1 Introduction

We appreciate the collaboration with Professor Span, particularly when we worked together on the viscosity correlation for ethane [1]. Professor Span's group performed measurements and modeling of the viscosity of nitrogen-carbon dioxide [2] and methane-ethane mixtures [3] which we used to derive diffusion coefficients. Previous measurements of the viscosity \(\eta\) for two samples of natural gas with different compositions, carried out by Schley et al. [4] using a vibrating-wire viscometer with freely suspended weight and additional measurements of temperature T and pressure p for the calculation of the required density \(\rho\) applying the International Standard ISO 12213 [5], have been re-evaluated. The re-evaluation concerns the determination of the wire radius by means of an improved calibration and the density calculation. In this work, the density is calculated using the equation of state by Kunz and Wagner [6].

During and since the investigation by Schley et al. [4], viscosity measurements on natural gases or on mixtures with at least three components of natural gases were performed by Nabizadeh and Mayinger [7], Assael et al. [8], Langelandsvik et al. [9], Atilhan et al. [10], Kashefi et al. [11], Jarrahian et al. [12], Nazeri et al. [13], and Al Ghafri et al. [14,15,16]. The re-evaluated data of this work are reported as \(\eta \rho pT\) values and can be used together with the other data to update the database for natural gases. Moreover, mixture models using the current database for natural gases were discussed in the literature, e.g., by Chichester and Huber [17], by Heidaryan and Jarrahian [18], by Yang et al. [19], by Theran-Becerra et al. [20], or by Xiong et al. [21], and should enable to generate a prospective viscosity correlation for natural gases.

2 Re-evaluation of the Data

The re-calibration of the applied vibrating-wire viscometer was performed such that the radius of the wire was newly determined using previous measurements on argon [22]. The original calibration employed an experimentally based reference value of Kestin and Leidenfrost [23] nowadays considered as obsolete. For the re-calibration, we applied a today accepted value for the zero-density viscosity coefficient of argon, derived by Vogel et al. [24] from an ab initio potential on the basis of the kinetic theory of dilute gases and upgraded by Hellmann [25], to be η0,Ar,298.15K = 22.5534 μPa·s with a standard uncertainty of 0.07 %. The wire radius amounts to \(R=12.7548~\upmu \textrm{m}\) using the new reference value for argon.

The results reported in Tables III and IV of the previous paper of Schley et al. [4] were restricted to \(\eta \rho p\) triples along the measured isotherms. In this work, we include more details in order to make the information comparable to recent viscosity measurements. The individual points were not exactly measured at the nominal temperature of an isotherm \(T_{{\textrm{nom}}}\), but were kept within small deviations from the nominal temperature each. The experimental viscosity data were adjusted to \(\eta _{T_{{\textrm{nom}}}}\) values at the nominal temperature employing a Taylor series expansion restricted to the first power in temperature. For this, the experimentally determined value of the temperature coefficient \(\left( \partial \eta /\partial T\right) _\rho = (0.020~\text{ to }~0.038)\) μPa·s·K−1 calculated from the fit of a double polynomial in temperature and in density to the re-evaluated experimental viscosity data for natural gas was used. In addition, it is supposed that the density values \(\rho _{{\textrm{eos}}}(T,p)\) computed from the measured data for p and T applying the equation of state by Kunz and Wagner [6] via their implementation in REFPROP (Lemmon et al. [26]) and those for the isotherms are the same. As a result of this, the pressures \(p(T_{\textrm{nom}},\rho _{\textrm{eos}})\) at the nominal temperature changed and were recalculated from the densities. Furthermore, it has to be stated that the original composition of the two samples natural gas H and L, given by Schley et al. in Table I of their paper including the uncertainties of the underlying gas analysis, were used in this work, but with some combinations of components to yield 20 components as input parameters for the mixture in REFPROP. Table 1 shows the mole fractions of the components of the two natural gas samples H and L that means the composition stated by Schley et al. and that used for the re-evaluation of the density and viscosity data in this work.

Table 1 Components and their mole fractions of the natural gas samples H and L given by Schley et al. [4] and used for the re-evaluated viscosity measurements of this work
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Schley et al. [4] summed up the mole fractions of oxygen and argon, whereas Table 1 reveals that we split the mole fractions in two parts with equal values and treated separately these components. To meet the requirement of REFPROP (Lemmon et al. [26]) to limit the components to the maximum number of 20, we added the mole fractions of the xylenes (0.0006 mol% for natural gas H and 0.0010 mol% for natural gas L) to that of toluene (an additional 0.0001 mol% was added to the mole fraction of toluene in the case of natural gas H to obtain unity) and the mole fraction of neopentane to that of isopentane (0.0032 mol% for natural gas H and 0.0056 mol% for natural gas L). Moreover, the composition for the sample of natural gas L given by Schley et al. does not yield unity. Therefore, we normalized the mole fractions of all components of this sample as shown in Table 1. Finally, the molar masses of each sample were recalculated, but the numbers do not change compared with the values given by Schley et al.

The improved experimental \(\eta \rho pT\) data of this work for the previous measurements of Schley et al. [4] on the two samples of natural gas at four isotherms (260 K, 280 K, 300 K, and 320 K) are summarized in Tables 2, 3, 4, and 5 for natural gas H (highly caloric) and in Tables 6, 7, 8, and 9 for natural gas L (lowly caloric), in which the data are given in the sequential arrangement of the original measurements. The experimental points at pressures below 0.19 MPa are influenced by the slip effect, they are marked in Tables 2, 3, 4, 5, 6, 7, 8, and 9. The relative combined expanded (\(k=2\)) uncertainty \(U_{\textrm{c,r}}(\eta )\) is estimated to be 1.0 % for the 260 K isotherms and 0.8 % for the 280 K, 300 K, and 320 K isotherms of the natural gas samples H and L. We note that the uncertainty reported by Schley et al. (0.5 %) was a standard uncertainty (\(k=1\)) and could be slightly improved in this work due to the application of a more accurate calibration value and an improved density calculation, except for the 260 K isotherms influenced by the retrograde behavior.

The experimental data of each nominal isotherm for natural gas samples H and L were correlated as a function of the reduced density \(\delta\) by means of a power series representation restricted to the fourth power:

$$\begin{aligned} \eta (\tau , \delta )= &{} \sum _{i=0}^{4} \eta _i (\tau )\,\delta ^i,\hspace{0.8cm}\delta = \frac{\rho }{\rho _{\textrm{c,}j}},\hspace{0.8cm}\tau = \frac{T}{T_{\textrm{c,}j}}, \end{aligned}$$
(1)
$$\begin{aligned}\textrm{with}{} & {}\,\, j=\textrm{H}~~~\mathrm {for~natural~gas~H,~~~}j=\textrm{L}~~~\mathrm {for~natural~gas~L,} \\ \textrm{and}{} & {}\, \rho _{\textrm{c,H}}=208.45~\textrm{kg}\!\cdot\! \hbox {m}^{-3}, \hspace{0.8cm} T_{\textrm{c,H}}=213.26~\textrm{K}~~~\mathrm {for~natural~gas~H,} \\{} & {} \rho _{\textrm{c,L}}=186.21~\textrm{kg}\!\cdot\! \hbox {m}^{-3}, \hspace{0.8cm} T_{\textrm{c,L}}=191.88~\textrm{K}~~~\mathrm {for~natural~gas~L},\end{aligned}$$

where \(\delta\) is the reduced density, whereas \(\tau\) is the reduced temperature. The values of the critical densities \(\rho _{\textrm{c,H}}\) and \(\rho _{\textrm{c,L}}\) as well as of the critical temperatures \(T_{\textrm{c,H}}\) and \(T_{\textrm{c,L}}\) were determined using REFPROP (Lemmon et al. [26]). Weighting factors \(w_i=100\,\eta ^{-2}_{{\textrm{exp}},i}\) were applied in the multiple linear least-squares regression to minimize the weighted sum of squares \(\sigma =\sum _i w_i(\eta _{\textrm{cor},i}-\eta _{{\textrm{exp}},i})^2\) as criterion for the quality of the representation of the considered isotherm. The coefficients \(\eta _i (\tau )\) of Eq. 1 including their standard deviations \(\mathrm {s.d.}_{\eta _i}\) and the weighted sum of squares \(\sigma\) for each isotherm are given in Table 10 for natural gas H and in Table 11 for natural gas L.

Table 2 Re-evaluated experimental \(\eta \rho pT\) data for natural gas H at 260 K
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Table 3 Re-evaluated experimental \(\eta \rho pT\) data for natural gas H at 280 K
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Table 4 Re-evaluated experimental \(\eta \rho pT\) data for natural gas H at 300 K
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Table 5 Re-evaluated experimental \(\eta \rho pT\) data for natural gas H at 320 K
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Table 6 Re-evaluated experimental \(\eta \rho pT\) data for natural gas L at 260 K
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Table 7 Re-evaluated experimental \(\eta \rho pT\) data for natural gas L at 280 K
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Table 8 Re-evaluated experimental \(\eta \rho pT\) data for natural gas L at 300 K
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Table 9 Re-evaluated experimental \(\eta \rho pT\) data for natural gas L at 320 K
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Table 10 Coefficients of Eq. 1 for the re-evaluated viscosity measurements on the sample of natural gas H
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Table 11 Coefficients of Eq. 1 for the re-evaluated viscosity measurements on the sample of natural gas L
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3 Discussion of the Results

Humberg et al. [3] recently stated that the experimental values published by Schley et al. [4] for methane are assumed to be about 0.15 % too high due to the use of the former reference value for the viscosity of argon given by Kestin and Leidenfrost [23] for the calibration of the viscometer used by Schley et al. The same calibration value was employed for the measurements of the two natural gas samples discussed herein. Therefore, the re-evaluated values reported in this work are expected to be about 0.15 % lower than the older ones. Figures 1 and 2 show deviations of the experimental viscosity data for the two samples natural gas H and L, respectively, given by Schley et al. from the re-evaluated viscosity data reported in this work.

Fig. 1
figure 1

Comparison of the re-evaluated viscosity data of this work \(\eta _{\mathrm {exp,re-eval}}\) for the sample natural gas H with the data \(\eta _{\textrm{exp,Schley}}\) for the four isotherms, as a function of density \(\rho\). Data: \(\bigcirc\), 260 K; \(\vartriangle\), 280 K; \(\square\), 300 K; and \(\Diamond\), 320 K

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Fig. 2
figure 2

Comparison of the re-evaluated viscosity data of this work \(\eta _{\mathrm {exp,re-eval}}\) for the sample natural gas L with the data \(\eta _{\textrm{exp,Schley}}\) for the four isotherms, as a function of density \(\rho\). Data: \(\bigcirc\), 260 K; \(\vartriangle\), 280 K; \(\square\), 300 K; and \(\Diamond\), 320 K

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In connection with Figs. 1 and 2, it has to be noted that the density values of the re-evaluated data were re-calculated from the experimental data for pressure and temperature employing the equation of state by Kunz and Wagner [6] via their implementation in REFPROP (Lemmon et al. [26]). Figure 1, which illustrates the deviations between the original Schley data and the re-evaluated data as function of density for the four isotherms of natural gas sample H, does not show constant values near to − 0.15 % as expected due to the re-calibration of the radius of the wire. The isotherm at 260 K is characterized by the largest oscillations in the deviations around − 0.15 %. The oscillations decrease with increasing temperature. The maximum and minimum values occur at \(\rho \approx 75\) kg\(\cdot\)m\(^{-3}\) and \(\rho \approx 180\) kg\(\cdot\)m\(^{-3}\), respectively. They are certainly caused by the change in the calculation of the densities from the ISO 12213 [5] to the equation of state by Kunz and Wagner.

The corresponding deviations between the original Schley data and the re-evaluated data for the natural gas sample L are displayed for the four isotherms in Fig. 2. The figure shows, similar to Fig. 1, also oscillations of the deviations around − 0.15 %. These oscillations are less at lower densities (\(\rho <100\) kg\(\cdot\)m\(^{-3}\)) and comparable at higher densities (\(\rho >120\) kg\(\cdot\)m\(^{-3}\)) compared with natural gas H. The oscillations in the deviations decrease again with increasing temperature.

Both figures indicate that the differences of the original Schley data and the re-evaluated data amount to extra ± 0.2 % due to the changed density calculation dependent on the density itself. Figures 1 and 2 show that the re-calibration of the radius of the vibrating wire results in a reduction in the viscosity of − 0.15 %, particularly at low densities.

In addition, a remark concerns the outliers in Figs. 1 and 2. The reason for these deviations are typing errors in Tables III and IV of Schley et al. [4]. In Table III (natural gas H) at the isotherm 260 K, the viscosity value at ρ = 4.3077 mol·L−1 (ρ = 77.4 kg·m−3) should read 12.315 μPa·s instead of 12.335 μPa·s and in Table IV (natural gas L) at the isotherm 320 K, the viscosity value at ρ = 5.3865 mol·L−1 (ρ = 99.3 kg·m−3) should read 15.910 μPa·s instead of 15.900 μPa·s.

Moreover, we compare our re-evaluated data with the extended corresponding states (ECS) model [27] implemented in REFPROP [26]. This is done in Figs. 3 and 4, which present deviations between the re-evaluated data for the two samples of natural gas H and L and the calculated values for the ECS model. The zero line corresponds to the ECS model developed by Ely and Hanley. The calculation of the ECS values results from the experimental temperature and pressure data. We note that for natural gas sample H as well as sample L no viscosity values applying the ECS model could be calculated at 260 K in low density ranges, for sample H at \(4<\rho/\)kg·m−3 < 84 and for sample L at \(2<\rho/\)kg·m−3 < 103, respectively. This is due to the retrograde behavior of the samples at low temperatures.

Fig. 3
figure 3

Comparison of the re-evaluated viscosity data of this work \(\eta _{\textrm{exp}}\) for the sample natural gas H with values \(\eta _{\textrm{REFPROP}}\) calculated from the approach of the ECS model developed by Ely and Hanley [27] using REFPROP [26] for the four isotherms, as a function of density \(\rho\). Data: \(\bigcirc\), 260 K; \(\vartriangle\), 280 K; \(\square\), 300 K; and \(\Diamond\), 320 K

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Fig. 4
figure 4

Comparison of the re-evaluated viscosity data of this work \(\eta _{\textrm{exp}}\) for the sample natural gas L with values \(\eta _{\textrm{REFPROP}}\) calculated from the approach of the ECS model developed by Ely and Hanley [27] using REFPROP [26] for the four isotherms, as a function of density \(\rho\). Data: \(\bigcirc\), 260 K; \(\vartriangle\), 280 K; \(\square\), 300 K; and \(\Diamond\), 320 K

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Figure 3, in which the deviations of the re-evaluated data for the natural gas sample H from the values calculated for the ECS model [27] are illustrated, makes evident that for the higher isotherms 280 K, 300 K, and 320 K the deviations are between \(+\)0.09 % at the lowest densities and \(\approx\)− 0.45 % up to high densities. For the isotherm 260 K, the largest deviation amounts to \(+\)0.84 % at \(\rho =85\) kg\(\cdot\)m\(^{-3}\). The deviations decrease with increasing density for this isotherm, pass through zero, and become negative with a minimum of − 0.45 % at \(\rho =181\) kg\(\cdot\)m\(^{-3}\). The deviations at \(\rho >181\) kg\(\cdot\)m\(^{-3}\) decrease to − 0.15 % on the average, corresponding to the re-calibration of the wire.

Figure 4, which displays the deviations of the re-evaluated data for the natural gas sample L from the values calculated for the ECS model [27], shows that larger deviations occur compared to natural gas sample H. For the isotherms 280 K, 300 K, and 320 K, the deviations at the lowest densities amount to \(+\)0.80 %, \(+\)1.23 %, and \(+\)1.44 %, respectively. They decrease with increasing density down to \(+\)0.60 % (280 K) and to \(\approx 1.0\) % (300 K and 320 K) at \(\rho \approx 100\) kg\(\cdot\)m\(^{-3}\), followed by a further decrease to \(+\)0.67 % (320 K) and to \(+\)0.33 % (300 K) at the highest density each. The deviation of the 260 K isotherm at \(\rho =104\) kg\(\cdot\)m\(^{-3}\) amounts to \(+\)0.63 %, whereas the further deviations and those for the isotherm 280 K show the similar course with increasing density. They pass through zero at \(\rho \approx 180\) kg\(\cdot\)m\(^{-3}\) and become negative (− 0.13 % for 280 K and − 0.18 % for 260 K at the highest density each).

4 Conclusion

A new reference value for argon at 298.15 K and at zero density, which was derived by Vogel et al. [24] and further updated by Hellmann [25], was used to re-evaluate viscosity measurements of Schley et al. [4] for two natural gas samples H (highly caloric) and L (lowly caloric) for the four isotherms each. Schley et al. carried out relative measurements using a vibrating-wire viscometer and calibrated their measuring device employing a nowadays outdated reference value of Kestin and Leidenfrost [23].

Furthermore, Schley et al. [4] applied the international standard ISO 12213 [5] for calculating the densities from the experimental pressure and temperature data. In this work, the densities were computed employing the equation of state of Kunz and Wagner [6] from the pressure and temperature data reported by Schley et al.

The re-evaluated experimental data of each nominal isotherm for the two natural gas samples H and L were correlated as a function of the reduced density \(\delta\) by means of a power-series representation restricted to the fourth power. Finally, the re-evaluated viscosity data were discussed in comparison with values calculated using the approach of the ECS model by Ely and Hanley [27] via its implementation in REFPROP [26]. The re-evaluated data of this work reported as \(\eta \rho pT\) data should be employed to update the database for natural gases.