Viscosity and Density of Normal Butane Simultaneously Measured at

Article pubs.acs.org/jced

Viscosity and Density of Normal Butane Simultaneously Measured at Temperatures from (298 to 448) K and at Pressures up to 30 MPa Incorporating the Near-Critical Region Sebastian Herrmann† and Eckhard Vogel*,‡ †

Fachgebiet Technische Thermodynamik, Hochschule Zittau/Görlitz, Theodor-Körner-Allee 16, D-02763 Zittau, Germany Institut für Chemie, Universität Rostock, Albert-Einstein-Straße 3a, D-18059 Rostock, Germany

‡

ABSTRACT: A vibrating-wire viscometer and a single-sinker densimeter were simultaneously used to determine accurate ηρpT data for normal butane. Seven isotherms were measured between (298.15 and 448.15) K up to maxima of 91 % of the saturated vapor pressure for the subcritical and of 30 MPa for the supercritical isotherms. The combined expanded uncertainty (k = 2) in density is 0.2 %, except for the lowdensity region. The data agree within ±0.4 % with densities for the equation of state by Bücker and Wagner, excluding the near-critical region due to the strong influence of allocation errors for temperature and pressure. The deviations for the near-critical isotherm 428.15 K amount to +8.5 %, distinctly higher than the total expanded uncertainty of the density data (2.4 %). The viscosity measurement is less influenced in the near-critical region by allocation errors for temperature and density so that the total expanded uncertainty is 0.6 %. The new data were compared with viscosity correlations of Younglove and Ely, Vogel et al., and Quiñones-Cisneros and Deiters. Maximum deviations between +16 % and −7 % at 428.15 K and densities 150 < ρ/kg·m−3 < 300 exceed seriously the expanded uncertainties of 6.0 % for the three correlations. The effect of about +1 % for the critical enhancement is obvious for 428.15 K. The present data should be used to generate a new viscosity correlation.

1. INTRODUCTION Normal butane, also known as R-600, is an important working fluid in the petrochemistry and in the natural gas industry. It is of importance as power gas in aerosols, as fuel gas in its liquified state, and as refrigerant in the cooling industry due to its low global warming and negligible ozone depletion potentials. Reliable thermophysical properties of normal butane are required when technical objects are designed, operated, maintained, or retrofitted and where this substance is concerned. Experimentally determined thermophysical properties are significant not only for process simulations and for computational fluid dynamics but also for a comparison with theoretically computed values. Moreover, exactly measured data are strongly needed for a reference equation of state1 and a reliable viscosity surface correlation.2 The equipment designed by Seibt3 for the simultaneous determination of viscosity and density in this work was first applied by Seibt et al.4 for measurements on helium and nitrogen. The used vibrating-wire viscometer represents an advanced version of an instrument developed and employed for measurements on gases by Wilhelm et al.5,6 To infer the viscosity from the observed damped oscillation curves of the vibrating wire on the basis of the respective working equation, density values of an uncertainty as low as possible are necessary. Although the evaluation of the parameters of the recorded oscillation curves enables, in principle, deduction of viscosity © XXXX American Chemical Society

and density simultaneously, the resulting density values are characterized by a comparably large uncertainty. The needed density values were calculated using concurrently measured values for pressure and temperature in combination with a reliable equation of state. The uncertainties of the measured temperatures and pressures influence not only the calculated densities but also the uncertainty of the finally derived viscosity values. Seibt et al. improved the experimental equipment of Wilhelm et al. by integrating an accurate single-sinker densimeter and, in so doing, they reduced the impact of an increased uncertainty in density, particularly in the near-critical region. Finally, the relative combined standard uncertainty (k = 1) of the density measurement amounts to 0.1 % so that the relative combined standard uncertainty in the viscosity is 0.3 %, with the exception of the low-density and the near-critical regions. A program concerned with the investigation of the thermophysical properties density and viscosity of alkanes, using the measuring equipment under discussion, was initiated. Within the framework of this program, Seibt et al.7 performed Special Issue: Memorial Issue in Honor of Anthony R. H. Goodwin Received: July 29, 2015 Accepted: October 9, 2015

A

DOI: 10.1021/acs.jced.5b00654 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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precise resistance measuring bridge to determine the temperature T. The combined standard uncertainty of the temperature measurement is 19 mK in the range between 235 and 505 K, resulting from the standard uncertainties of the thermometer (18 mK), of the measuring bridge (2 mK), and of the control accuracy of the thermostat (2 mK). A two-stage thermostat is used to control the temperature of the measuring cell. It consists of a custom-built oil-filled double-wall thermostat including a vacuum vessel for the insulation of the measuring cell. The outer stage of the thermostat is regulated using an oil-thermostat with a performance of ±(10 to 50) mK, whereas a heating tape operates as inner stage governed by means of a precise temperature controller within ±10 mK. The relative combined standard uncertainty in temperature amounts to ur,c(T) = 0.0001. The measurement of the pressure p is carried out by means of four absolute pressure transmitters with ranges of (41.4, 13.8, 2.76, 0.689) MPa. They are characterized by relative standard uncertainties of 0.01 % of full scale as well as of 0.03 % and 0.05 %, respectively, from reading. The transmitters are installed in a separate nitrogen-filled gas system to protect them from the condensable fluid. For that purpose, the nitrogen-filled and the measuring fluid-filled systems are separated by a high-precision differential pressure transducer which can detect pressure differences of ±5 kPa. The transducer is calibrated with a relative standard uncertainty of 0.04 % from reading which contributes 0.002 % to the relative standard uncertainty in pressure in the case of measurements at pressures p > 0.15 MPa. When the apparatus is operated for supercritical temperatures at pressures lower than the critical one but higher than the saturation pressure, the measuring fluid is condensing in the connecting tubes to the measuring cell. The same is observed in the tubes for subcritical isotherms at pressures higher than the saturation pressure. The reason for the condensation is that the measuring cell is situated within the thermostat at a higher temperature, while the tubes are at ambient temperature. Therefore, a liquid-level indicator is placed at a definite position of the connecting tubes, the temperature of which is adjusted to a certain saturation temperature and simultaneously to a definite pressure in the measuring cell (see ref 3 for more details). The relative standard uncertainty of the hydrostatic pressure correction is 0.001 % of the measured pressure value. Summarizing, the relative combined standard uncertainty in pressure is estimated to be ur,c(p) = 0.0005. 2.4. Error Analysis. The basic error analysis for the combined viscometer−densimeter conducted by Seibt et al.4 in the case of their measurements on helium and nitrogen was extended by Seibt et al.7 to ethane and propane and by Herrmann et al.8 to isobutane. For these substances, the situation is much more complex, because the investigation was expanded to the respective near-critical region. The same applies to normal butane. 2.4.1. Density. The relative total standard uncertainty, ur,tot(ρ), for the pρT data comprises the relative combined standard uncertainty of the density itself, ur,c(ρ), and the contributions of the allocation errors resulting from the combined standard uncertainties of the temperature and pressure measurements, uc(T) and uc(p):

density and viscosity measurements on ethane and propane, including the near-critical region. Then, Herrmann et al.8 continued and published the results of the determination of density and viscosity for isobutane. In this paper, new density and viscosity measurements on normal butane are evaluated. Their results are applied to verify the performance of the reference equation of state1 and of the available viscosity surface correlations2,9,10 for normal butane.

2. EXPERIMENTAL SECTION The arrangement of the combined apparatus for the simultaneous measurements of density and viscosity has already been given in Figure 2 of ref 4. 2.1. Vibrating-Wire Viscometer. The details of the vibrating-wire viscometer, which operates in the transient mode using the decay rate of the oscillation after a short initiation pulse, were specified by Seibt et al.3,4 The wire with a nominal diameter of 25 μm consists of Chromel, by reason for its comparably smooth surface. The 90 mm long wire is symmetrically arranged in a magnetic field with the length of 60 mm so that all even and the third harmonic oscillation modes are practically eliminated. A sinusoidal voltage pulse with a frequency near the resonant frequency of the wire is applied to initiate the first oscillation mode. The decaying oscillation is then detected as a function of time by amplifying the induced voltage. To improve the signal-to-noise ratio, it proved useful to perform data acquisitions of a hundred of oscillations within a run and to average the recorded curves. Information on the layout and arrangement of the three incorporated viscometers including the encased magnets were also reported by Seibt et al.7 and Herrmann et al.8 A nonlinear regression algorithm is applied to derive the parameters logarithmic decrement Δ and frequency ω from the oscillation curves. The viscosity η is calculated using the working equations given by Seibt et al.4 Although both the density and the radius of the wire are unproved, its density is assumed to be established. Then the radius is inferred by means of a calibration using a value for the zero-density viscosity coefficient of helium, derived by Bich et al.11 from an ab initio potential for the helium dimer on the basis of the kinetic theory of dilute gases (η0,4He,293.15K = 19.600 μPa·s, distinguished by a relative standard uncertainty of 0.02 %). The thermal expansion of the wire radius and of the wire density was taken into account (see ref 3). 2.2. Single-Sinker Densimeter. The single-sinker densimeter, suitable for measurements in the range of 1 ≤ ρ/kg·m−3 ≤ 2000, is based on the buoyancy principle. A measuring device analogous to a densimeter, developed at the Ruhr-Universität Bochum, Bochum, Germany, and advanced by Klimeck et al.,12 was installed. The forces on the sinker situated within the pressure-tight measuring cell are transmitted without direct contact by means of an electronically controlled magnetic suspension coupling13,14 to a calibrated microbalance, which is placed under ambient conditions above the thermostat of the measuring cell. Note the magnetic fields of the suspension coupling and of the vibrating-wire viscometer are arranged in a comparably large distance to avoid interactions between them. A sinker made of quartz glass was applied for the measuring series on normal butane, because absorption on quartz glass does not affect the measurements as in the case of helium (see ref 4). 2.3. Temperature and Pressure Measurements. A 25 Ω platinum resistance thermometer is employed together with a B



ur,tot(ρ) =

2 ur,c (ρ )

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Gerling Holz & Co. (GHC) Handels GmbH, Hamburg, Germany, supplied the used sample of normal butane, which was characterized with a certified purity of xn‑C4H10 ≥ 0.9999 (normal butane 4.0). First, hypothetical samples of normal butane, contaminated with the individual impurities at the highest possible mole fraction each, were examined. Second, a sample with impurities according to the product specification of the supplier, but with the restriction that the sum of all mole fractions equals 1 (see caption of Figure 2), was studied. The

2 ⎛ ∂ρ ⎞2 uc2(T ) ⎛ ∂ρ ⎞ uc2(p) +⎜ ⎟ +⎜ ⎟ ⎝ ∂T ⎠ p ρ2 ⎝ ∂p ⎠T ρ2

(1)

ur,c(ρ) is given for the single-sinker apparatus as ur,c(ρ) = 0.000526 +

0.0007 ρ /(10 kg·m−3)

(2)

To evaluate the allocation errors related to the uncertainties of the measured temperatures T and pressures p, an equation of state is needed to compute the values of the thermal expansion coefficient (∂ρ/∂T)p and of the isothermal compressibility coefficient (∂ρ/∂p)T. For helium and nitrogen, Seibt et al.4 found the relative total standard uncertainty of the density determination in wide thermodynamic regions to be 0.1 % for ρ > 15 kg·m−3 due to the low uncertainties uc(T) and uc(p) as well as to the small values of (∂ρ/∂T)p and (∂ρ/∂p)T. It is apparent that the influence of the allocation errors is small in large thermodynamic regions. However, this influence becomes more significant when the values of (∂ρ/∂T)p and of (∂ρ/∂p)T increase. Thus, a significant impact on the relative total standard uncertainty in the density yields in the near-critical region caused by comparably high values for the partial derivatives (∂ρ/∂T)p and (∂ρ/∂p)T. In the case of normal butane, the allocation errors were calculated applying the reference equation of state by Bücker and Wagner1 to compute the needed (∂ρ/∂T)p and (∂ρ/∂p)T values. For the near-critical isotherm at 428.15 K, Figure 1

Figure 2. Comparison of calculated values ρeos,cont for the density of contaminated samples of normal butane (equation of state by Kunz and Wagner15) at 428.15 K with computed densities ρeos,pure for pure normal butane (equation of state by Span and Wagner16):  , xC2H6 = 90 × 10−6; − −, xC3H8 = 90 × 10−6; −·−, xi‑C4H10 = 90 × 10−6; −  −, xn‑C5H12 = 90 × 10−6; −··−, xN2 = 20 × 10−6; ······, xO2 = 5 × 10−6;  −, GHC n-butane 4.0 (xN2 = 15 × 10−6, xO2 = 5 × 10−6, xH2O = 3 × 10−6, xS = 0 × 10−6, xC2H6 = 22 × 10−6, xC3H8 = 25 × 10−6, xi‑C4H10 = 30 × 10−6); deviations, Δ = 100·(ρeos,cont − ρeos,pure)/ρeos,pure.

densities of the contaminated normal butane samples were determined applying the equation of state generated for natural gases by Kunz and Wagner.15 For the isotherm at 428.15 K, these densities are compared in Figure 2 with values obtained from the equation of state for pure normal butane developed by Span and Wagner.16 Instead of the reference equation of state by Bücker and Wagner,1 the equation of Span and Wagner was employed in this comparison because Kunz and Wagner used this type of equations in their model. The figure illustrates that a sample with normal pentane as impurity has a positive deviation of +0.20 %, whereas ethane and propane as impurities lead to negative deviations of −0.45 % and −0.21 %. For the sample used in the measurements, a maximum difference of −0.43 % is to be expected. As the purity of the supplied samples is often better than that given in the product specification each (see ref 7 for propane), this maximum difference may be a conservative estimation. Nevertheless, highly pure samples are recommended for high-precision measurements, even though in the near-critical region the influence of impurities is only one-third compared to the contributions of the allocation errors to the relative total standard uncertainty. 2.4.2. Viscosity. The relative total standard uncertainty in the viscosity, ur,tot(η), considers the relative combined standard

Figure 1. Allocation errors in temperature and pressure as a function of density ρ for isotherms of pρT measurements on normal butane using the equation of state by Bücker and Wagner.1

illustrates a maximum of −0.60 % for the allocation error related to the uncertainty in temperature and of +1.04 % for the allocation error referred to the uncertainty in pressure. Therefore, the relative total standard uncertainty of experimental pρT data (eq 1) increases up to 1.2 %, and the relative total expanded uncertainty (k = 2) is estimated to be 2.4 % in maximum for this particular isotherm. The studies of the effect of impurities in the measured samples of ethane,7 of propane,7 and of isobutane8 revealed that their impact on the uncertainty of the pρT data should be examined, in particular in combination with the large influence of the allocation errors due to the uncertainties in temperature and pressure for the near-critical isotherm at 428.15 K. The C



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uncertainty in the viscosity measurement itself, ur,c(η), and the allocation errors related to the combined standard uncertainties of the temperature and density measurements, uc(T) and uc(ρ), in combination with the temperature and density dependencies of the viscosity ur,tot(η) =

2 ur,c (η)

is discharged in each case so that the pressure is lowered corresponding to the measuring program. To regain thermodynamic equilibrium after discharging measuring fluid, a specific period has to elapse, several hours at the highest densities and in the near-critical region, whereas half an hour is adequate at low densities. An experimental point implies the determination of the logarithmic decrement Δ and of the frequency ω of the damped oscillation followed by three or more weighings of the sinker mass ms,fluid in the fluid as well as concurrent measurements of temperature T and pressure p. For a sound evaluation of Δ and ω, runs of a hundred oscillations, started at different initial amplitudes, are recorded. An isothermal series of measurements is finished by repeating the determination of the logarithmic decrement and of the sinker mass in vacuo. The measurements for the individual experimental points could not accurately be performed at the nominal temperature of a specific isotherm, Tnom, but within small differences from it. The experimental density ρexp, directly measured with the single-sinker densimeter at the temperature T and the pressure p, was applied to evaluate the experimental viscosity η. The densities ρexp correspond to those for the isotherm. Their total expanded uncertainties Utot(ρexp) were calculated according to the error analysis discussed in Section 2.4. Then, the originally determined viscosity data η were converted to ηTnom values at the nominal temperature using a Taylor series expansion restricted to the first power in temperature. Values for the temperature coefficient (∂η/∂T)ρ required for the Taylor series expansion were deduced from a double polynomial with respect to reciprocal reduced temperature and to reduced density, which was fitted to the experimentally determined viscosity data η. Note the values for the supercritical isotherms 428.15 K and 448.15 K could only be used for densities ρ < 100 kg·m−3. At higher densities ρ > 100 kg·m−3, the values for (∂η/∂T)ρ at ρ = 99.3 kg·m−3 (for 428.15 K) and ρ = 92.9 kg·m−3 (for 448.15 K) were utilized. A comparison with the values (∂η/∂T)ρ=0 = (0.024 to 0.025) μPa·s·K−1, derived from re-evaluated viscosity data in the limit of zero density originally measured by Kü chenmeister and Vogel,17 revealed close agreement. Furthermore, the pressure pTnom,ρexp at the nominal temperature Tnom does not correspond to the pressure p at the experimental temperature T. Hence, it was recalculated from the experimental density ρexp using the equation of state by Bücker and Wagner1 for normal butane. In the end, density values ρeos(T,p) were derived from the measured values for temperature T and pressure p applying again the equation of state. As mentioned above, a sample (normal butane 4.0) supplied by Gerling Holz & Co. (GHC) Handels GmbH, Hamburg, Germany, with a certified purity of xn‑C4H10 ≥ 0.9999 was employed. The product specification indicates the following maximum values for the most important impurities: xN2 ≤ 20 × 10−6, xO2 ≤ 5 × 10−6, xH2O ≤ 3 × 10−6, xS ≤ 1 × 10−6, xCnHm ≤ 90 × 10−6. Five isotherms were studied at subcritical temperatures (298.15 K, 323.15 K, 348.15 K, 373.15 K, and 423.15 K) and two at supercritical ones (428.15 K and 448.15 K). The subcritical isotherms cover ranges up to 91 % of the saturated vapor pressure, while the supercritical isotherms were performed up to 30 MPa. Note the first supercritical isotherm is only 3.025 K away from the critical temperature Tc = 425.125 K. The results of the seven isothermal series of measurements on normal butane, including all the data and values discussed before, are summarized in Tables 1 to 7. It is to note that some

2 ⎛ ∂η ⎞2 uc2(T ) ⎛ ∂η ⎞ uc2(ρ) ⎜ ⎟ + +⎜ ⎟ ⎝ ∂T ⎠ ρ η2 ⎝ ∂ρ ⎠T η2

(3)

The standard uncertainty of the viscosity measurement itself has already extensively been discussed in refs 3 and 4. Because the density is needed for the evaluation of the viscosity measurements, the relative total standard uncertainty in viscosity is enlarged, if the density values are derived from concurrently measured temperature and pressure data using an equation of state. The reason for this is the increase of the allocation errors connected with the uncertainties of the temperature and pressure measurements (see above), particularly in the near-critical region. Seibt et al.4 overcame this problem by incorporating a single-sinker densimeter in the experimental equipment so that a simultaneous measurement of density and viscosity could be conducted. Consequently, the relative combined standard uncertainty of the viscosity determination itself was considerably reduced. Seibt et al. performed an analysis of multivariate functions resulting in a relative combined standard uncertainty of ur,c(η) = 0.0028, in which a relative combined standard uncertainty of ur,c(ρ) = 0.001 is assumed for the density. Starting on the assumption that the relative combined standard uncertainties of temperature and density amount to ur,c(T) = 0.0001 and ur,c(ρ) = 0.001, the allocation errors were inferred for experimentally obtained values of temperature and density, considering the coefficients (∂η/∂T)ρ and (∂η/∂ρ)T for normal butane. These partial derivatives were derived from the available viscosity surface correlation of normal butane.2 Due to comparably small values of (∂η/∂T)ρ and (∂η/∂ρ)T, the influence of the allocation errors on the total uncertainty of the viscosity is very small. The reason that the impact of the allocation errors, even in the near-critical region, is minor consists in that the critical enhancement of viscosity is small in the investigated thermodynamic region. The contribution of the critical enhancement to the total viscosity amounts to about 1 % for the critical density, if the temperature is about 1 % higher than the critical temperature. Moreover, the viscosity surface correlation of normal butane by Vogel et al.2 does not imply a contribution for the critical enhancement. Finally, the analysis with eq 3 leads to a relative total standard uncertainty ur,tot(η) = 0.003 for the experimental ηρT data, only a little more than the uncertainty in the viscosity measurement itself. Hence, the relative total expanded uncertainty (k = 2) is estimated to be Ur,tot(η) = 0.006. But experimental difficulties in the near-critical region can provoke an increase of the uncertainty in the viscosity for this specific region.

3. MEASUREMENTS AND RESULTS Each isothermal series of measurements is always started by determining the logarithmic decrement Δ0 and the sinker mass ms,vac in vacuo. Then, the measuring-fluid system is filled with normal butane up to the highest pressure and the first measuring point is taken. For recording the further experimental points, a certain amount of the measuring fluid D



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Table 1. Experimental ηρpT Data for Normal Butane at 298.15 Ka T

p

p298.15K,ρexp

K

MPa

MPa

298.122 298.124 298.140 298.142 298.143 298.144 298.114 298.114 298.124 298.128 298.133 298.135 298.140 298.140 298.142 298.143 298.143 298.144 298.150 298.150

0.22139 0.22135 0.19675 0.19677 0.17529 0.17529 0.16281 0.16282 0.14216 0.14217 0.12343 0.12344 0.10290 0.10291 0.082370 0.082383 0.061794 0.061803 0.057183 0.057184

0.22142 0.22137 0.19676 0.19677 0.17530 0.17530 0.16283 0.16284 0.14218 0.14219 0.12343 0.12345 0.10290 0.10291 0.082372 0.082385 0.061796 0.061804 0.057183 0.057184

ρexp kg·m

Utot(ρexp)

−3

kg·m

5.5598 5.5591 4.9073 4.9087 4.3401 4.3415 4.0109 4.0116 3.4791 3.4777 3.0002 3.0016 2.4873 2.4851 1.9772 1.9779 1.4751 1.4737 1.3637 1.3637

−3

0.0208 0.0208 0.0199 0.0199 0.0191 0.0191 0.0187 0.0187 0.0180 0.0180 0.0174 0.0174 0.0168 0.0168 0.0162 0.0162 0.0156 0.0156 0.0155 0.0155

ρeos(T,p) −3

kg·m

5.5728 5.5716 4.9091 4.9096 4.3414 4.3415 4.0157 4.0160 3.4823 3.4826 3.0049 3.0052 2.4887 2.4889 1.9794 1.9797 1.4756 1.4758 1.3635 1.3636

η

η298.15K

μPa·s

μPa·s

7.3337 7.3332 7.3425 7.3407 7.3470 7.3465 7.3528 7.3538 7.3587 7.3593 7.3653 7.3634 7.3691 7.3698 7.3747 7.3720 7.3765 7.3782 7.3766b 7.3780

7.3344 7.3339 7.3428 7.3409 7.3472 7.3467 7.3537 7.3547 7.3594 7.3599 7.3657 7.3638 7.3694 7.3701 7.3749 7.3722 7.3767 7.3784 7.3766b 7.3780

a

The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006. bInfluenced by slip.

the equation of state is only 0.02 % in this region. The measured density data for the isotherms at 348.15 K, 373.15 K, and 423.15 K are reproduced within the uncertainty of the equation of state (0.2 % and 0.4 %, respectively). In contrast, large differences with a maximum of +8.47 % occur for the near-critical isotherm at 428.15 K in the density range 159 ≤ ρ/ kg·m−3 ≤ 322 including the critical density (ρc = 228.0 kg·m−3). These deviations exceed distinctly the relative total expanded uncertainty for this isotherm, even if it would be increased by additional 0.9 % considering the possible impact of impurities of the sample. The measured density data for the isotherm 448.15 K are interestingly characterized by large deviations up to +1.47 % near the critical density, too. In contrast, the deviations at lower and higher densities for this isotherm are similar to that of the other ones. To clarify the reason for the large differences at 428.15 K and 448.15 K, the comparison of the experimental density data was extended to values computed with further equations of state. Figure 4 shows the deviations of the experimental density data at 428.15 K from values calculated for the equation of state of Younglove and Ely9 as well as for the fundamental equations of state of Miyamoto and Watanabe,19 of Span and Wagner,16 and of Bücker and Wagner.1 The deviations are within ±0.65 % for densities ρ < 120 kg·m−3 and ρ > 340 kg·m−3, and the differences between the four equations are small. The maximum deviations occur next to the critical density ρc = 228.0 kg·m−3: +7.74 % at ρ = 251 kg·m−3 for Younglove and Ely, +5.98 % at ρ = 252 kg·m−3 for Miyamoto and Watanabe, +6.18 % at ρ = 218 kg·m−3 for Span and Wagner, and +8.47 % at ρ = 247 kg·m−3 for Bücker and Wagner. The likewise deviations for the different equations of state lead to the assumption that similar data sets with nearly the same weighting were used when generating the equations of state. It is to state that the expanded uncertainty in density of 0.4 % for the equation of state of Bücker and Wagner indicated in the figure is not representative for this isotherm in the near-critical

experimental points at the lowest densities could be influenced by the slip effect. They are marked and should not be utilized for generating viscosity correlations.

4. EVALUATION OF DATA AND DISCUSSION 4.1. Density. The experimental density data inferred from the measurements on normal butane are checked against the values computed for the equation of state of Bücker and Wagner.1 This equation is valid from the triple-point temperature (Ttr,n‑C4H10 = 134.895 K) up to 575 K and for pressures up to 69 MPa. The equation is distinguished by a relative expanded uncertainty in density of about 0.02 % (k = 2) for temperatures up to 340 K and pressures up to 12 MPa, essentially based on measurements performed by Glos et al. with a two-sinker densimeter at the Ruhr-Universität Bochum.18 Relative expanded uncertainties in density of 0.2 % for pressures p ≤ 0.6 MPa and 0.4 % up to 30 MPa were reported for the remaining thermodynamic region studied up to 450 K in this work, apart from the extended critical region for which an expanded uncertainty in pressure of 0.5 % was given. Figure 3 illustrates the deviations of the measured density data of this work from the values calculated for the concurrently measured temperatures and pressures by applying the equation of state by Bücker and Wagner.1 In addition, the relative total expanded uncertainty of the density measurement at 428.15 K (2.4 %), and the expanded uncertainty of the equation of state at supercritical pressures (0.4 %) are indicated. The figure reveals generally a good agreement with the equation of state within ±0.2 %, except for the low-density and the extended critical regions. The deviations for the isotherms at 298.15 K and 323.15 K amount to ±0.17 % so that they are within the expanded uncertainty of the density measurement (0.2 %), apart from two measuring points near the phase boundary (−0.23 % at 298.15 K). However, the expanded uncertainty of E



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p

p323.15K,ρexp

K

MPa

MPa

322.982 322.997 323.007 323.061 323.077 323.089 323.110 323.111 323.117 323.124 323.125 323.126 323.126 323.126 323.126 323.132 323.134 323.137 323.139 323.141 323.143 323.145 323.146 323.147 323.146 323.146 323.142 323.143 323.143 323.142 323.143 323.143 323.143 323.144 323.143 323.143 323.143 323.143 323.143 323.143

0.45039 0.45069 0.45089 0.39210 0.39250 0.39290 0.38078 0.38074 0.38076 0.33012 0.33036 0.33063 0.30154 0.30339 0.30532 0.26118 0.26121 0.26128 0.21587 0.21565 0.21563 0.18424 0.18435 0.18436 0.15003 0.15004 0.13243 0.13244 0.13246 0.11293 0.11294 0.11296 0.090508 0.090523 0.066379 0.066382 0.066387 0.058204 0.058199 0.058206

0.45069 0.45096 0.45115 0.39224 0.39261 0.39299 0.38084 0.38080 0.38081 0.33015 0.33039 0.33066 0.30156 0.30341 0.30535 0.26120 0.26122 0.26130 0.21588 0.21566 0.21563 0.18424 0.18435 0.18436 0.15003 0.15005 0.13243 0.13244 0.13246 0.11294 0.11295 0.11296 0.090510 0.090525 0.066381 0.066384 0.066388 0.058206 0.058201 0.058207

ρexp kg·m

Utot(ρexp)

−3

kg·m

11.024b 11.032b 11.037b 9.4260b 9.4391b 9.4492b 9.1236b 9.1236b 9.1272b 7.8092b 7.8164b 7.8215b 7.0807b 7.1241b 7.1762b 6.0223 6.0230 6.0252 4.9169 4.9111 4.9096 4.1645 4.1652 4.1667 3.3622 3.3622 2.9506 2.9492 2.9499 2.5057 2.5035 2.5057 1.9979 1.9986 1.4575 1.4567 1.4553 1.2744 1.2766 1.2766

−3

0.0286b 0.0286b 0.0286b 0.0262b 0.0262b 0.0262b 0.0257b 0.0257b 0.0257b 0.0238b 0.0238b 0.0238b 0.0228b 0.0228b 0.0229b 0.0213 0.0213 0.0214 0.0199 0.0199 0.0199 0.0189 0.0189 0.0189 0.0179 0.0179 0.0174 0.0174 0.0174 0.0168 0.0168 0.0168 0.0162 0.0162 0.0156 0.0156 0.0156 0.0154 0.0154 0.0154

ρeos(T,p) −3

kg·m

10.968 10.975 10.980 9.3758 9.3859 9.3960 9.0727 9.0717 9.0719 7.7512 7.7572 7.7642 7.0239 7.0706 7.1194 6.0178 6.0184 6.0202 4.9153 4.9099 4.9092 4.1615 4.1640 4.1643 3.3602 3.3605 2.9533 2.9537 2.9540 2.5069 2.5072 2.5074 1.9985 1.9989 1.4576 1.4576 1.4577 1.2757 1.2755 1.2757

η

η323.15K

μPa·s

μPa·s

7.9534 7.9513 7.9528 7.9577 7.9545 7.9604 7.9581 7.9578 7.9578 7.9671 7.9669 7.9646 7.9703 7.9665 7.9703 7.9704 7.9726 7.9684 7.9741 7.9762 7.9758 7.9788 7.9802 7.9792 7.9820 7.9821 7.9905 7.9888 7.9886 7.9902 7.9907 7.9903 7.9873 7.9898 7.9914 7.9936 7.9937 7.9942 7.9883c 7.9896c

7.9580 7.9555 7.9567 7.9601 7.9564 7.9620 7.9592 7.9588 7.9587 7.9678 7.9676 7.9652 7.9709 7.9671 7.9709 7.9709 7.9730 7.9687 7.9744 7.9764 7.9760 7.9789 7.9803 7.9793 7.9821 7.9822 7.9907 7.9890 7.9888 7.9904 7.9909 7.9905 7.9875 7.9900 7.9916 7.9938 7.9939 7.9944 7.9885c 7.9898c

a The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006. bSuffered from a problem with the single-sinker densimeter. cInfluenced by slip.

restricted to 77 % of the saturation pressure corresponding to ρ = 73.9 kg·m−3. Because the values of Gupta and Eubank were not corrected for adsorption, Bücker and Wagner did only use data points above 0.1 MPa, but the data from the pressure range 0.5 to 3 MPa could not adequately be represented by the equation of state. Nevertheless, Bücker and Wagner left the data of Gupta and Eubank in their primary data set, because no other reliable data were available in this region. Furthermore, the new data measured in the liquid state at 424 K by Miyamoto and Uematsu22 and at 420 K by the same authors23 were not available, when the equations of state were established. Figure 5 shows very similar deviations of the experimental data of Beattie et al., of Miyamoto and Uematsu, and also of this work from the values for the equation of state of Bücker and Wagner. For the data from the literature,20,22,23 the deviations are within +(2 to 7) % at supercritical densities 270

region. However, a value of 5 %, as a result of an estimation starting with the uncertainty in pressure in the near-critical region (0.5 %) and multiplying by ten, should be too high. Because the deviations are very large at all, a straight comparison of the experimental data used for generating the equations of state may be promising. Hence, the experimental density data from the literature determined in the near-critical region are compared with values calculated for the equation of state by Bücker and Wagner1 in Figure 5. Note the data by Beattie et al.20 at 423.15 K in the unsaturated vapor and at 425.16 K for supercritical densities had only been used as primary data for developing the equation of state by Younglove and Ely. 9 The results of the measurements by Gupta and Eubank21 in the vapor phase at 425 K were used for the generation of the equation of state by Bücker and Wagner, but the pressure range was regrettably F



Article


p

p348.15K,ρexp

K

MPa

MPa

348.027 348.028 348.029 348.130 348.134 348.125 348.125 348.139 348.143 348.147 348.148 348.152 348.153 348.152 348.152 348.152 348.151 348.143 348.143 348.144 348.145 348.150 348.150

0.81399 0.81413 0.81385 0.76025 0.76034 0.67752 0.67782 0.61824 0.61834 0.51676 0.51669 0.45498 0.45521 0.34091 0.33838 0.33786 0.33788 0.29319 0.29373 0.093820 0.093833 0.063283 0.063298

0.81442 0.81455 0.81426 0.76031 0.76039 0.67758 0.67788 0.61826 0.61836 0.51676 0.51670 0.45498 0.45521 0.34091 0.33838 0.33786 0.33788 0.29320 0.29374 0.093822 0.093835 0.063283 0.063298

ρexp

ρeos(T,p)

Utot(ρexp)

−3

−3

−3

kg·m

kg·m

kg·m

19.581 19.586 19.576 17.999 18.002 15.682 15.691 14.097 14.099 11.497 11.497 9.9811 9.9833 7.3031 7.2445 7.2315 7.2315 6.2173 6.2296 1.9124 1.9131 1.2845 1.2860

0.0425 0.0425 0.0424 0.0397 0.0397 0.0358 0.0358 0.0332 0.0332 0.0292 0.0292 0.0269 0.0269 0.0231 0.0230 0.0230 0.0230 0.0216 0.0216 0.0161 0.0161 0.0154 0.0154

19.582 19.587 19.578 17.999 18.002 15.685 15.693 14.097 14.099 11.498 11.496 9.9821 9.9877 7.2986 7.2407 7.2288 7.2292 6.2164 6.2285 1.9152 1.9155 1.2848 1.2851

η

η348.15K

μPa·s

μPa·s

8.6479 8.6493 8.6517 8.6400 8.6372 8.6312 8.6314 8.6214 8.6277 8.6176 8.6154 8.6098 8.6099 8.5993 8.5994 8.6032 8.6070 8.6021 8.6018 8.5995 8.5967 8.5885b 8.5908b

8.6513 8.6527 8.6551 8.6406 8.6376 8.6319 8.6321 8.6217 8.6279 8.6177 8.6155 8.6098 8.6098 8.5993 8.5994 8.6032 8.6070 8.6023 8.6020 8.5997 8.5968 8.5885b 8.5908b

a

The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006. bInfluenced by slip.

< ρ/kg·m−3 < 300, whereas at densities ρ < 150 kg·m−3 and ρ > 430 kg·m−3, the deviations do not exceed the relative expanded uncertainty of the equation of state (0.4 %). One data point of Beattie et al. is hallmarked by an extremely large deviation (+16.2 %) near the critical density. The data of Gupta and Eubank are in good agreement with the other data from the literature as well as with the data of this work. The maximum deviation of their experimental data from the calculated values amounts to −0.24 % at the highest density ρ = 73.9 kg·m−3. Because the uncertainty in pressure should be preferred in the near-critical region, the experimentally determined pressures of this work are compared with pressure values calculated for the four discussed equations of state using measured temperature and density data. Figure 6 illustrates for the isotherm 428.15 K that at densities ρ < 200 kg·m−3 the deviations of the experimental pressure data from the calculated values for all four equations of state amount to ±0.3 % and at densities ρ ≤ 325 kg·m−3 also only up to −0.8 %. Afterward, the deviations for the equation of state of Younglove and Ely9 decrease with rising density, change the sign, and increase to +1.01 % at ρ = 440 kg·m−3. For the equation of state by Miyamoto and Watanabe,19 the deviations enlarge at high densities up to −2.47 % at ρ = 420 kg·m−3. In the case of the equation of state by Bücker and Wagner,1 the deviations decrease again at higher densities ρ > 325 kg·m−3, whereas for the equation of state by Span and Wagner,16 deviations of −0.75 % occur on average. When comparing Figures 4 and 6, the trend of the deviations in pressure is different from that of the deviations in density. In the end, experimental pressure data of Beattie et al.24 measured in the very close vicinity of the critical point should be compared with pressure values computed for the equation of state by Bü cker and Wagner,1 applying the measured

temperature and density data. Beattie et al. carried out pρT measurements for eight isotherms within ±0.1 K around the critical temperature Tc = 425.125 K in the density range 166 < ρ/kg·m−3 < 281. Remarkably, Bücker and Wagner included these data points into their primary data set when developing their equation of state. In contrast, Younglove and Ely,9 Miyamoto and Watanabe,19 as well as Span and Wagner16 did not consider the data of Beattie et al. to be primary. Figure 7 reveals that the deviations of the experimental pressure data by Beattie et al. from the calculated pressure values are very small (±0.04 %) and do not exceed at all the experimental uncertainty of 0.5 % in pressure. It is evident that the behavior of the equation of state of Bücker and Wagner in the immediate vicinity of the critical point is distinctly influenced by the large number of measuring points of Beattie et al. and their use as primary data, in which their weighting does not play any role, because they are the only ones in this specific region. Consequently, data not measured directly at the critical point, but in the near-critical region are characterized by large deviations from the values calculated for the equation of state by Bücker and Wagner. A similar situation was found for isobutane.8 4.2. Viscosity. The experimental data at the nominal temperature Tnom of each isotherm were correlated as a function of the reduced density δ using a power-series representation limited to the sixth or a lower power depending on the density range and the reciprocal reduced temperature τ: n

η (τ , δ ) =

∑ ηi(τ)δ i , i=0

τ=

Tc ρ , δ= , ρc T

with Tc = 425.125 K, ρc = 228.0 kg· m−3 G

(4)



Article


p

p373.15K,ρexp

K

MPa

MPa

373.150 373.149 373.145 373.144 373.136 373.139 373.135 373.135 373.140 373.147 373.144 373.141 373.143 373.142 373.142 373.140 373.139 373.148 373.147 373.149 373.151 373.152 373.144 373.143 373.147 373.145 373.139 373.135 373.130 373.132 373.134 373.138 373.141 373.143

1.3664 1.3671 1.2627 1.2630 1.0954 1.0955 0.91645 0.91645 0.83834 0.75055 0.75104 0.66981 0.67000 0.57254 0.57260 0.49381 0.49367 0.43963 0.44123 0.39271 0.34433 0.34435 0.27338 0.27472 0.21037 0.21039 0.15775 0.15777 0.10693 0.10695 0.10698 0.057687 0.057722 0.057756

1.3664 1.3671 1.2627 1.2631 1.0954 1.0956 0.91651 0.91650 0.83837 0.75055 0.75105 0.66983 0.67002 0.57256 0.57261 0.49383 0.49369 0.43963 0.44124 0.39271 0.34433 0.34435 0.27339 0.27473 0.21037 0.21039 0.15775 0.15778 0.10693 0.10696 0.10699 0.057689 0.057724 0.057757

ρexp

Utot(ρexp)

−3

−3

ρeos(T,p) −3

kg·m

kg·m

kg·m

33.460 33.484 30.019 30.031 24.953 24.958 20.053 20.052 18.049 15.881 15.893 13.955 13.959 11.724 11.728 9.9744 9.9707 8.8018 8.8343 7.8028 6.7909 6.7909 5.3326 5.3593 4.0602 4.0609 3.0215 3.0215 2.0348 2.0363 2.0370 1.0901 1.0901 1.0909

0.0687 0.0688 0.0613 0.0613 0.0513 0.0513 0.0426 0.0426 0.0393 0.0358 0.0358 0.0327 0.0328 0.0294 0.0294 0.0268 0.0268 0.0251 0.0252 0.0237 0.0223 0.0223 0.0204 0.0204 0.0188 0.0188 0.0175 0.0175 0.0163 0.0163 0.0163 0.0152 0.0152 0.0152

33.442 33.466 29.999 30.009 24.933 24.937 20.036 20.036 18.033 15.867 15.879 13.947 13.952 11.715 11.717 9.9688 9.9658 8.7952 8.8296 7.7968 6.7839 6.7843 5.3275 5.3547 4.0608 4.0611 3.0216 3.0222 2.0334 2.0338 2.0344 1.0894 1.0900 1.0907

η

η373.15K

μPa·s

μPa·s

9.5141 9.5176 9.4635 9.4641 9.3936 9.3903 9.3291 9.3342 9.3148 9.2978 9.2934 9.2804 9.2832 9.2632 9.2589 9.2474 9.2546 9.2467 9.2430 9.2405 9.2298 9.2339 9.2304 9.2265 9.2281 9.2251 9.2201 9.2171 9.2107 9.2130 9.2100 9.2112 9.2139 9.2071b

9.5141 9.5176 9.4636 9.4643 9.3940 9.3906 9.3295 9.3346 9.3151 9.2979 9.2936 9.2806 9.2834 9.2634 9.2591 9.2477 9.2549 9.2468 9.2431 9.2405 9.2298 9.2339 9.2306 9.2267 9.2282 9.2252 9.2204 9.2175 9.2112 9.2134 9.2104 9.2115 9.2141 9.2073b

a The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006. bInfluenced by slip.

standard deviations σ are somewhat enlarged at higher temperatures for several reasons: measurements near the phase boundary at 423.15 K, in the near-critical region at 428.15 K, and in a high-temperature range (for the experimental equipment) at 448.15 K. For the near-critical isotherm at 428.15 K, it was found that, when all the data from the complete density range ρ ≤ 498 kg· m−3 were considered for the correlation, the best representation was obtained with a power series of the sixth order. In addition, a reasonable value for the coefficient of the initial density dependence, η1, in relation to the values for the other investigated isotherms, resulted (see Table 8). However, it became evident that the data from the density range 0.81 ≤ δ ≤ 1.21 corresponding to 185 ≤ ρ/kg·m−3 ≤ 276 impair considerably the representation of the complete data set. Consequently, these particular data were excluded from the fit of the power series of sixth order to the data set of the isotherm. At last, the viscosity of normal butane for the 428.15 K isotherm is reproduced in Figure 8, as a function of density. The inset of the figure illustrates distinctly the effect of the critical enhancement of the viscosity in the near-critical region. The viscosity data excluded from the fit and depicted as

The values of the critical temperature Tc and of the critical density ρc are those recommended by Bücker and Wagner.1 The application of the reciprocal reduced temperature τ instead of the reduced temperature is of no relevance with regard to eq 4, but τ is utilized in the fundamental equation of state and presumably in a viscosity surface correlation to be generated in 2 the future. Weighting factors of (100·η−1 exp) were used to minimize the relative deviations in the multiple linear leastsquares regression. The adequate representation of the data of a particular isotherm was inspected by means of the weighted standard deviation σ as criterion. For that purpose, the order of the power-series expansion according to eq 4 was successively increased and the quality of the representation was checked. The coefficients ηi(τ) of eq 4 together with their standard deviations σηi and the weighted standard deviation σ are listed in Table 8. The table reveals that for subcritical isotherms, a power series up to the third order is adequate to describe appropriately the viscosity data, whereas for supercritical isotherms, a power series of the sixth order is required. The higher order in the case of the supercritical isotherms results from the larger density range. The values of the weighted H



Article


p

p423.15K,ρexp

K

MPa

MPa

423.476 423.469 424.215 424.213 423.276 423.202 423.189 423.201 423.197 423.198 423.109 423.123 423.134 423.169 423.166 423.189 423.241 423.248 423.260 423.258 423.198 423.190 423.166 423.167 423.163 423.161

3.2808 3.2809 3.1768 3.1770 3.1714 3.1702 3.0060 3.0058 2.7816 2.7811 2.5325 2.5340 2.5356 2.2723 1.4902 1.4900 1.0161 1.0159 0.88412 0.88438 0.65348 0.65426 0.45237 0.45091 0.19554 0.19559

3.2735 3.2737 3.1553 3.1556 3.1689 3.1692 3.0053 3.0049 2.7809 2.7804 2.5330 2.5343 2.5358 2.2721 1.4902 1.4898 1.0158 1.0156 0.88384 0.88410 0.65340 0.65419 0.45235 0.45089 0.19553 0.19558

ρexp kg·m

Utot(ρexp)

−3

kg·m

96.642 96.663 88.401 88.415 89.266 89.280 79.774 79.773 69.058 69.059 59.203 59.251 59.303 50.310 29.071 29.063 18.650 18.648 15.981 15.987 11.519 11.533 7.8076 7.7808 3.2893 3.2893

−3

0.2779 0.2780 0.2284 0.2285 0.2351 0.2353 0.1905 0.1905 0.1510 0.1509 0.1217 0.1218 0.1220 0.0994 0.0569 0.0568 0.0396 0.0396 0.0355 0.0355 0.0289 0.0289 0.0237 0.0236 0.0178 0.0178

ρeos(T,p) kg·m

−3

96.702 96.720 88.577 88.593 89.439 89.459 79.892 79.873 69.107 69.086 59.188 59.238 59.291 50.257 29.014 29.006 18.620 18.616 15.956 15.961 11.502 11.517 7.8001 7.7736 3.2886 3.2895

η

η423.15K

μPa·s

μPa·s

13.063 13.066 12.722 12.728 12.735 12.730 12.341 12.329 11.913 11.912 11.565 11.576 11.574 11.288 10.780 10.779 10.598 10.591 10.555 10.554 10.494 10.496 10.448 10.448 10.400 10.403

13.051 13.054 12.684 12.690 12.730 12.728 12.340 12.327 11.911 11.910 11.566 11.576 11.574 11.287 10.779 10.778 10.596 10.589 10.552 10.551 10.493 10.495 10.447 10.447 10.399 10.403

a The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006.

filled symbols differ from the power series by +0.91 % at most, near to the critical density. The differences of all other data points from the fitted values amount to ±0.06 % at densities ρ < 100 kg·m−3 and are within ±0.11 % at higher densities. Measurements on hydrocarbons in the near-critical region with the aim to identify the critical enhancement of the viscosity are rather limited. Nonetheless, it is well known that large fluctuations in the order parameterin this case, in the density ρbecome dominant in the vicinity of the vapor-liquid critical point of a pure fluid and that the viscosity η and the thermal conductivity λ diverge at the critical point. However, the effect of the critical enhancements of both transport properties is different. The thermal conductivity exhibits a very strong divergence, whereas that of the viscosity is weak and occurs only in a small range of temperatures and densities near to the critical point.25 Two theoretical approaches (see ref 26) are applied to treat the critical enhancement of the transport properties. The dynamic renormalization-group theory enables to predict the asymptotic behavior of the transport properties at the critical point and supports additionally the understanding of the dynamic scaling and of the universality classes. The modecoupling theory of critical dynamics assists rather the handling of the critical enhancement a little outside the critical region and the modeling of the crossover from the critical to the nearcritical region. A simplified closed-form solution of the modecoupling equations was developed by Bhattacharjee et al.27 and recently used for the calculation of the critical enhancement of the viscosity of ethane in the framework of developing a new viscosity surface correlation for ethane.28 This model is also

applicable in the case of the critical enhancement of the viscosity for further hydrocarbons like normal butane. The experimental viscosity data of this work are contrasted with values calculated for the viscosity surface correlations of Younglove and Ely,9 Vogel et al.,2 and of Quiñones-Cisneros and Deiters.10 These comparisons are performed on the basis of the relative standard uncertainties (k = 1), as the correlations are characterized by them. First of all, the viscosity surface correlation by Vogel et al. is discussed because this one is preferably recommended in the software package REFPROP, Version 9.1, from NIST.29 For computing the viscosity values of this correlation, the measured temperature data and assigned values for density are applied. The correlation of Vogel et al. implies the modified Benedict−Webb−Rubin equation of state generated by Younglove and Ely,9 so that the assigned density values were determined from the experimental temperature and pressure data utilizing this equation of state. In addition, the surface correlation by Vogel et al. is based on the residual quantity concept, in which the viscosity is assumed to consist of three contributions η(T , ρ) = η0(T ) + ηR (T , ρ) + ηc(T , ρ)

(5)

In eq 5, η0 is the viscosity in the limit of zero density, which is only a function of temperature, whereas ηR and ηc are the residual viscosity and the critical enhancement of the viscosity, respectively. The viscosity surface correlation of Vogel et al. (see also Küchenmeister and Vogel17) incorporates a term for the viscosity in the limit of zero density but does not include a contribution for the critical enhancement. The residual viscosity I



Article


p

p428.15K,ρexp

ρexp

Utot(ρexp)

ρeos(T,p)

η

K

MPa

MPa

kg·m−3

kg·m−3

kg·m−3

μPa·s

μPa·s

30.023 30.022 30.018 22.974 22.974 22.973 17.077 17.075 12.773 12.732 12.735 9.6611 9.6623 7.4662 7.4671 7.4675 5.9489 5.9496 5.9496 5.1502 5.1499 4.6259 4.6266 4.6268 4.3120 4.3124 4.3124 4.1459 4.1462 4.0673 4.0675 4.0300 4.0301 3.9901 3.9901 3.9848 3.9849 3.9763 3.9761 3.9649 3.9648 3.9535 3.9533 3.9387 3.9392 3.8841 3.8838 3.8051 3.8065 3.7938 3.7909 3.6552 3.6557 3.4271 3.4272 3.2638 3.2637 3.1086 3.1078

30.028 30.026 30.023 22.954 22.955 22.955 17.078 17.077 12.772 12.731 12.734 9.6595 9.6602 7.4632 7.4645 7.4655 5.9514 5.9521 5.9526 5.1583 5.1585 4.6323 4.6318 4.6319 4.3092 4.3095 4.3094 4.1472 4.1472 4.0676 4.0674 4.0265 4.0266 3.9914 3.9914 3.9868 3.9870 3.9748 3.9746 3.9664 3.9659 3.9516 3.9518 3.9365 3.9369 3.8823 3.8822 3.7982 3.7979 3.7921 3.7897 3.6557 3.6560 3.4273 3.4271 3.2637 3.2636 3.1072 3.1065

498.21 498.20 498.20 480.02 480.02 480.02 460.12 460.12 440.28 440.05 440.06 420.21 420.22 399.67 399.68 399.69 378.00 378.01 378.02 360.25 360.26 341.66 341.66 341.65 322.22 322.24 322.25 304.60 304.61 289.00 288.98 274.89 274.90 251.58 251.63 246.50 246.49 230.48 230.45 218.37 218.33 199.12 199.18 185.24 185.30 159.21 159.22 138.83 138.80 137.58 137.62 119.22 119.23 99.322 99.316 88.856 88.854 80.486 80.455

0.5436 0.5436 0.5436 0.5248 0.5248 0.5248 0.5047 0.5047 0.4852 0.4850 0.4850 0.4669 0.4669 0.4509 0.4509 0.4510 0.4416 0.4416 0.4416 0.4482 0.4482 0.4896 0.4895 0.4895 0.6202 0.6200 0.6199 0.9057 0.9052 1.3937 1.3937 2.1047 2.1038 3.9042 3.9013 4.3158 4.3153 5.1063 5.1106 5.2882 5.2764 4.1729 4.1879 2.9757 2.9811 1.3262 1.3270 0.7195 0.7173 0.7001 0.7008 0.4371 0.4371 0.2779 0.2778 0.2226 0.2226 0.1870 0.1869

498.23 498.23 498.22 479.83 479.84 479.84 459.80 459.79 439.94 439.71 439.73 419.99 420.00 399.64 399.66 399.67 378.11 378.12 378.13 360.21 360.22 341.02 341.00 341.01 320.07 320.09 320.09 299.94 299.94 280.73 280.67 262.01 262.04 232.24 232.25 227.25 227.39 213.50 213.38 204.46 203.99 191.02 191.21 180.60 180.87 157.94 157.90 138.91 138.84 137.84 137.42 119.48 119.50 99.574 99.564 89.043 89.041 80.594 80.559

94.979 94.858 95.050 85.034 84.919 84.953 75.611 75.508 67.536 67.406 67.436 60.451 60.474 54.115 54.179 54.130 48.458 48.373 48.440 44.278 44.250 40.382 40.349 40.327 36.731 36.725 36.685 33.737 33.726 31.293 31.316 29.342b 29.340b 26.372b 26.377b 25.759b 25.770b 23.949b 23.937b 22.620b 22.627b 20.635b 20.634b 19.315b 19.326b 17.125 17.119 15.672 15.685 15.567 15.577 14.434 14.442 13.376 13.371 12.886 12.884 12.529 12.521

94.980 94.858 95.051 85.032 84.917 84.951 75.611 75.508 67.536 67.406 67.436 60.451 60.474 54.115 54.179 54.129 48.459 48.374 48.441 44.280 44.252 40.384 40.350 40.329 36.730 36.724 36.683 33.737 33.726 31.293 31.315 29.340b 29.338b 26.373b 26.377b 25.761b 25.772b 23.948b 23.937b 22.620b 22.628b 20.634b 20.633b 19.314b 19.324b 17.123 17.117 15.665 15.676 15.565 15.576 14.435 14.442 13.376 13.371 12.886 12.884 12.526 12.519

428.139 428.138 428.137 428.207 428.206 428.204 428.146 428.146 428.152 428.156 428.156 428.157 428.159 428.165 428.163 428.160 428.135 428.135 428.132 428.094 428.091 428.099 428.109 428.109 428.176 428.177 428.178 428.136 428.140 428.146 428.151 428.196 428.197 428.131 428.130 428.117 428.118 428.176 428.174 428.124 428.130 428.187 428.178 428.195 428.195 428.194 428.190 428.343 428.391 428.197 428.184 428.132 428.142 428.141 428.152 428.156 428.153 428.229 428.222

J

η428.15K



Article

Table 6. continued T

p

p428.15K,ρexp

ρexp

Utot(ρexp)

ρeos(T,p)

η

K

MPa

MPa

kg·m−3

kg·m−3

kg·m−3

μPa·s

η428.15K μPa·s

428.139 428.131 428.129 428.134 428.122 428.120 428.102 428.106 428.209 428.213 428.179 428.176 428.147 428.142 428.107 428.103 428.097 428.100 428.122 428.131 428.134 428.137 428.133 428.136 428.091 428.085 428.105 428.109 428.178 428.172 428.160

2.8628 2.8629 2.6247 2.6246 2.3062 2.3062 1.9641 1.9643 1.5398 1.5401 1.0985 1.0991 1.0407 1.0412 0.90486 0.90490 0.79614 0.79619 0.68503 0.68454 0.57867 0.57876 0.48091 0.48081 0.19434 0.19436 0.13161 0.13164 0.062104 0.062101 0.062099

2.8630 2.8632 2.6250 2.6248 2.3065 2.3065 1.9644 1.9646 1.5395 1.5398 1.0984 1.0990 1.0407 1.0412 0.90497 0.90502 0.79626 0.79630 0.68509 0.68457 0.57870 0.57878 0.48093 0.48083 0.19437 0.19439 0.13163 0.13165 0.062099 0.062098 0.062097

69.471 69.476 60.360 60.353 49.889 49.887 40.163 40.167 29.634 29.642 20.025 20.038 18.847 18.856 16.141 16.138 14.038 14.041 11.939 11.929 9.9774 9.9803 8.2126 8.2112 3.2308 3.2322 2.1741 2.1741 1.0176 1.0183 1.0176

0.1490 0.1491 0.1230 0.1230 0.0976 0.0976 0.0772 0.0772 0.0577 0.0577 0.0417 0.0418 0.0399 0.0399 0.0357 0.0357 0.0326 0.0326 0.0295 0.0295 0.0267 0.0267 0.0242 0.0242 0.0177 0.0177 0.0164 0.0164 0.0151 0.0151 0.0151

69.490 69.499 60.330 60.325 49.816 49.817 40.083 40.088 29.570 29.577 19.989 20.001 18.816 18.826 16.120 16.120 14.020 14.021 11.925 11.915 9.9664 9.9679 8.2040 8.2020 3.2283 3.2287 2.1740 2.1744 1.0194 1.0194 1.0193

12.092 12.091 11.766 11.773 11.438 11.439 11.172 11.176 10.935 10.932 10.755 10.755 10.734 10.730 10.690 10.693 10.663 10.659 10.632 10.631 10.610 10.609 10.590 10.590 10.534 10.537 10.524 10.524 10.512 10.510 10.513

12.092 12.092 11.767 11.773 11.439 11.440 11.174 11.178 10.933 10.930 10.754 10.754 10.734 10.730 10.691 10.694 10.665 10.660 10.632 10.632 10.611 10.609 10.590 10.591 10.536 10.539 10.525 10.525 10.512 10.509 10.512

a The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006. bInfluenced by the near-critical region.

for all subcritical isotherms and increase with rising density. The deviations at the lowest densities ρ ≤ 2.91 kg·m−3 amount to −(0.38 to 0.65) % for all isotherms, mostly larger than the relative standard uncertainty of the correlation in this region (0.4 %). In the adjacent density range 2.91 < ρ/kg·m−3 ≤ 5.81, the deviations are −0.76 % at most and, hence, lower than the estimated relative standard uncertainty in this region (1.0 %). Only two measuring points at 373.15 K (−3.08 % and −3.11 % at ρ = 33.5 kg·m−3) and all experimental points at 423.15 K with densities ρ ≥ 69 kg·m−3 (up to −4.16 %) exceed the relative standard uncertainty of the viscosity correlation (3.0 %). The experimental data of the first supercritical isotherm at 428.15 K are distinguished by alternating deviations with a minimum at ρ = 138 kg·m−3 (−4.25 %) and a maximum at ρ = 252 kg·m−3 (+7.99 %), whereas the deviation at the highest density ρ = 498 kg·m−3 amounts to +0.26 % only. It is additionally demonstrated in Figure 9 that, in the density range 185 ≤ ρ/kg·m−3 < 305, the deviations of this isotherm overshoot considerably the critical enhancement of the viscosity of about +1 %, indicated prior to this in the inset of Figure 8. This strong enlargement can not only be explained by the critical enhancement of the viscosity. A substantial reason for this consists also in the density, whose experimental data determined in this work differ distinctly from the values calculated using the equation of state of Younglove and Ely9

is separated into a term for the initial-density dependence and higher-density contributions consisting of double power-series expansions in reduced density and reciprocal reduced temperature and of a free-volume term with a temperature-dependent close-packed density. The correlation was established from the triple-point temperature (Ttr,n‑C4H10 = 134.895 K) to 500 K and up to pressures of 70 MPa in accordance with the equation of state of Younglove and Ely and up to temperatures of 600 K in the limit of zero density. The relative standard uncertainty (k = 1) of the viscosity surface correlation was stated to be 0.4 % in the region 298 ≤ T/K ≤ 600 and ρ ≤ 2.32 kg·m−3 as well as 1.0 % for 273 ≤ T/K ≤ 600 and 2.32 < ρ/kg·m−3 ≤ 5.81. In further regions for which both the equation of state by Younglove and Ely was developed and primary experimental data were determined, the uncertainty was assessed to be 3.0 %. These uncertainties were estimated in accord with those assumed for the selected primary experimental data sets. Moreover, the relative standard uncertainty was supposed to be 6.0 %, when primary experimental data were not available but the equation of state is valid. At last, the uncertainty was further raised to 8.0 % in those ranges in which the equation of state is not applicable. Figure 9 shows the comparison between the experimental data of this work and values calculated from the viscosity surface correlation by Vogel et al.2 Negative deviations appear K



Article


p

p448.15K,ρexp

ρexp

Utot(ρexp)

ρeos(T,p)

η

K

MPa

MPa

kg·m−3

kg·m−3

kg·m−3

μPa·s

μPa·s

448.150 448.136 448.168 448.164 448.142 448.145 448.155 448.167 448.150 448.154 448.171 448.171 448.163 448.163 448.139 448.135 448.162 448.161 448.111 448.111 448.279 448.274 448.146 448.146 448.177 448.173 448.133 448.132 448.113 448.112 448.090 448.129 448.227 448.226 448.172 448.184 448.127 448.129 448.184 448.185 448.184 448.182 448.153 448.156 448.159 448.159 448.168 448.162 448.157 448.154 448.169 448.168 448.156 448.146 448.148 448.147 448.158 448.161 448.163

29.472 29.460 23.154 23.152 18.184 18.187 14.319 14.321 11.528 11.529 9.5386 9.5374 8.1406 8.1394 7.1910 7.1898 6.5514 6.5511 6.1448 6.1450 5.7851 5.7846 5.5519 5.5537 5.3467 5.3466 5.2032 5.2035 5.0548 5.0545 4.8462 4.8470 4.7271 4.7270 4.4983 4.4981 4.1616 4.1616 3.7478 3.7479 3.4339 3.4337 3.0733 3.0733 2.5118 2.5121 2.0730 2.0737 1.6622 1.6619 1.1584 1.1584 0.80776 0.80760 0.61860 0.61843 0.50850 0.50969 0.43017

29.472 29.465 23.148 23.148 18.186 18.188 14.318 14.317 11.528 11.528 9.5349 9.5337 8.1386 8.1374 7.1925 7.1918 6.5500 6.5499 6.1487 6.1490 5.7738 5.7737 5.5522 5.5540 5.3449 5.3451 5.2043 5.2047 5.0568 5.0566 4.8490 4.8479 4.7239 4.7239 4.4975 4.4969 4.1622 4.1622 3.7471 3.7472 3.4333 3.4332 3.0732 3.0732 2.5117 2.5120 2.0729 2.0736 1.6622 1.6619 1.1583 1.1584 0.80774 0.80761 0.61860 0.61844 0.50849 0.50968 0.43016

478.91 478.91 460.09 460.08 440.74 440.74 420.46 420.46 400.23 400.23 380.13 380.12 359.99 359.98 340.37 340.36 321.12 321.12 304.18 304.17 281.66 281.65 263.05 263.02 240.34 240.34 222.02 222.03 201.47 201.47 174.16 174.10 160.07 160.07 138.72 138.68 114.81 114.81 92.905 92.910 79.739 79.735 66.943 66.941 50.321 50.329 39.287 39.305 30.086 30.078 19.932 19.932 13.456 13.453 10.141 10.135 8.2591 8.2800 6.9362

0.5248 0.5248 0.5058 0.5058 0.4867 0.4867 0.4677 0.4677 0.4506 0.4506 0.4369 0.4369 0.4297 0.4297 0.4344 0.4344 0.4580 0.4580 0.4998 0.4998 0.5814 0.5816 0.6625 0.6628 0.7427 0.7428 0.7745 0.7746 0.7543 0.7543 0.6357 0.6346 0.5483 0.5483 0.4189 0.4186 0.2956 0.2956 0.2088 0.2088 0.1673 0.1673 0.1330 0.1330 0.0956 0.0956 0.0741 0.0741 0.0579 0.0579 0.0414 0.0414 0.0316 0.0316 0.0269 0.0269 0.0243 0.0243 0.0225

478.75 478.73 459.73 459.73 440.29 440.30 420.07 420.06 400.10 400.10 380.15 380.13 360.11 360.09 340.25 340.23 320.37 320.37 302.56 302.57 278.72 278.71 259.41 259.59 236.86 236.88 219.32 219.37 199.97 199.95 173.94 173.81 160.11 160.10 139.01 138.96 115.07 115.07 93.010 93.014 79.739 79.734 66.875 66.874 50.217 50.225 39.192 39.209 30.010 30.004 19.889 19.890 13.438 13.436 10.128 10.125 8.2499 8.2700 6.9352

84.890 84.969 75.926 76.013 68.113 68.129 61.015 60.934 54.780 54.772 49.405 49.478 44.661 44.736 40.611 40.552 36.942 36.946 34.096 34.096 30.664 30.635 28.113 28.113 25.338 25.331 23.312 23.331 21.240 21.224 18.865 18.847 17.759 17.770 16.265 16.252 14.809 14.794 13.675 13.673 13.079 13.074 12.570 12.570 12.009 12.000 11.687 11.689 11.460 11.459 11.254 11.254 11.147 11.142 11.093 11.100 11.072 11.075 11.062

84.890 84.969 75.926 76.013 68.113 68.129 61.015 60.934 54.780 54.772 49.405 49.478 44.661 44.736 40.611 40.552 36.942 36.946 34.096 34.096 30.664 30.635 28.113 28.113 25.338 25.331 23.312 23.331 21.240 21.224 18.865 18.847 17.759 17.770 16.265 16.252 14.809 14.794 13.674 13.672 13.078 13.073 12.570 12.570 12.009 12.000 11.686 11.688 11.460 11.459 11.254 11.254 11.146 11.142 11.093 11.100 11.072 11.074 11.061

L

η448.15K



Article

Table 7. continued T

p

p448.15K,ρexp

ρexp

Utot(ρexp)

ρeos(T,p)

η

K

MPa

MPa

kg·m−3

kg·m−3

kg·m−3

μPa·s

μPa·s

448.163 448.154 448.156

0.43102 0.22017 0.22038

0.43100 0.22017 0.22037

11.063 11.020 11.012

11.063 11.020 11.012

6.9500 3.4915 3.4965

0.0225 0.0180 0.0180

6.9492 3.4917 3.4950

η448.15K

a

The relative combined expanded uncertainties (k = 2) in temperature and pressure are Ur,c(T) = 0.0002 and Ur,c(p) = 0.001. The relative total expanded uncertainty (k = 2) in viscosity is Ur,tot(η) = 0.006.

Figure 3. Comparison of the experimental density data ρexp of this work for normal butane with values ρeos calculated for the equation of state by Bücker and Wagner1 using the measured temperatures and pressures, as a function of density ρ: ▼, 298.15 K; △, 323.15 K; ▽, 348.15 K; ★, 373.15 K; ○, 423.15 K; ◇, 428.15 K; □, 448.15 K; ······, relative total expanded uncertainty (k = 2) in the density measurement for normal butane at 428.15 K;  , relative expanded uncertainty in density of the equation of state by Bücker and Wagner; deviations, Δ = 100·(ρexp − ρeos)/ρeos.

Figure 4. Comparison of the experimental density data ρexp of this work for normal butane at 428.15 K with values ρeos calculated for different equations of state using the measured temperatures and pressures, as a function of density ρ: □, equation of state by Younglove and Ely;9 △, equation of state by Miyamoto and Watanabe;19 ▽, equation of state by Span and Wagner;16 ◇, equation of state by Bücker and Wagner;1······, relative total expanded uncertainty (k = 2) in the density measurement for normal butane at 428.15 K;  , relative expanded uncertainty in density of the equation of state by Bücker and Wagner; deviations, Δ = 100·(ρexp − ρeos)/ρeos.

(see also Figure 4). It is to take into account that the viscosity surface correlation by Vogel et al. utilizes density values calculated from the equation of state of Younglove and Ely. For the isotherm at 448.15 K in accordance to that at 428.15 K, Figure 9 displays alternating deviations with a minimum at ρ = 50.3 kg·m−3 (−0.84 %) and a maximum at ρ = 282 kg·m−3 (+4.43 %) followed by a comparably small deviation at the highest density ρ = 479 kg·m−3 (+0.79 %). The deviations for the two supercritical isotherms do not exceed the relative standard uncertainties of the viscosity surface correlation, estimated to be 3.0 % and 6.0 % in the respective regions, apart from the data for 428.15 K at densities 88.9 ≤ ρ/kg·m−3 < 159 and 185 ≤ ρ/kg·m−3 < 305. The present experimental data were also checked against viscosity values of the older correlation by Younglove and Ely,9 calculated using the experimental temperatures and density values computed from the measured temperature and pressure data applying the equation of state by Younglove and Ely. This viscosity correlation was characterized by a relative standard uncertainty (k = 1) of 2 %, overall in the fluid region except for the near-critical region, for which 5 % were supposed. The validity range of the correlation extends from the triple-point temperature to 600 K and to pressures of 70 MPa in accordance with the concurrently proposed equation of state. The viscosity correlation based on the residual quantity concept includes a contribution for the viscosity in the limit of zero

density but not a specific term for the critical enhancement. The residual viscosity implies a single temperature-independent value as contribution of the initial-density dependence and an empirical approach with exponential terms as higher-density contributions. Figure 10 reveals that for the four subcritical isotherms at 298.15 K, 323.15 K, 348.15 K, and 373.15 K, only negative deviations of the experimental data from the calculated values appear, increasing with rising density. At 373.15 K, the maximum deviation is −7.08 % at ρ = 33.5 kg·m−3, which is approximately twice the deviation for the correlation of Vogel et al.2 The last subcritical isotherm at 423.15 K is characterized by negative deviations with a maximum of −5.84 % at ρ = 59.2 kg·m−3 and with −4.64 % at the highest density ρ = 96.6 kg· m−3. The two supercritical isotherms 428.15 K and 448.15 K exhibit alternating deviations with minima at ρ = 60.4 kg·m−3 (−5.44 %) and at ρ = 50.3 kg·m−3 (−4.64 %) and with maxima at ρ = 252 kg·m−3 (+16.3 %) and ρ = 263 kg·m−3 (+7.33 %). It is obvious that the maxima occur at higher densities than the critical one (ρc = 228 kg·m−3). The large difference of about +9 % in the maxima of the deviations for both supercritical isotherms does not only account for the critical enhancement, which is not considered in the viscosity correlation by Younglove and Ely but is to a certain degree due to the calculated densities using the equation of state by Younglove and Ely (see the paragraph above). The deviations at high M



Article

Figure 7. Comparison of the experimental pressure data pexp of Beattie et al.24 for normal butane in the immediate vicinity of the critical point with values peos calculated for the equation of state by Bücker and Wagner1 using the measured temperatures and densities, as a function of density ρ. ×, 425.04 K; #, 425.09 K; $, 425.14 K; %, 425.15 K; &, 425.16 K; §, 425.18 K; ?, 425.21 K; ß, 425.24 K; deviations, Δ = 100· (pexp − peos)/peos.

Figure 5. Comparison of experimental density data ρexp for normal butane at temperatures near the critical one with values ρeos calculated for the equation of state by Bücker and Wagner1 using the measured temperatures and pressures, as a function of density ρ. Beattie et al.:20 ⊕, 423.13 K; ⊞, 425.15 K. Gupta and Eubank:21 □, 425 K. Miyamoto and Uematsu:22 △, 424 K; ▲, 424 K, boiling curve. Miyamoto and Uematsu:23 ▽, 420 K. This work: ○, 423.15 K; ◇, 428.15 K. ······, relative total expanded uncertainty (k = 2) in the density measurement for normal butane at 428.15 K;  , relative expanded uncertainty in density of the equation of state by Bücker and Wagner. Deviations, Δ = 100·(ρexp − ρeos)/ρeos.

by Quiñones-Cisneros et al.30 and based on different equations of state. In this concept, the viscosity in the limit of zero-density is separated from the remaining f riction term. QuiñonesCisneros and Deiters utilized the equation of state by Span and Wagner16 to correlate the viscosity of normal butane in the temperature range 145 K to 500 K at pressures up to 68 MPa. They adjusted 12 parameters of their Friction-Theory model using the database recommended by Zéberg-Mikkelsen,31 which differs partly from that used by Vogel et al.2 Unfortunately, Quiñones-Cisneros and Deiters did not state an uncertainty for their viscosity correlation. In fact, they provided the information that the absolute average deviation (AAD) of the viscosities, calculated for their correlation, from the recommended data of Zéberg-Mikkelsen is 0.80 % with a maximum value of 5 %. Because the AAD value does only characterize the quality of the fit when generating the correlation but not the uncertainty of the applied experimental data, the relative standard uncertainty is probably not lower than that for the correlation of Vogel et al. (3 %). The deviations for the four lowest isotherms between 298.15 K and 373.15 K increase with rising density and are between −0.80 % at the lowest densities and +2.49 % at the higher densities up to ρ = 25.0 kg·m−3. In contrast, the opposite behavior was found for the two other viscosity correlations (see Figures 9 and 10). The last subcritical isotherm at 423.15 K is distinguished by that the deviations enlarge to a maximum of +3.08 % at ρ = 29.1 kg·m−3 followed by a decrease to −2.85 % at the highest density ρ = 96.6 kg·m−3. The two supercritical isotherms at 428.15 K and 448.15 K exhibit alternating deviations with maxima and minima, such as for 428.15 K: +3.60 % at ρ = 40.2 kg·m−3, −6.73 % at ρ = 159 kg·m−3, and +2.78 % at ρ = 322 kg· m−3. The deviations decrease at further increasing density. The more strongly alternating deviations at 428.15 K in the density range 100 ≤ ρ/kg·m−3 < 300 compared to those at 448.15 % are not due to the critical enhancement of the viscosity because no maximum occurs near the critical density. This unexpected behavior of the deviations may be related to the viscosity data which were used in this region by Quiñones-Cisneros and Deiters to adjust the parameter of their Friction Theory model.

Figure 6. Comparison of the measured pressures pexp of this work for normal butane at 428.15 K with values peos calculated for different equations of state using the measured temperatures and densities, as a function of density ρ: □, equation of state by Younglove and Ely;9 △, equation of state by Miyamoto and Watanabe;19 ▽, equation of state by Span and Wagner;16 ◇, equation of state by Bücker and Wagner;1  , relative expanded uncertainty (k = 2) in pressure of the equation of state by Bücker and Wagner; deviations, Δ = 100·(pexp − peos)/peos.

densities for the isotherms at 428.15 K and 448.15 K are within the assumed relative standard uncertainty of this viscosity correlation (2 %). Figure 11 illustrates the deviations of the experimental data of this work from values derived from the viscosity correlation by Quiñ ones-Cisneros and Deiters10 using experimental pressure and temperature data. Quiñones-Cisneros and Deiters used a generalization of the Friction Theory concept developed N



Article

Table 8. Coefficients of Equation 4 for the Viscosity Measurements on Normal Butane T

n

−3

K 298.15 323.15 348.15 373.15 423.15 428.15 448.15

ρmax kg·m

1 2 2 2 3 6 6

5.56 10.98 19.59 33.48 96.66 498.2 478.9

η0 ± ση0 μPa·s 7.395 8.002 8.598 9.210 10.378 10.505 10.987

± ± ± ± ± ± ±

η1 ± ση1

η2 ± ση2

η3 ± ση3

μPa·s

μPa·s

μPa·s

± ± ± ± ± ±

−1.960 ± 0.748 −2.286 ± 0.698 −8.482 ± 0.927

−2.437 −1.354 −0.082 0.442 1.635 1.868 1.892

0.001 0.001 0.002 0.001 0.004 0.002 0.004

± ± ± ± ± ± ±

0.072 0.126 0.109 0.045 0.092 0.051 0.079

7.866 8.223 11.165 11.930 11.331 13.722

2.360 1.060 0.307 0.499 0.306 0.429

T

η4 ± ση4

η5 ± ση5

η6 ± ση6

K

μPa·s

μPa·s

μPa·s

298.15 323.15 348.15 373.15 423.15 428.15 448.15

3.568 ± 0.700 9.317 ± 0.926

−2.289 ± 0.315 −4.591 ± 0.430

σ (weighted)

0.761 ± 0.053 1.098 ± 0.075

0.025 0.027 0.031 0.032 0.050 0.052 0.059

Figure 9. Comparison of experimental viscosity data ηexp for normal butane with calculated values ηcor using the correlation of Vogel et al.2 and measured values for temperature and assigned density values, calculated from experimental pressures and temperatures using the equation of state of Younglove and Ely:9 ▼, 298.15 K; ▲, 323.15 K; ▽, 348.15 K; △, 373.15 K; ○, 423.15 K; ◇, 428.15 K; □, 448.15 K;  , relative standard uncertainty (k = 1) of the correlation of Vogel et al.; deviations, Δ = 100·(ηexp − ηcor)/ηcor.

Figure 8. Viscosity of normal butane at 428.15 K, as a function of density ρ: ○, measured data ηexp; ●, measured data ηexp influenced by the critical enhancement; , fitted values ηfit according to a powerseries expansion of sixth order in the reduced density δ (eq 4, Table 8); deviations in the inset, Δ = 100·(ηexp − ηfit)/ηfit.

Finally, Figure 12 shows a comparison of the viscosity data of this work extrapolated to the limit of zero density (η0 values of Table 8) with values obtained from the viscosity surface correlation of Vogel et al.2 The plotted error bars represent the relative standard uncertainty (k = 1) of the present experimental data (0.3 %), whereas the dashed lines at ±0.4 % mark the relative standard uncertainty of the correlation in the low-density region. Experimental data and correlated values agree within their mutual uncertainties, as the deviations are between −0.32 % and −0.55 %. Furthermore, re-evaluated data of Küchenmeister and Vogel17(see Table B.11 in ref 32), likewise extrapolated to this limit, are displayed in the figure. These values differ by about −0.2 % from the correlated values of Vogel et al. but feature a very good agreement with the temperature function of the correlated values. This is attributed to the fact that Vogel et al., when developing their viscosity surface correlation, included the data of Küchenmeister and Vogel extrapolated to the limit of zero density, but not

corrected. The re-evaluation of the data measured by Küchenmeister and Vogel applies to the calibration of the employed oscillating-disk viscometer, which was performed originally with a reference value of Kestin et al.33 from 1972, nowadays out of date. Today, a specific value for the zerodensity viscosity of argon at room temperature should be utilized. This value is theoretically calculated with a relative standard uncertainty (k = 1) of 0.1 % inferred from an ab initio potential energy curve for the argon atom pair on the basis of the kinetic theory of dilute gases by Vogel et al.34 The shift of the re-evaluated data is due to the difference between old and new reference values at 298.15 K (−0.217 %). In addition, Figure 12 presents deviations of values calculated for the correlations by Younglove and Ely9 as well as by QuiñonesCisneros and Deiters10 from those for the correlation of Vogel O



Article

Figure 10. Comparison of experimental viscosity data ηexp for normal butane with calculated values ηcor using the correlation of Younglove and Ely9 and measured values for temperature and assigned density values, calculated from experimental pressures and temperatures using the equation of state of Younglove and Ely:9 ▼, 298.15 K; ▲, 323.15 K; ▽, 348.15 K; △, 373.15 K; ○, 423.15 K; ◇, 428.15 K; □, 448.15 K.  , relative standard uncertainty (k = 1) of the correlation of Younglove and Ely; deviations, Δ = 100·(ηexp − ηcor)/ηcor.

Figure 12. Comparison of experimental viscosity data ηexp for normal butane and of viscosity values calculated from correlations of the literature ηcor at low densities and in the limit of zero density, respectively, with calculated values ηcor,Vogel using the correlation of Vogel et al.,2 as a function of temperature T. Experimental data: □, Abe et al.39 (atmospheric pressure); ◇, Küchenmeister and Vogel17 (zero density), corrected (see Table B.11 in ref 32); ●, this work (zero density). Correlations: −·−, Younglove and Ely;9 −··−, QuiñonesCisneros and Deiters;10  , relative standard uncertainty (k = 1) for the correlation of Vogel et al. at low density; deviations, Δ = 100· (ηexp/cor − ηcor,Vogel)/ηcor,Vogel. Error bars: 0.3 %, relative total standard uncertainty (k = 1) in the viscosity measurement of this work.

Vogel et al. The distinctly different temperature functions of the correlations are not alone due to the experimental data of Abe et al., used by Quiñones-Cisneros and Deiters to adjust the coefficients of their zero-density correlation, but also to the coupling of the zero-density correlation with a specific linear Friction Theory model.

5. CONCLUSIONS A vibrating-wire viscometer and a single-sinker densimeter were applied to carry out simultaneous measurements of density and viscosity on gaseous normal butane. Seven isothermal series of measurements were taken between 298.15 K and 448.15 K at pressures up to 91 % of the saturation pressure for subcritical isotherms or up to 30 MPa for supercritical isotherms. The relative total expanded uncertainties (k = 2) are 0.2 % at ρ > 15 kg·m−3 for density and 0.6 % for viscosity. With regard to the viscosity, it is advantageous to determine simultaneously the density, as its values are required in evaluating the viscosity measurements. Thus, a low uncertainty in viscosity yields from one in density. The new density data were checked against values computed for the reference equation of state of Bücker and Wagner1 utilizing the concurrently measured values for pressure and temperature. The deviations in density from the reference equation of state are within ±0.4 %, excluding the low-density and the near-critical regions. However, deviations up to +8.5 % arise in the near-critical region of the isotherm at 428.15 K. They overrun the relative total expanded uncertainty of the single-sinker densimeter, considering the allocation errors for temperature and pressure at this temperature (2.4 %). The experimental data at 428.15 K were also compared with density values derived from the older equations of state of Younglove and Ely,9 of Miyamoto and Watanabe,19 and of Span and Wagner.16 Experimental data and correlated values agree within

Figure 11. Comparison of experimental viscosity data ηexp for normal butane with calculated values ηcor using the correlation of QuiñonesCisneros and Deiters10 and measured values for temperature and assigned density values, calculated from experimental pressures and temperatures using the equation of state of Span and Wagner:16 ▼, 298.15 K; ▲, 323.15 K; ▽, 348.15 K; △, 373.15 K; ○, 423.15 K; ◇, 428.15 K; □, 448.15 K.  , relative standard uncertainty (k = 1) of the correlation of Quiñones-Cisneros and Deiters (assumed in this work); deviations, Δ = 100·(ηexp − ηcor)/ηcor.

et al. Note the correlation of Younglove and Ely, generated at NIST, Boulder, U.S.A., was widely based on data determined in the group of Kestin et al. (see ref 35 to 40). The deviations in Figure 12, plotted for the data measured by Abe et al.39 at atmospheric pressure, affirm roughly this finding. The temperature functions of the correlations of Younglove and Ely as well as of Vogel et al. are mutually consistent within the standard uncertainty (0.4 %) between room temperature and 600 K. Differences of −0.70 % (250 K) to +1.38 % (500 K) appear between the values for the viscosity surface correlation by Quiñones-Cisneros and Deiters and the correlated values of P



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±0.65 % at densities ρ < 120 kg·m−3 and ρ > 340 kg·m−3, whereas in the near-critical region large deviations of +(6.0 to 7.7) % appear. New experimental density data of Miyamoto and Uematsu,22 determined at 424 K after developing the reference equation of state by Bücker and Wagner, yield deviations from the reference equation in accord with the data of this work. Older experimental data of Beattie et al.,24 measured in the immediate vicinity of the critical point and used by Bücker and Wagner as primary data, have a strong influence on the reference equation of state in this specific thermodynamic region. This is evident from the fact that deviations of ±0.04 % result for most of the pressure data of Beattie et al. when comparing with pressures computed from the reference equation of state. In contrast, the expanded uncertainty in pressure of 0.5 % estimated for the equation of state by Bücker and Wagner in the near-critical region is ten times larger than these deviations. The weighting of the data of Beattie et al. was obviously overestimated, as they were the only in the immediate vicinity of the critical point. Consequently, other experimental data somewhat away from the critical point are not appropriately described, in particular data that were not available when developing the equation of state. The new data of Miyamoto and Uematsu22 and of this work as well as the handling of the data of Beattie et al. lead to the conclusion that a further improvement of the presently applied reference equation of state could be reasonable. A critical enhancement of about +0.9 % at maximum, adequate with regard to sign and magnitude, was found for the isotherm 428.15 K. A fit of a power-series expansion of sixth order in the reduced density δ to the experimental viscosity data of this isotherm revealed that the values in the density range 0.81 ≤ δ ≤ 1.21 had to be excluded to represent appropriately the remaining data set. The new viscosity data of this paper were compared with the viscosity surface correlations of Vogel et al.,2 of Younglove and Ely,9 and of Quiñones-Cisneros and Deiters.10 The correlations of Vogel et al. and of Younglove and Ely utilize the modified Benedict−Webb−Rubin equation of state, proposed by Younglove and Ely9 and today out of date, whereas the correlation of Quiñones-Cisneros and Deiters applies the equation of state of Span and Wagner.16 None of them uses the reference equation of state by Bücker and Wagner,1 which should be an essential component in a new viscosity surface correlation. All three correlations do not include a separate contribution for the critical enhancement of the viscosity. The missing critical enhancement, established in this work, is a further reason for developing a new viscosity correlation. The residual quantity concept was employed for the correlations of Vogel et al. and of Younglove and Ely, whereas the correlation of QuiñonesCisneros and Deiters was based on the Friction Theory concept. The deviations of the present data, for the isotherms 423.15 K and 428.15 K at densities 70 < ρ/kg·m−3 < 170 as well as for the isotherms 428.15 K and 448.15 K at densities ρc < ρ/ kg·m−3 < 350, from the values for the viscosity surface correlation of Vogel et al.2 exceed the relative standard uncertainty (k = 1) of the correlation (3 %) in this thermodynamic region. The main reason for the large deviations, particularly for the first supercritical isotherm 428.15 K, consists in that only a few primary data points were available in the range 100 < ρ/kg·m−3 < 300 when this correlation was generated. With respect to the correlation of Younglove and Ely,9 the deviations indicate an analog pattern but with still higher differences. This is completely inconsistent

with the relative standard uncertainty of 2 % supposed for this correlation. The viscosity correlation of Quiñones-Cisneros and Deiters10 differs distinctly from the other two correlations, particularly in the low-density and in the near-critical regions. The deviations of the present viscosity data from the values for this correlation are characterized by an alternating behavior with maxima up to +4.3 % and minima down to −6.7 % in the density range 100 < ρ/kg·m−3 < 300. It is obvious that the three viscosity correlations differ from one another. None of them is suitable to represent appropriately the experimental data of this work which enable to improve the reliability and to reduce the uncertainty of a prospective viscosity surface correlation for normal butane.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Bücker, D.; Wagner, W. Reference Equations of State for the Thermodynamic Properties of Fluid Phase n-Butane and Isobutane. J. Phys. Chem. Ref. Data 2006, 35, 929−1019. (2) Vogel, E.; Küchenmeister, C.; Bich, E. Viscosity Correlation for nButane in the Fluid Region. High Temp. - High Pressures 1999, 31, 173−186. (3) Seibt, D. Schwingdrahtviskosimeter mit integriertem Ein-SenkkörperDichtemessverfahren für Untersuchungen an Gasen in größeren Temperatur- und Druckbereichen; Fortschr.-Ber. VDI, Reihe 6, Nr. 571, VDIVerlag: Düsseldorf, 2008. (4) Seibt, D.; Herrmann, S.; Vogel, E.; Bich, E.; Hassel, E. Simultaneous Measurements on Helium and Nitrogen with a Newly Designed Viscometer-Densimeter over a Wide Range of Temperature and Pressure. J. Chem. Eng. Data 2009, 54, 2626−2637. (5) Wilhelm, J.; Vogel, E.; Lehmann, J. K.; Wakeham, W. A. A Vibrating-Wire Viscometer for Dilute and Dense Gases. Int. J. Thermophys. 1998, 19, 391−401. (6) Wilhelm, J.; Vogel, E. Viscosity Measurements on Gaseous Argon, Krypton, and Propane. Int. J. Thermophys. 2000, 21, 301−318. (7) Seibt, D.; Voß, K.; Herrmann, S.; Vogel, E.; Hassel, E. Simultaneous Viscosity-Density Measurements on Ethane and Propane over a Wide Range of Temperature and Pressure Including the Near-Critical Region. J. Chem. Eng. Data 2011, 56, 1476−1493. (8) Herrmann, S.; Hassel, E.; Vogel, E. Viscosity and Density of Isobutane Measured in Wide Ranges of Temperature and Pressure Including the Near-Critical Region. AIChE J. 2015, 61, 3116−3137. (9) Younglove, B. A.; Ely, J. F. Thermophysical Properties of Fluids. II. Methane, Ethane, Propane, Isobutane, and Normal Butane. J. Phys. Chem. Ref. Data 1987, 16, 577−798. (10) Quiñones-Cisneros, S. E.; Deiters, U. K. Generalization of the Friction Theory for Viscosity Modeling. J. Phys. Chem. B 2006, 110, 12820−12834. (11) Bich, E.; Hellmann, R.; Vogel, E. Ab Initio Potential Energy Curve for the Helium Atom Pair and Thermophysical Properties of the Dilute Helium Gas. II. Thermophysical Standard Values for LowDensity Helium. Mol. Phys. 2007, 105, 3035−3049. (12) Klimeck, J.; Kleinrahm, R.; Wagner, W. An Accurate SingleSinker Densimeter and Measurements of the (p,ρ,T) Relation of Argon and Nitrogen in the Temperature Range from (235 to 520) K at Pressures up to 30 MPa. J. Chem. Thermodyn. 1998, 30, 1571−1588. (13) Lösch, H. W. Entwicklung und Aufbau von neuen Magnetschwebewaagen zur berührungsfreien Messung vertikaler Kräfte; Fortschr.Ber. VDI, Reihe 3, Nr. 138, VDI-Verlag: Düsseldorf, 1987. (14) Lösch, H. W.; Kleinrahm, R.; Wagner, W. Neue Magnetschwebewaagen für gravimetrische Messungen in der Verfahrenstech-

Q



Article

nik; In: VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen (GVC). Jahrbuch; VDI-Verlag: Düsseldorf, 1994; pp 117−137. (15) Kunz, O.; Wagner, W. The GERG-2008 Wide-Range Equation of State for Natural Gases and Other Mixtures: An Expansion of GERG-2004. J. Chem. Eng. Data 2012, 57, 3032−3091. (16) Span, R.; Wagner, W. Equations of State for Technical Applications. II. Results for Nonpolar Fluids. Int. J. Thermophys. 2003, 24, 41−109. (17) Küchenmeister, C.; Vogel, E. The Viscosity of Gaseous n-Butane and Its Initial Density Dependence. Int. J. Thermophys. 1998, 19, 1085−1097. (18) Glos, S.; Kleinrahm, R.; Wagner, W. Measurement of the (p,ρ,T) Relation of Propane, Propylene, n-Butane, and Isobutane in the Temperature Range from (95 to 340) K at Pressures up to 12 MPa Using an Accurate Two-Sinker Densimeter. J. Chem. Thermodyn. 2004, 36, 1037−1059. (19) Miyamoto, H.; Watanabe, K. Thermodynamic Property Model for Fluid-Phase n-Butane. Int. J. Thermophys. 2001, 22, 459−475. (20) Beattie, J. A.; Simard, G. L.; Su, G.-J. The Compressibility of and an Equation of State for Gaseous Normal Butane. J. Am. Chem. Soc. 1939, 61, 26−27. (21) Gupta, D.; Eubank, P. T. Density and Virial Coefficients of Gaseous Butane from 265 to 450 K at Pressures to 3.3 MPa. J. Chem. Eng. Data 1997, 42, 961−970. (22) Miyamoto, H.; Uematsu, M. Measurements of Vapour Pressures and Saturated-Liquid Densities for n-Butane at T = (280 to 424) K. J. Chem. Thermodyn. 2007, 39, 827−832. (23) Miyamoto, H.; Uematsu, M. The (p,ρ,T,x) Properties of (x1 Propane + x2 n-Butane) with x1 = (0.0000, 0.2729, 0.5021, and 0.7308) over the Temperature Range from (280 to 440) K at Pressures from (1 to 200) MPa. J. Chem. Thermodyn. 2008, 40, 240− 247. (24) Beattie, J. A.; Simard, G. L.; Su, G.-J. The Vapor Pressure and Critical Constants of Normal Butane. J. Am. Chem. Soc. 1939, 61, 24− 26. (25) Sengers, J. V. Transport Properties of Fluids Near Critical Points. Int. J. Thermophys. 1985, 6, 203−232. (26) Hohenberg, P. C.; Halperin, B. I. Theory of Dynamic Critical Phenomena. Rev. Mod. Phys. 1977, 49, 435−479. (27) Bhattacharjee, J. K.; Ferrell, R. A.; Basu, R. S.; Sengers, J. V. Crossover Function for the Critical Viscosity of a Classical Fluid. Phys. Rev. A: At., Mol., Opt. Phys. 1981, 24, 1469−1475. (28) Vogel, E.; Span, R.; Herrmann, S. Reference Correlation for the Viscosity of Ethane. J. Phys. Chem. Ref. Data 2015, 44, 043101. (29) Lemmon, E. W.; Huber, M. L.; McLinden, M. O. REFPROP: Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference Database 23, Version 9.1; Standard Reference Data Program; NIST: Gaithersburg, MD, 2013. (30) Quiñones-Cisneros, S. E.; Zéberg-Mikkelsen, C. K.; Stenby, E. H. The Friction Theory (F-Theory) for Viscosity Modeling. Fluid Phase Equilib. 2000, 169, 249−276. (31) Zéberg-Mikkelsen, C. K. Viscosity Study of Hydrocarbon Fluids at Reservoir Conditions. Modeling and Measurements; Ph.D. Dissertation, Technical University of Denmark, Department of Chemical Engineering, Lyngby, Denmark, 2001. (32) Herrmann, S. Viskositäts- und Dichtemessungen an n-Butan und Isobutan in größeren Temperatur- und Druckbereichen; Fortschr.-Ber. VDI, Reihe 6, Nr. 615, VDI-Verlag: Düsseldorf, 2015. (33) Kestin, J.; Ro, S. T.; Wakeham, W. A. Viscosity of the Noble Gases in the Temperature Range 25−700°C. J. Chem. Phys. 1972, 56, 4119−4124. (34) Vogel, E.; Jäger, B.; Hellmann, R.; Bich, E. Ab Initio Pair Potential Energy Curve for the Argon Atom Pair and Thermophysical Properties for the Dilute Argon Gas. II. Thermophysical Properties for Low-Density Argon. Mol. Phys. 2010, 108, 3335−3352. (35) Kestin, J.; Yata, J. Viscosity and Diffusion Coefficient of Six Binary Mixtures. J. Chem. Phys. 1968, 49, 4780−4791.

(36) Kestin, J.; Ro, S. T.; Wakeham, W. A. Reference Values of the Viscosity of Twelve Gases at 25°C. Trans. Faraday Soc. 1971, 67, 2308−2313. (37) Kestin, J.; Khalifa, H. E.; Wakeham, W. A. The Viscosity of Five Gaseous Hydrocarbons. J. Chem. Phys. 1977, 66, 1132−1134. (38) Abe, Y.; Kestin, J.; Khalifa, H. E.; Wakeham, W. A. The Viscosity and Diffusion Coefficients of the Mixtures of Four Light Hydrocarbon Gases. Physica 1978, 93A, 155−170. (39) Abe, Y.; Kestin, J.; Khalifa, H. E.; Wakeham, W. A. The Viscosity of Normal Butane, Isobutane and Their Mixtures. Physica 1979, 97A, 296−305. (40) Abe, Y.; Kestin, J.; Khalifa, H. E.; Wakeham, W. A. The Viscosity and Diffusion Coefficients of the Mixtures of Light Hydrocarbons with Other Polyatomic Gases. Ber. Bunsenges. Phys. Chem. 1979, 83, 271− 276.

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Viscosity and Density of Normal Butane Simultaneously Measured at

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