Reference Values and Reference Correlations for the Thermal Conductivity and Viscosity of Fluidsa)
In this paper, reference values and reference correlations for the thermal conductivity and viscosity of pure fluids are reviewed. Reference values and correlations for the thermal conductivity and the viscosity of pure fluids provide thoroughly evaluated data or functional forms and serve to help calibrate instruments, validate or extend models, and underpin some commercial transactions or designs, among other purposes. The criteria employed for the selection of thermal conductivity and viscosity reference values are also discussed; such values, which have the lowest uncertainties currently achievable, are typically adopted and promulgated by international bodies. Similar criteria are employed in the selection of reference correlations, which cover a wide range of conditions, and are often characterized by low uncertainties in their ranges of definition.
Introduction<!>Thermal Conductivity<!><!>Thermal Conductivity<!>Reference values for the thermal conductivity of liquids<!>Reference values for the thermal conductivity of gases<!>Viscosity<!>Reference values for the viscosity of liquids<!>Reference values for the viscosity of gases<!>Reference Correlations<!><!>Reference correlations for the viscosity and thermal conductivity of molten metals<!>Reference correlations for the viscosity of high-viscosity liquids<!>Conclusions
<p>In this work, we review reference values and correlations for two important fluid transport properties: thermal conductivity and viscosity. Internationally accepted "reference values" (known also as "standard reference values") serve two primary purposes: first, they can provide a means of confirming the operation and experimental uncertainty of any new absolute apparatus and the stability and reproducibility of existing absolute measurement equipment. Second, in the case of instruments operating in a relative way, they provide the basis to calibrate one or more unknown constants in the working equation. Reference values refer to the properties specified at a fixed state condition (specific temperature, pressure and composition) or at a small number of such states. These values are often characterized by the lowest uncertainty possible at the time of their acceptance.</p><p>"Reference correlations" for pure-fluid transport properties often cover a wide range of conditions - typically from the triple-point temperature to 1000 K, and up to 100 MPa pressure - and are developed to achieve the lowest possible uncertainties (although perhaps higher than those of reference values). In between these two categories, there exist "restricted reference correlations" that refer to a limited range of conditions, often with lower uncertainty than wide-range reference correlations, and may be of specific industrial or scientific interest. When appropriate, the reference correlations or restricted reference correlations are constrained to agree with any reference values that may have been established for the fluid of interest.</p><p>The current paper emphasizes the work of three main bodies that remain active in the field of reference values and correlations for transport properties. The National Institute of Standards and Technology (NIST) in Boulder, CO, has been involved in the development of wide-range reference correlations for thermal conductivity and viscosity to extend the capabilities of the reference software they develop. The International Association for Transport Properties (IATP), formerly known as the Subcommittee on Transport Properties of IUPAC, has been proposing mostly reference values. Finally, we should also mention the International Association for the Properties of Water and Steam that since 1929 has been the body that proposes the reference correlations and values for the properties of water and steam, including transport properties. These three organizations often collaborate on both reference data and correlations for transport properties.</p><!><p>The thermal conductivity of a fluid, λ, has proven to be one of the most difficult thermophysical properties to measure accurately. It is important to recall that the thermal conductivity, λ(T, P), is the state-dependent proportionality constant in Fourier's Law relating heat flow to an infinitesimally small temperature gradient. It was not until 1951 that any proposal was made for standard reference values for this fluid property. At that time Riedel1 suggested that the thermal conductivity of liquid toluene (a liquid that can easily be obtained at high purity) be adopted as a reference value at 293.15 K and 0.1 MPa.</p><p>The inherent difficulty in the measurement of the thermal conductivity for both liquids and gases arises from the impossibility of decoupling the processes associated with heat transfer: conduction, convection, and radiation. The absence of a gravitational field (e.g., spacebased measurements) can mitigate convective heat flow, and radiative heat flow is generally less of a problem at low temperatures.</p><p>In 1986, in view of the rapid developments in the measurement of the thermal conductivity, primarily of fluids in the liquid phase, Nieto de Castro et al.2 proposed a complete reappraisal of reference values for thermal conductivity. The work was carried out under the auspices of the Subcommittee on Transport Properties (since 2001 known as the International Association for Transport Properties, IATP) of the International Union of Pure and Applied Chemistry, IUPAC.</p><!><p>Measurements must have been made with a primary experimental apparatus, i.e., an essentially complete working equation must be available.</p><p>The form of the working equation should be such that the sensitivity of thermal conductivity to the principal variables does not magnify the random errors of measurement.</p><p>All principal variables should be measureable to a high degree of precision.</p><p>The published work should include some description of purification methods for pure fluids and a validated assessment of purity (or an appropriate characterization of a mixture).</p><p>The data reported must be unsmoothed data. While graphs and fitted equations are useful summaries for the reader, they are not sufficient for standardization purposes.</p><p>Explicit quantitative estimates of the uncertainty of reported values should be given, based on the extant guidelines for the expression of uncertainty in measurements (GUM)3 and taking into account all sources of uncertainty of the experimental measurements including possible systematic components of uncertainty.</p><p>Owing to the desire to produce high accuracy reference values, limits are usually imposed on the two-sigma expanded relative uncertainty of the primary data sets; these are usually required to be better than 1.5%.</p><!><p>Only primary data are used, when adequate, to develop a reference correlation, or to develop recommended reference values. These recommendations for assessing literature data for transport properties have continued to guide subsequent work by IATP on both reference data and correlations.</p><!><p>Toluene and water have been proposed as thermalconductivity reference liquids. Nieto de Castro et al.2 recommended in 1986 the following reference values, which are still valid today:</p><p>for toluene, at 298.15 K and 0.1 MPa,
(1)λ = 0.1311 ± 0.0026 W m−1 K−1.
and for water, at 298.15 K and 0.1 MPa,
(2)λ = 0.6067 ± 0.0122 W m−1 K−1.</p><p>These expanded uncertainties were reported at the 95 % confidence level.</p><p>The temperature dependence of the thermal conductivity of liquid toluene at 0.1 MPa was represented2 by the following equations, still valid today, where T* = (T/298.15 K) , and λ*=(λ(T)/λ(298.15 K)) : temperature range 230 K ≤ T ≤ 360 K,
(3)λ* = 1.68182 − 0.682022 T*,
and extended range 189 K ≤ T ≤ 360 K,
(4)λ* = 1.45210 − 0.224229 T* − 0.225873 T*2.</p><p>Considering the uncertainty of the primary experimental data, the relative expanded uncertainty of the thermal conductivity from Eq. (3) is 2.2 % and from Eq. (4) is 2.6 %, at the 95 % confidence level.</p><p>In the case of water, the IAPWS recommendation of 20124 proposed the following equation as a function of temperature, valid at 0.1 MPa, over the temperature range of 273.15 K ≤ T ≤ 383.15 K, as
(5)λ(T) / 1mW m−1K−1=1.663Tr−1.15−1.7781Tr−3.4+1.1567Tr−6−0.432115Tr−7.6
where Tr = T/(300 K). The expanded relative uncertainty thermal conductivity from this equation was 1.5 % at the 95 % confidence level.</p><p>Other liquid thermal-conductivity reference values and limited correlations are of slightly higher uncertainty, e.g. n-heptane,2 and benzene.5</p><!><p>Significant progress has been made over the last two decades in establishing reference values for the thermal conductivity of the noble gases at low densities. This is principally a result of theoretical advances in the ab-initio determination of intermolecular pair potential energy surfaces for the noble gases, which have been made possible by rapid increases in computational power. Cencek et al.6 developed the most accurate pair potential energy surface to date and used it to calculate the thermosphysical properties of helium in the dilute-gas state over the temperature range from (1 to 104) K with uncertainties up to nearly two orders of magnitude smaller than those of the most accurate measurements. The calculations of Cencek et al.6 lead to the following reference value for the thermal conductivity of helium at 25 °C, 0.1 MPa, λHe = 0.155 000 8 ±0.000 001 5 Wm−1 K−1 . May et al.7, 8 have shown that for noble gases the ratio of the thermal conductivity to the viscosity can be calculated very accurately using a modern ab-initio pair potential function with classical kinetic theory. Thus, by combining reference experimental viscosity ratios with accurate (λ/η) ratios calculated using the potentials of Hellmann and coworkers,9–12 the ab-initio properties of helium can be leveraged to determine recommended values for the thermal conductivity of neon, argon, krypton, and xenon at 25 °C, 0.1 MPa, that we present in Table 1. In Table 1 the error bounds correspond to the expanded (k = 2) uncertainty at a 95% confidence interval.</p><p>Dilute-gas transport properties evaluated from abinitio pair potential functions are strictly evaluated in the limit of zero density. For comparisons with experiment, values obtained from theory should be adjusted to match the pressure at which the measurement was made using the initial density dependence of that transport property. This initial density dependence is temperature dependent and can be estimated using the Rainwater-Friend corresponding-states theory;13 such adjustments are negligible at 0.1 MPa for helium at 25 °C. The values of λNe, λAr, λKr, and λXe at 25 °C listed in Table 1 have already had this adjustment applied using the initial density dependence from Bich et al.,9 which amounts to approximately 0.04%, 0.23 %, 0.41 %, and 0.71 %, respectively.</p><p>Remarkably, the modern reference values for the noble gases at 25 °C and 0.1 MPa listed above differ from values for the thermal conductivity of noble gases (Table 2) recommended by Kestin et al.14 in 1980 by less than 0.42 %. This is essentially within the expanded relative uncertainty estimated by Kestin et al. 14 (0.6% in the range 25–200 °C, and 1% in the range 200 – 500 °C, at the 95% confidence level). We recommend using the values presented above in Table 1 for the thermal conductivity of gases at 298.15 K and 0.1 MPa. For higher temperatures, Table 2 can still be employed directly to obtain the thermal conductivity of the gas.</p><p>For higher pressures, up to 30 MPa and at a temperature of 27.5 °C, the thermal conductivity of argon gas is represented by the equation,14 (corrected for the current valid equation of state for argon15)
(6)λ = 17.743 + 21.440×10−3ρ + 28.321×10−6ρ2
where λ is measured in mW m−1 K−1 and ρ in kg m-3.</p><p>Wide-range reference correlations for these gases will yield slightly different values when evaluated at the fixed points of these reference values. At 25 °C and 0.1 MPa, the correlations in the NIST REFPROP version 10 database16 yield (0.15531 and 0.017745) W m−1 K−1 for helium17 and argon18, respectively. Compared to Table 1, we note differences of 0.2 % and 0.4 %, respectively, for these noble gases. While these comparisons indicate mutual agreement within the stated uncertainties, they serve to re-emphasize several points. Reference values are often more accurate than reference correlations, and reference values are preferred for calibrations when they are sufficient. The dates and sources associated with a specific reference value and reference correlation are important considerations when selecting the best source for application.</p><!><p>In this work, we only consider the dynamic Newtonian viscosity—the coefficient of the linear response to an infinitesimally small shear velocity.</p><!><p>The viscosity of water is one of the most important standards for viscometry, and the International Association for the Properties of Water and Steam (IAPWS) maintains current international consensus standards for the fluid. Consistent with the IAPWS reference correlation, ISO/TR 3666:199819 provides the internationally agreed standard value for the viscosity of liquid water at 20 °C and atmospheric pressure (0.101 325 MPa): the consensus reference value is
(7)η = 1.0016 mPa s</p><p>This value has an expanded relative uncertainty of 0.17 % (at the 95 % confidence level). The reference value is largely based on the experimental value of 1.0019 mPa s reported by Swindells et al.20 in 1952, which was also the basis of ISO/TR 3666:1977. The standard value given in Eq.(7) accounts for the difference between the ITS-48 and ITS-90 temperature scales.</p><p>The 2009 IAPWS reference equation21 for liquid water at 0.1 MPa from 253.15 K to 383.15 K is
(8)η / 1 μPa s=280.68Tr−1.9+511.45Tr−7.7+61.131Tr−19.6+0.45903Tr−40.0
with Tr= T/(300 K). The expanded relative uncertainty of this equation is 1.5 % at the 95 % confidence level. This reference correlation gives a value of 1.0016 mPa s at 20 °C and 0.1 MPa that is consistent with the ISO/TR 3666:199819 standard reference value.</p><p>In a recent paper,22 a reference correlation for toluene was proposed. In the specific range of 263 to 273 K and 0.1 MPa, the viscosity values proposed had an uncertainty of 0.7% (at the 95% confidence level). The following equation fits those values with an uncertainty of less then 0.1%.
(9)η / 1 μPa s = 47.53−304.968Tr + 75.285Tr2 − 838.095Tr3 + 353.120Tr4
where, in this case, Tr = T/Tc and Tc is the critical temperature of toluene22 (= 591.75 K).</p><!><p>In the case of the viscosity of gases, the theoretical advances discussed in Section 2.1.2 allow the calculation of helium's thermophysical properties at low densities and result in the most accurately known viscosity standard, with a relative uncertainty around 10−5 at ambient temperatures.6 By combining reference values for viscosity ratios derived from measurements with the ab- initio viscosity of helium, the viscosities of H2, CH4, Ne, N2, C2H6, Ar, C3H8, Kr, Xe, and SF6 can also be determined at 25 °C and 0.1 MPa (using the initial density dependencies reported by Berg and Moldover23) and are listed in Table 3.</p><p>Reference values for the viscosity of the noble gases at a pressure of 0.1 MPa and temperatures up to 500 °C recommended by Wakeham et al.24 are shown in Table 4. These values differ from the viscosity values listed for the noble gases at 25 °C and 0.1 MPa in Table 3, by less than 0.33 %, which is essentially consistent with the uncertainty estimates reported by Wakeham et al.24 (0.2 % in the range 25 – 200 °C, and 0.4 % in the range 200 – 500 °C). In principle, the uncertainty of these values could be reduced several fold by combining the reference values listed in Table 3 with the ratio of the gas' viscosity at temperature T to that at 25 °C calculated using an abinitio pair potential,9, 10 together with a small correction for the initial density dependence from Bich et al.11</p><p>Finally, we note that the NIST REFPROP16 program discussed in the next section uses a variety of individual wide-ranging correlations from different authors, developed at different times, to compute viscosity values, and does not necessarily reproduce the recommended values in Table 3 to within the stated uncertainty in Table 3. For instrument calibration, the values in Table 3 are preferable to those found in the correlations in REFPROP16, and one should always check to see if there are newer recommendations in place.</p><p>Gas viscosities at slightly higher pressures, with uncertainties comparable to those now achievable at 0.1 MPa could in principle be obtained using an adaptation of the two-capillary viscometer method as described by Berg et al.25 This proposed approach exploits helium's small viscosity virial coefficient, which means the value at pressure differs only marginally from the dilute-gas value calculated ab initio. At 20 °C, the viscosity of helium is 19.598 μPa s with an expanded relative uncertainty of 0.3 % at pressures to 10 MPa and an expanded relative uncertainty of 0.5 % at higher pressures up to 25 MPa.26</p><p>For now, more empirical correlations are needed for other gases at higher pressures: for example up to 30 MPa, and at a temperature of 25 °C, the viscosity of nitrogen can be employed to produce reference values by the equation24
(10)η=0.17763×10−4+0.86870×10−8ρ+0.14240×10−9ρ2
where η is measured in Pa s and ρ in kg m-3. Note that this equation does not yield the exact reference value given in Table 3.</p><!><p>Since wide-ranging reference correlations are often connected with the work on thermophysical properties at NIST, it is worthwhile presenting some historical information. The National Bureau of Standards (NBS) was founded in 1901, and in 1988 became the National Institute of Standards and Technology (NIST). NIST began producing and distributing tabulations of thermophysical properties early in its history, but the dissemination of computerized databases for thermophysical properties dates to the 1980's with the release of programs27 such as REFPROP (NIST23), DDMIX (NIST14), MIPROPS (NIST12), and Supertrapp (NIST4). The roots of such computer programs follow from earlier work on property tabulations and collaborations with NASA which included both thermodynamic and transport properties of hydrogen.</p><p>In 1989 NBS/NIST released the computer program REFPROP (REFrigerant PROPerties).16 Its scope was refrigerants, while the other NIST thermophysical properties program at the time covered hydrocarbon fluids and cryogens, in support of the NASA space programs, the natural gas sector, and the more general petrochemical industry. In 1991, the program included the calculation of transport properties (viscosity and thermal conductivity) with an extended corresponding states model. The usage of "reference equations or correlations" for transport properties appears to have arisen from the ability to use a correlation as a reference fluid in the context of a corresponding states model. In 2002, the acronym for REFPROP16 was changed to stand for REFerence fluid PROPerties since by that time the program contained more than just refrigerants and had expanded to include reference-quality equations of state and transport correlations for many industrial fluids such as constituents of natural gas and cryogens.</p><p>In Tables 5 and 6 we summarize wide-ranging correlations for viscosity and thermal conductivity, many of which have appeared in the Journal of Physical and Chemical Reference Data, and are also implemented in REFPROP. These correlations are typically formulated in terms of dilute-gas contribution that is a function only of temperature, a residual term that is a function of both temperature and density, and a critical enhancement term. The critical enhancement term is also a function of temperature and density; for viscosity correlations if often is ignored since it is significant only in a small region very near the critical point. For thermal conductivity correlations it typically is included since it is active over a wider region. Since these correlations are expressed in terms of temperature and density, a high-accuracy equation of state is typically used to provide the density for a given temperature and pressure. The publication containing the correlation should state what method is used to obtain density, and if an alternative method is used care should be taken to check that the results are not changed significantly. The viscosity at low temperatures near the triple point and the thermal conductivity near the critical region are especially sensitive to changes in density.</p><!><p>Viscosity and thermal conductivity of molten metals</p><p>Viscosity of high-viscosity liquids</p><!><p>Following the need for reference values of the viscosity and thermal conductivity of liquid metals identified over several years, a project was initiated by the International Association for Transport Properties, IATP, in 2006 to critically evaluate the density, the viscosity, and the thermal conductivity of selected liquid metals. Reference correlations developed based on critically evaluated experimental data so far, are shown in Tables 7 and 8.</p><!><p>For higher-viscosity liquids, a correlating equation for the representation of the viscosity of squalane as a function of temperature at atmospheric pressure has recently been proposed in conjunction with IATP.100, 101 Note that the measurements of Schmidt et al.102, published in 2015, were incorporated in the correlation of Mylona et al.101 as part of the unpublished data of M. Trusler. The viscosity of squalane covers a range of 0.5 to 140 mPa s. The temperature and pressure range covered is 273 K to 473 K with pressures to 200 MPa, and an uncertainty of 4.75% at the 95% confidence level.101</p><p>Very recently, a new reference correlation for the viscosity of Tris(2-ethylhexyl) trimellitate (TOTM), was proposed.103 The new correlation was designed to serve in industrial applications for the calibration of viscometers at elevated temperatures and pressures such as those encountered in the exploration of oil reservoirs and in lubrication. The correlation covers temperatures from (303 to 477) K, pressures from (0.1 to 200) MPa and viscosities from (1.6 to 760) mPa s. The uncertainty in the data provided is of the order of 3.2 % at a 95 % confidence level which was proposed by IATP as adequate for most industrial applications.</p><!><p>In this paper, we discussed and presented reference values and reference correlations for the thermal conductivity and viscosity of many important fluids. The criteria employed for the development of thermal conductivity and viscosity reference values and reference correlations were also discussed.</p><p>Although it seems that there exist reference correlations for many fluids covering a very wide range of conditions, a lot of work still needs to be done. In particular, consistency and consensus for reference quantities should be established. New measurements should concentrate in extreme conditions of temperatures and pressures and in fluids not covered in the tables presented in this work to meet modern industrial demands. For low-pressure gases, theoretical advances in the ability to calculate ab-initio pair potentials will become increasingly important for an increasingly wide range of substances. Theoretical progress is needed, however, to extend these results to higher pressures by considering, for example, the effects of three-body collisions on transport properties, and to establish and validate more general liquid-state predictive models. Similarly, theoretical advances are needed to extend the use of ab initio calculations of thermal conductivity beyond the noble gases and/or to higher densities.</p>