6839 1.0617 1.4535 1.3796 1.3949 1.4490
1.2074 [18] 1.2976 [18] 1.690b [21] 1.05858 [23] 1.46118 [24] 1.3808 [26] 1.3947 [27] 1.4487 [29]
1.4798 1.4377 1.4165 1.5094 1.4255 1.4295 1.4229 1.4432
1.4794 [19] 1.43756 [20] 1.4165 [22] 1.5089 [20] 1.42632 [25] 1.42928 [26] 1.42302 [28] 1.44379 [30]
a Standard uncertainties are: u(T) ¼ 0.005 K, u(r) ¼ 0.001 g cm3 and u(nD) ¼ 0.001. b Measured at 297 K.
Table 1 Supplier mass fraction purity (w), source and purification method of the samples. Chemical name
Abbreviation
w
Origin
Purification method
Hexane 1-butyl-3-methylimidazolium methylsulfate 1-butyl-3-methylimidazolium trifluoromethanesulfonate 1-butyl-3-methylimidazolium hexafluoroantimonate 1-butyl-3-methylimidazolium dicyanamide Triethylsulfonium bis(trifluormethylsulfonyl)imide 1-butyl-1-methyl-piperidinium bis(trifluormethylsulfonyl)imide 1-butyl-1-methyl-pyrrolidinium bis(trifluormethylsulfonyl)imide 1-butylpyridinium bis(trifluormethylsulfonyl)imide
C6H14 [Bmim][MeSO4] [Bmim][TfO] [Bmim][SbF6] [Bmim][N(CN)2] [S222][NTf2] [Bmpip][NTf2] [Bmpyrr][NTf2] [Bpyr][NTf2]
>0.99 >0.99 >0.99 >0.99 >0.98 >0.99 >0.99 >0.99 >0.99
Carlo Erba Solvent Innovation Solvent Innovation Solvent Innovation Io-li-tec Io-li-tec Io-li-tec Io-li-tec Io-li-tec
none Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum Vacuum
> > > > > > > >
48 48 48 48 48 48 48 48
h h h h h h h h
at at at at at at at at
333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15
K K K K K K K K
and and and and and and and and
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Pa Pa Pa Pa Pa Pa Pa Pa
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
3
Table 3 Fitting parameters of Eq. (4) and standard deviation s.
a0/W$m1$K1 a1/W$m1$K2 b1/W$m1$K1$MPa1 b2/W$m1$K1$MPa2 s/W$m1$K1
a0/W$m1$K1 a1/W$m1$K2 b1/W$m1$K1$MPa1 b2/W$m1$K1$MPa2 s/W$m1$K1
[Bmim][MeSO4]
[Bmim][TfO]
[Bmim][SbF6]
[Bmim][N(CN)2]
0.290760 4.0415$104 4.1800$104 2.6335$106 0.0018
0.202567 2.0758$104 3.6303$104 2.1603$106 0.0012
0.207965 2.4537$104 3.1967$104 1.7807$106 0.0010
0.299712 4.3300$104 5.6780$104 4.1710$106 0.0010
[S222][NTf2]
[Bmpip][NTf2]
[Bmpyrr][NTf2]
0.152298 1.2091$104 3.3693$104 2.1339$106 0.0009
0.154884 1.5546$104 2.9731$104 1.5927$106 0.0011
0.145106 1.1499$104 3.5287$104 2.1309$106 0.0014
Values in the (0.1e0.2) W$m1$K1 interval were obtained for all ionic liquids over the whole temperature and pressure range. Thermal conductivity is much more strongly influenced by the anion than by the cation. Ionic liquids with the NTf-2 anion shows quite similar l values, all around 0.12 W m1 K1 whereas changing the anion has an important effect over l, changing from around 0.18 for [Bmim][N(CN)2] to 0.14 W m1 K1 for [Bmim][SbF6]. Thermal conductivity increases with pressure and decreases with temperature, as usual for most liquids. Moreover, by simple inspection of the coefficients given in Table 3, one can conclude that temperature and pressure dependences are quite similar to those of molecular compounds [13]. Ionic liquids with larger thermal conductivity also show stronger temperature and pressure dependencies. This behaviour has been also found for other liquids [13]. Thermal conductivity at atmospheric pressure was previously determined for [Bmim][TfO], [Bmim][N(CN)2], [Bmpyrr][NTf2], and [Bmim][MeSO4] [2e6], which allows comparison with literature to be carried on. To quantify it, we have used the relative deviation D:
l lmeas D ¼ 100$ lit llit
(5)
where llit and lmeas are the literature and measured thermal conductivity, respectively. The results of this comparison show mean relative deviation values of: 1.2% for [Bmim][N(CN)2] from Ref. [4], 1.2% and 4.6% for [Bmim][TfO] from Refs. [2,3], 7.9%, 12.9% and 4.8% for [BmPyrr][NTf2] from Refs. [2,3,5] respectively, and, for [Bmim] [MeSO4], 4.4% from Ref. [6], similar or below experimental uncertainty for most cases. Fig. 3 shows the obtained relative deviations against temperature. Error bars correspond to the uncertainty of the difference between literature and measured values; they have been estimated as the quadratic combination of the declared uncertainties for each reference and those of the present work. The main discrepancies are found for [Bmimpyrr][NTf2]; which shows not only large differences with our measurements but also between literature data. Comparison at the highest temperature for [Bmim] [MeSO4] also yields a quite poor agreement, around 10%. Different impurity levels of the ionic liquid samples could be the origin of the observed discrepancies; it is a well-established fact that impurities have a very strong effect over their physical properties. Thermal conductivity against pressure has been previously reported for other ionic liquids. Tomida and coworkers [8e11] measured l for a set of alkyl-imidazolium and alkyl-pyridinium ionic liquids with the [BF4] and [PF6] anions in the temperature and pressure intervals (294.3e334.3) K and (0.1e20) MPa. Their reported thermal conductivity data were between 0.14 and 0.17 W m1 K1, quite similar to those obtained in this work. They also observed the stronger effect of changing the anion: the systems with the BF4 anion show l values all around 0.165 W m1 K1, whereas for PF6-based ones, l
decreases to around 0.146; the change in the cation has a quite milder effect. Moreover, they obtained temperature and pressure dependencies that follows the observed tendencies for the liquids of the present study. 3.2. Theoretical analysis As above said, first-principles theories for thermal conductivity can be only applied for the simplest fluids; obviously ionic liquids are not simple, but quite complex liquids. Therefore, the semi empirical approximation is the only way to achieve some clues about the main microscopic factors that affect thermal conductivity. Two of these models are applied to the experimental data: Bridgman and a heuristic modification of Enskog theory (HMET). Both models need density as input parameter at the same pressures and temperatures as thermal conductivity. Unfortunately, density in the covered p, T interval is not available for the studied ionic liquids. Therefore, we had to estimate it using an approximate method, as shown below. 3.2.1. Density estimation Since density as a function of pressure is only available for [Bmim][MeSO4] [31] and [Bmim][TfO] [32], and in a rather narrow pressure interval, high pressure density data has been estimated using an iterative algorithm, based on the isothermal compressibility kT, calculated through Mayer equation:
kT ¼ ks þ
Tva2p Cp
(6)
being ap the isobaric thermal expansivity, Cp the isobaric molar heat capacity, v the molar volume and kS the isentropic compressibility, obtained from the Laplace equation:
ks ¼
1
(7)
rc2
where r denotes density and c speed of sound. For calculating ks, speed of sound was taken from Ref. [22], and density was obtained from the previous algorithm step. ap was calculated from density data [21,22,31,32,34,35] as a function of temperature using its formal definition:
ap ¼
1 vr r vT p
(8)
whereas isobaric molar heat capacity, Cp, was taken from literature [18,33,34]. For [Bmim][SbF6] and [S222][NTf2], Cp is not available in literature. It is a well-known fact the heat capacity per unit mass for
4
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
per unit mass [36,37]. Therefore, we have estimated Cp for [Bmim] [SbF6] assuming heat capacity per unit of mass to be the same as that of [Bmim][PF6], since their molecular structure is quite similar. For [S222][NTf2], we have taken the mean value of cp for the systems with the same anion, i.e. [Bmpip] [NTf2] and [Bmpyrr][NTf2]. From the isothermal compressibility calculated using Eq. (6), density at high pressure is estimated from:
rðT; pi Þ ¼ rðT; pi1 ÞekT ðpi pi1 Þ yrðT; pi1 Þð1 þ kT ðpi pi1 ÞÞ
(9)
where i denotes the i-th algorithm step. Once r is calculated at the selected pressure, the new value was inserted in Eq. (7) and the next step of the algorithm was carried out. This procedure was applied for liquids with well-known density behaviour against pressure and temperature and maximum differences around 0.3% were obtained, which is an important deviation considering the precision that can be achieved in density measurements. However, it is well below the requested precision, taking into account both the uncertainty of the thermal conductivity data and accuracy of the used models. 3.2.2. Bridgman theory The Bridgman equation [38] establishes a link between thermal conductivity, density, and speed of sound. It was originally developed by supposing the heat to travel at the speed of sound through the liquid layers. It can be expressed as:
l ¼ KkB
NA v
2=3 c
(10)
where l is the thermal conductivity, NA the Avogadro's number, kB the Boltzmann's constant, v the molar volume and c the speed of sound. K is a dimensionless constant that measures the total transferred heat, and it is temperature and pressure independent. It was fixed to values around 3 for most liquids [39e41]. We have applied this equation to the studied ionic liquids. Molar volumes were calculated from density data, estimated in the previous section, using:
v¼
M
(11)
r
where M denotes molar mass. The speed of sound data as a function of temperature and pressure were obtained from Ref. [22]. We have found that a value around 3 for the K constant of Eq. (10) does not correctly reproduce the thermal conductivity. Therefore, it was fitted to the experimental data. Fig. 4 shows the measured values for three ionic liquids as well as theoretical curves for each temperature as a function of pressure, whereas Table 4 lists the obtained values and the mean relative absolute deviations d, obtained from the relative deviation d through:
d ¼ 100 Fig. 1. Experimental data against temperature and pressure for (a) [S222][NTf2], (b) [BmPyrr][NTf2], (c) [BmPip][NTf2], and: Black d: fitted curves using Eq. (4) for 283.15 K, 293.15 K, 303.15 K and 313.15 K. Experimental values: Blue 283.15 K; Green 293.15 K; Red 303.15 K; Purple 313.15 K. Error bars represent the standard uncertainty of each experimental point.
ionic liquids, cp ¼ Cp/M, being M the molar mass, is very similar for those with similar molecular structure [7]. Moreover, it has been found that the anion chemical nature plays a major role in cp; systems with a common anion show quite similar heat capacities
d¼
lexp lcalc lexp
n 1 X jd i j n i¼1
(12)
(13)
where lexp and lcalc are the experimental and calculated thermal conductivity. With the aim of comparison with other compounds, this methodology was also applied to a set of molecular liquids with assorted chemical nature, determined in a previous work [13], being the experimental data needed for their K estimation obtained from Refs. [13,42e52]. These results are also shown in Table 4. Bridgman equation correctly fits the thermal conductivity data for
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
5
Fig. 3. Relative differences (in %) of measured conductivity data in comparison with references: Blue △ [2], Green △ [3] and Red △ [5] for [BmPyrr][NTf2]; Blue D [2] and Green D [3] for [Bmim][TfO]; Orange D [4] for [Bmim][N(CN)2]; Magenta > [6] for [Bmim][MeSO4]. The error bars represent the combined uncertainties of this work and literature data.
ionic liquids, but it shows a quite worse performance for most nonionic ones. The K values for ionic liquids are large, going even up to almost 4, whereas for most molecular compounds K is well under 3; only alkanes present K near 3.
3.2.3. Heuristic modification of the Enskog theory (HMET) In several papers [53e55], an Enskog theory variation has been proposed to fit thermal conductivity data with high accuracy. A scaled thermal conductivity l* is defined as:
l* ¼
l r0 2=3
(14)
r
l0
where l0 is the thermal conductivity in the zero density limit, r0 is the closed packed density of the fluid, and l and r are the thermal conductivity and density, respectively. Applying Enskog theory to the van der Waals model, it is possible to conclude that l* is a function only of the ratio r/r0, this is:
l* ¼ f
r r0
(15)
Correlating conductivity against density for several fluids, this general relation was confirmed [54,55]. Indeed, a general law that relates l* versus r/r0 was found, given by: Fig. 2. Experimental data against temperature and pressure (a) [Bmim][N(CN)2], (b) [Bmim][SbF6], (c) [Bmim][TfO], (d) [Bmim][MeSO4] and: Black d: fitted curves using Eq. (4) for 283.15 K, 293.15 K, 303.15 K and 313.15 K. Experimental values: Blue 283.15 K; Green 293.15 K; Red 303.15 K; Purple 313.15 K. Error bars represent the standard uncertainty of each experimental point.
r Lnðl* Þ ¼ a þ bLn
r0
(16)
where a and b are fitting coefficients. It must be noted that the zerodensity thermal conductivity l0 depends only on temperature, following the potential law:
6
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
Fig. 4. Experimental data against temperature and pressure for (a) [Bmim][N(CN)2], (b) [Bmim][SbF6], (c) [S222][NTf2], and comparison with predictions from Bridgman equation (10) (left), and from HMET equation (18) (right). Blued 283.15 K, Green d 293.15 K, Red d 303.15 K, Purple d 313.15 K. Experimental values: Blue 283.15 K; Green 293.15 K; Red 303.15 K; Purple 313.15 K. Error bars represent the standard uncertainty of each experimental point.
l0 ¼ K0 T 1=2
(17)
where K0 is a constant. Substituting (16) and (17) into Eq (14), a general relation for the thermal conductivity behaviour against density and temperature is obtained:
l ¼ A þ BLn
r
$ kg$m3
T þ 0:5Ln K
(18)
where A and B are the parameters of the model, which characterize the p, T behaviour of the liquid.
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573 Table 4 Mean relative deviations d, defined in Eq. (13) for the fits to experimental data for the Bridgman and HMET theories [Eqs. (10) and (18)], as well as theory parameters. Substance
Bridgman
HMET
d/%
K
d/%
A
B
[Bmim][MeSO4] [Bmim][TfO] [Bmim][N(CN)2] [Bmim][SbF6] [BmPyrro][Ntf2] [BmPip][Ntf2] [S222][Ntf2]
1.13 1.47 1.08 1.10 1.87 2.44 2.00
3.63 3.66 3.38 3.89 3.91 3.83 3.91
1.27 0.93 1.18 1.07 0.99 0.96 0.77
6.40 4.47 6.17 4.60 4.46 4.54 3.99
50.02 36.84 47.63 39.04 37.35 37.92 34.09
Methanol Ethanol 1-Propanol Toluene 1-Nitropropane Heptane Octane 1,3-Dichloropropane 1,2-Dichloropropane Propionitrile Propionic Acid 1-Propylamine
3.85 2.93 3.23 2.20 2.35 3.79 3.36 1.84 2.73 2.92 2.62 4.37
1.94 2.04 2.16 2.18 2.30 2.85 3.04 2.11 2.06 2.21 2.15 2.45
0.99 0.93 0.79 0.66 0.92 1.01 0.94 0.72 0.70 1.00 0.70 0.75
2.80 3.34 3.50 3.54 3.30 3.22 3.41 3.59 3.12 2.97 3.62 2.44
23.16 26.94 28.14 28.86 27.53 25.94 27.30 30.29 27.04 24.39 29.75 20.67
7
The HMET theory was applied to the studied ionic liquids, by fitting Eq. (18) to temperature and density. Fig. 4 shows the comparison between experimental and theoretical values for HMET. The A and B parameters are given in Table 4. HMET fits very well the experimental data, clearly better than the Bridgman equation. For the sake of comparison, this methodology was also applied to other fluids as Table 4 shows. Experimental data for carrying on this comparison were obtained from Refs. [13,44e52]. For ionic liquids, A is within the interval (3.99e6.40) W$m1$K1 and B within (50.02, 34.09) W$m1$K1, whereas for molecular compounds A is within (2.44e3.62) W$m1$K1 and B within (23.16, 30.29) W$m1$K1 intervals. Therefore, a clear difference is observed between the A and B parameters for ionic liquids compared with those for molecular compounds.
3.2.4. Predictive capability of Bridgman and HMET theories Determining l at high pressure is a quite complex and difficult task. Therefore, it would be interesting to have a reliable equation that would correctly predict this quantity basing on quantities easier to be experimentally determined. The Bridgman and HMET theories have only one and two adjustable parameters, respectively, for fitting thermal conductivity against temperature and
Table 5 Predictive capabilities of Bridgman and HMET models. d/% as defined in Eq (12). HMET [Bmim][MeSO4] p/MPa 10 30 60
d/% [Bmim][TfO] p/MPa 10 30 60
d/% [Bmim][N(CN)2] p/MPa 10 30 60
d/% [Bmim][SbF6] p/MPa 10 30 60
d/% [BmPyrro][NTf2] p/MPa 10 30 60
d/% [BmPip][NTf2] p/MPa 10 30 60
d/% [S222][NTf2] p/MPa 10 30 60
d/%
Bridgman
T/K 283.15 0.80 5.76 10.53 3.56
293.15 0.78 3.53 7.69
303.15 1.52 0.85 6.17
313.15 1.20 0.78 5.58
T/K 283.15 2.86 0.12 1.32 1.17
293.15 1.27 0.70 0.12
303.15 1.76 1.58 0.33
313.15 0.48 1.39 1.45
T/K 283.15 1.11 1.16 2.04 1.29
293.15 0.58 0.23 1.66
303.15 0.13 0.62 2.50
313.15 0.11 1.28 3.15
T/K 283.15 0.39 2.22 3.18 1.52
293.15 0.26 0.41 2.37
303.15 0.09 0.28 2.91
313.15 0.28 1.37 3.20
T/K 283.15 0.51 0.48 3.33 1.56
293.15 0.61 0.23 3.45
303.15 0.74 1.08 3.52
313.15 0.85 1.91 3.73
T/K 283.15 2.67 3.16 2.65 1.82
293.15 1.89 3.03 1.78
303.15 1.08 2.99 0.90
313.15 0.18 2.92 0.07
T/K 283.15 1.23 0.55 2.93 1.51
293.15 0.72 0.28 2.50
303.15 0.11 0.05 2.97
313.15 0.56 1.67 4.91
T/K 283.15 0.71 0.86 2.79 1.11
293.15 0.68 0.05 1.69
303.15 0.43 0.70 1.62
313.15 0.05 0.62 2.97
T/K 283.15 0.29 2.10 6.00 2.37
293.15 0.66 1.64 4.09
303.15 1.74 1.13 5.29
313.15 3.16 1.39 1.71
293.15 0.88 1.01 2.48
303.15 1.86 0.54 3.58
313.15 3.11 1.91 0.08
T/K 283.15 2.51 6.94 10.52 4.51
293.15 2.01 4.29 9.53
303.15 0.35 1.95 8.50
313.15 1.01 1.64 6.43
T/K 283.15 0.94 4.31 7.38 2.93
293.15 0.97 2.09 6.57
303.15 0.73 0.29 5.91
313.15 0.66 0.51 4.18
T/K 283.15 1.72 0.25 1.20 1.00
293.15 1.49 0.86 0.98
303.15 0.95 1.70 0.82
313.15 0.60 0.93 2.11
T/K 283.15 0.38 3.51 5.67 2.26
293.15 0.36 1.38 4.40
303.15 0.69 0.34 3.33
313.15 1.15 0.38 3.72
283.15 0.50 1.55 4.57 1.82
8
F. Yebra et al. / Fluid Phase Equilibria 515 (2020) 112573
pressure. Moreover, they make use of density and speed of sound, quantities easy to be experimentally determined. Therefore, they could be useful tools for thermal conductivity prediction. We have fitted the values of the parameters K (for Bridgman) and A and B (for HMET) using only the thermal conductivity, speed of sound and density data at atmospheric pressure, to evaluate the predictive capabilities of both equations. These values are thus used for predicting the behaviour against temperature and pressure over the whole experimental range. In Table 5, relative deviations at selected temperatures and pressures are presented, as well as the mean relative deviations of the predictions. Bridgman predictions are better than HMET ones for most cases. This is quite surprising, since HMET fits better the experimental data if the whole experimental p, T range is considered (cfr. Table 4). Moreover, HMET needs one more parameter than Bridgman theory. Therefore, the Bridgman equation, in spite of its simplicity, seems to be a very useful tool for thermal conductivity prediction of ionic liquids. 4. Conclusion Thermal conductivity has been determined for eight ionic liquids as a function of pressure and temperature using a transient hot wire technique. The results at atmospheric pressure have been compared with the available literature data, obtaining satisfactory results. They have been also compared with the literature data for ionic liquids at high pressure, observing similar tendencies. The Bridgman and a heuristic modification of the Enskog theories have been applied to the experimental data. The adjustable parameters of both theories have been compared with those of molecular compounds. Important differences have been obtained, showing that the molecular mechanisms underlying thermal conduction for ionic liquids could differ significantly from those of molecular compounds. Finally, a methodology based on these theories has been developed for predicting the thermal conductivity at high pressure, once thermal conductivity at atmospheric pressure is known. Bridgman equation has shown, in spite of its simplicity, the best predictive performance. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Francisco Yebra: Methodology, Software, Investigation, Data curation, Writing - original draft. Jacobo Troncoso: Conceptualization, Writing - review & editing, Formal analysis. Luis Romaní: Supervision, Writing - review & editing. Acknowledgements This work was supported by Ministerio de Ciencia, Tecnología y Universidades under the Grant FIS2017-89361-C3-3-P. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2020.112573. References [1] P. Wasserscheid, T. Welton (Eds.), Ionic Liquids in Synthesis, second ed., Wiley-VCH Verlags GmbH & Co. KgaA, Weinheim, 2008.
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