for x = 0.9484 with a Vibrating-Wire Viscometer - ACS Publications

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Viscosity and Dew Point Measurements of {xCH4 + (1 − x)C4H10} for x = 0.9484 with a Vibrating-Wire Viscometer Clayton R. Locke,† Dan Fang,†,‡ Paul L. Stanwix,† Thomas J. Hughes,† Gongkui Xiao,† Michael L. Johns,† Anthony R. H. Goodwin,†,§ Kenneth N. Marsh,† and Eric F. May*,† †

Centre for Energy, School of Mechanical & Chemical Engineering, The University of Western Australia, Crawley, Western Australia 6009, Australia ‡ Key Laboratory of Thermo-Fluid Science and Engineering of MOE, Xi’an Jiaotong University, Xi’an 710049, China § Schlumberger Technology Corporation, Sugar Land, Texas 77478, United States ABSTRACT: The viscosity of {xCH4 + (1 − x)C4H10} with x = 0.9484 has been measured at temperatures and pressures in the range (200 to 423) K and (2 to 30) MPa, respectively, corresponding to densities between (20 and 371) kg·m−3. The measurements were made using a vibrating-wire-viscometer with the wire clamped at both ends and operated in steadystate mode with a combined relative uncertainty of 1 %. The viscometer was also used to investigate the ability of a vibrating-wire instrument to determine the upper and lower dew pressures of the mixture in the retrograde region at (263 and 273) K. The dew pressures were determined by identifying the point along an isothermal pathway at which the slope of the wire’s resonance half-width with pressure exhibited a discontinuity. At the upper dew pressures near 10 MPa the results were consistent to within 0.2 MPa of predictions made using the GERG-2008 equation of state (EOS), while at the lower dew pressures near 3 MPa the agreement was within 0.3 MPa. To facilitate future dew-point measurements, where it may be desirable to scan the mixture pressure rapidly, a novel ring-down technique was demonstrated utilizing the steady-state setup configuration, allowing accurate transient measurements without requiring fast data acquisition and providing flexibility for automated measurements.

1. INTRODUCTION Measurements of thermodynamic and transport properties of hydrocarbon mixtures are of immense importance to industry, particularly at conditions near and above the critical point. Accurate thermophysical property data for binary mixtures of hydrocarbons are necessary to develop and validate engineering models that underpin process design and simulation of the more complex multicomponent mixtures encountered in practice, with a degree of confidence that allows for economically reasonable design tolerances. In the production and processing of oil and gas, Goodwin1 identified the three most important thermophysical properties of hydrocarbon mixtures to be phase behavior, density, and viscosity; often these three properties are the basis for classifying the oil and gas mixture under consideration. Measurements of individual thermodynamic and transport properties are most often performed using purpose built instruments, optimized for the property under investigation. Instruments that are able to perform multiple property measurements, either in parallel or in series, are often desirable as they can significantly reduce overall sample preparation and measurement time. This is especially relevant for complex hydrocarbon mixtures, where sophisticated sample handling © XXXX American Chemical Society

systems capable of maintaining single-phase conditions are required. Furthermore, automated systems capable of optimized measurements of multiple properties covering a wide ranging parameter space would be highly beneficial. Although it is one of the most important constituent binaries of multicomponent hydrocarbon mixtures, thermophysical property data for the methane + butane system are surprisingly deficient, particularly at conditions of significant industrial interest. (Throughout this manuscript, we are referring to the isomer often called normal butane.) Recently May et al.2 reviewed the limitations of the phase behavior data available in the literature for this binary and presented new measurements that should assist in the development of more accurate equations of state for natural gas at low temperatures and high pressures. Similarly, there is a notable dearth of viscosity data for binary mixtures of methane + butane in the literature, with the number of measurements reported to our knowledge Special Issue: Memorial Issue in Honor of Anthony R. H. Goodwin Received: July 23, 2015 Accepted: September 23, 2015

A

DOI: 10.1021/acs.jced.5b00635 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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from a discontinuity in the slope of the fluid viscosity with pressure or temperature. Proof-of-concept dew-point measurements using the VWV technique are presented here for the same binary mixture for which viscosity results are presented, {xCH4 + (1 − x)C4H10} with x = 0.9484. The method employed involves continuously monitoring the half-width of the wire resonance at constant temperature as the pressure is varied from the single phase to the two-phase region with the dew-pressure identified where the measured half width exhibits a statistically significant deviation from the single-phase trend. Finally, we present a method for implementing the transient decay VWV technique16 that utilizes the same measurement system as the steady-state approach. We demonstrate the use of a “lock-in amplifier based transient measurement” to record the amplitude and phase of the decaying resonance signal from the wire, reducing the requirements for fast sampling. This approach has the potential to optimize measurement times by allowing the most appropriate mode of operation to be selected depending on the fluid under investigation without requiring any change in experimental configuration.

being limited to four data sets. Dolan et al. (1964)3 measured the viscosity of methane + butane mixtures using a capillary tube viscometer for methane mole fractions x = 0.25, 0.5, 0.7, and 0.9 over the density range of (13 to 533) kg·m−3. The work of Kestin and Yata (1968)4 included 41 measurements made using an oscillating-disk viscometer (with a stated precision of ± 0.05 % at 298 K) for a range of mixture compositions with x = (0.3553 to 0.8432) at 293 K and 303 K; however the measurement conditions were limited to the density range (0.95 to 6.4) kg·m−3. Subsequently, Abe et al. (1978)5 reported 15 viscosity data measured using the same apparatus as Kestin and Yata4 with a stated uncertainty of ± 0.3 % at densities below 2 kg·m−3. More recently Gozalpour et al. (2005)6 performed viscosity measurements of both single-phase and saturated methane + butane mixtures at 310.95 K using a capillary tube viscometer with an experimental uncertainty simply stated as “ ± 3 %”. This work presents new viscosity measurements of a binary mixture of {xCH4 + (1 − x)C4H10} with x = (0.9484 ± 0.0003), made using a vibrating-wire-viscometer (VWV). Figure 1 illustrates the range of conditions at which the

2. EXPERIMENTAL SECTION The VWV instrument and its operation have been described in detail in our previous paper,17 so only a brief summary is included here. The vibrating wire viscometer consisted of a centerless ground tungsten rod of length 40 mm and diameter 51 μm, clamped at both ends in the presence of a static magnetic field. The wire was driven at its resonance frequency by an alternating current, with the amplitude of its motion determined by observation of the electromagnetically induced voltage through demodulation with a lock-in amplifier. The fluid viscosity was determined by stepping the drive signal frequency f through the resonance and fitting the measured response to the two quadratures u( f) and v( f) of the hydrodynamic response function Vhydro18 Vhydro(f ) = u(f ) − iv(f )

Figure 1. Range of temperatures and pressure measured in this work: ●, single phase viscosity measurements; ■, dew point measurements, with numbers corresponding to the points listed in Table 3.

=

(A1 + iA 2 )f 2

i[f (1 + β) − f02 ] + f 2 (β′ + 2Δ0)

(1)

where A1 and A2 are the amplitudes of the complex response, β is the dimensionless added mass of the fluid surrounding the wire, β′ is the dimensionless viscous damping of the fluid surrounding the wire, and f 0 is the resonance frequency in vacuum. In eq 1, Δ0 is the logarithmic decrement in vacuum (vacuum damping) characterizing all damping mechanisms other than viscous damping, including those of the wire itself and the supporting structure, determined by measurements in vacuum. In principle the VWV can be operated as an absolute viscometer, since all operating parameters can be determined by independent means, including the radius of the wire. In practice, however, accurately determining the radius of very thin wires is technically challenging. Therefore, in this work the radius of the wire was determined by calibration with helium, for which the viscosity has been determined from ab initio calculations, and validated against the pure fluids nitrogen, methane and carbon dioxide to within the total measurement uncertainty.17,19 The viscometer calibration was retested prior to the measurements presented here using pure fluids helium and methane, as were measurements in vacuum. After one year of operation the calibration parameters were found to have

mixture was measured: 53 data were measured at temperatures between (200 to 423) K and pressures between (2 to 30) MPa, corresponding to viscosities between (11.6 and 56.5) μPa·s and densities in the range (20 to 370) kg·m−3. Measurements of a hydrocarbon mixture’s dew point are particularly important because of their sensitivity to vanishingly small fractions of higher alkanes in the fluid.7 Many techniques exist for the measurement of dew-points with visual observation of the first onset of liquid condensation probably remaining the most common in industrial practice.8 Visual methods of measuring dew points are known to suffer from systematic errors that arise from blind regions and dead volumes, which can in part be overcome in natural gas streams where fouling is unlikely through the use of a chilled mirror. Alternative nonvisual methods, in which blind spots and fouling can be less of an issue, usually monitor a fluid property that is discontinuous at a phase transition. These alternative methods generally measure one of four properties: density,9−11 refractive index,12 speed-of-sound,13 or dielectric constant.14,15 In principle, it should also be possible to detect a dew point B



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remained unchanged. The total Type B relative uncertainty20 contributions, which were dominated by uncertainties in wire radius calibration and nonlinear/out of plane motion, totaled 0.68 %. The binary mixture {xCH4 + (1 − x)C4H10} with x = (0.9484 ± 0.0003) was prepared gravimetrically from methane and butane (sources and purities of the mixture components are listed in Table 1).

3. RESULTS AND DISCUSSION: VISCOSITY The experimental viscosity data for the binary mixture of {xCH4 + (1 − x)C4H10} with x = (0.9484 ± 0.0003) at temperatures from (200 to 423) K and pressures up to 30 MPa are presented in Table 2 and plotted in Figure 2. The GERG2008 equation of state21 was used to calculate the fluid densities at the measurement temperature and pressure, as is needed to convert the measured wire resonance curves to fluid viscosities. The measured viscosities and results from the literature were compared with values calculated using the Extended Corresponding States (ECS) correlation;22 both the ECS model and the GERG-2008 EOS were as implemented in the software package REFPROP (REFerence fluid PROPerties database 23, version 9.1, NIST).23 The new data were also compared against a correlation based on molecular dynamics (MD) of LennardJones molecules24 that is predictive for mixtures. Each data point in Table 2 was determined at each T and p from an average of between 5 and 7 up−down frequency-sweep pairs. Each sweep direction required t ≈ 40 min, for a maximum measurement time per reported point of 4.7 h. Table 2 also lists the Type A uncertainties25 of the measured viscosity that were determined from the standard deviation of up to 7 measurements, together with the combined standard uncertainty at a coverage factor of k = 125 obtained by combining in quadrature the Type A uncertainties with the Type B uncertainties presented in Table 1 of Locke et al.17

Table 1. Chemical Sample Sourcesa chemical name

source

manufacturer’s purity in mole fraction (moles) purity

butane methane

BOC Coregas

0.99995 0.99995

a

Purity claimed by manufacturer and no further chemical analysis or purification was performed.

The relative uncertainty in the binary mixture’s methane mole fraction was determined from the relative contributions arising from the measurement of mass (0.030 %), mole fraction purity of the component gases (0.005 %), and dead volume (0.003 %) in the cylinder valve. The uncertainty in the methane mole fraction corresponds to an uncertainty in the calculated viscosity of 0.015 μPa·s (a relative uncertainty of 0.06 %).

Table 2. Viscosity η, Type A Uncertainty uA(η), and Combined Uncertainty uC(η) as a Function of Temperature T and Pressure p for {xCH4 + (1 − x)C4H10} with x = 0.9484a T/K

p/MPa

ρEOS/kg·m−3

η/μPa·s

uA(η)/μPa·s

uC(η)/μPa·s

T/K

p/MPa

ρEOS/kg·m−3

η/μPa·s

uA(η)/μPa·s

uC(η)/μPa·s

203.97 203.99 204.02 204.06 204.09 204.12 204.18 204.21 232.73 232.68 232.71 232.72 232.73 232.74 269.97 269.99 269.99 270.02 270.03 313.11 313.10 313.09 312.99 312.97 312.95 312.88 312.81 312.84 312.83 312.78 312.72

12.176 15.215 18.140 19.569 21.253 24.282 27.310 30.331 11.211 15.226 19.261 21.317 23.399 24.794 13.168 16.207 19.306 22.342 25.341 2.721 3.736 5.749 6.625 8.648 10.664 12.679 14.696 16.714 18.733 20.751 24.798

323.3 335.1 344.2 348.0 352.2 359.1 365.2 370.6 233.4 273.9 296.8 305.6 313.3 317.9 164.2 200.2 227.5 247.7 263.5 20.1 28.1 44.9 52.6 71.1 90.2 109.6 128.7 146.7 163.4 178.8 205.1

39.26 43.04 46.26 47.35 49.04 51.64 54.03 56.47 24.30 31.01 35.73 37.72 39.31 39.98 18.36 21.75 24.91 27.66 30.16 11.62 11.82 12.47 12.97 13.65 14.59 15.81 17.05 18.33 19.67 21.06 23.85

0.22 0.01 0.28 0.06 0.03 0.16 0.15 0.10 0.08 0.33 0.89 0.59 0.48 0.24 0.11 0.10 0.08 0.05 0.11 0.08 0.02 0.17 0.12 0.09 0.13 0.12 0.08 0.06 0.04 0.12 0.27

0.35 0.29 0.42 0.33 0.33 0.39 0.40 0.40 0.18 0.39 0.92 0.64 0.55 0.36 0.17 0.18 0.19 0.19 0.23 0.11 0.08 0.19 0.15 0.13 0.16 0.16 0.14 0.14 0.14 0.19 0.31

312.70 351.19 351.17 351.15 351.13 351.11 351.10 351.09 351.08 351.07 351.06 351.06 390.41 390.40 390.39 390.38 390.37 390.36 390.35 390.34 390.34 390.33 423.14 423.12 423.09 423.07 423.05

28.777 6.827 8.853 10.879 12.905 14.932 16.955 18.972 20.973 22.996 25.011 27.025 10.172 12.185 14.205 16.225 18.245 20.263 22.279 24.294 26.308 28.318 4.111 8.164 12.211 16.216 20.247

226.1 46.0 60.8 75.7 90.7 105.6 120.1 133.9 146.9 159.3 170.8 181.6 60.5 72.9 85.2 97.3 109.1 120.5 131.4 141.9 151.9 161.4 21.7 43.6 65.5 86.8 107.4

26.41 14.29 14.61 15.16 15.84 16.62 17.48 18.39 19.24 20.32 21.27 22.22 15.60 16.06 16.62 17.25 17.92 18.64 19.34 20.07 20.84 21.59 15.40 15.87 16.67 17.75 18.88

0.04 0.04 0.01 0.02 0.01 0.02 0.02 0.02 0.14 0.04 0.04 0.05 0.08 0.02 0.03 0.02 0.02 0.06 0.03 0.02 0.01 0.03 0.10 0.03 0.01 0.02 0.14

0.18 0.15 0.07 0.14 0.07 0.13 0.12 0.13 0.19 0.14 0.15 0.16 0.13 0.11 0.12 0.12 0.12 0.14 0.13 0.14 0.14 0.15 0.14 0.11 0.11 0.12 0.19

The density ρEOS used to determine the viscosity from the resonance curve was calculated from the GERG-2008 EOS21 implemented in REFPROP 9.1.23 Standard uncertainties u for temperature T and pressure p are u(T) = 0.05 K and u(p) = 0.004 MPa. a

C



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between (204 and 423) K have a relative root-mean-square (r.m.s.) deviation of 1.8 %, which at approximately twice the experimental uncertainty represents very good agreement and, as discussed below, is significantly better than the agreement with the other high-pressure viscosity data sets in the literature. For most of the isotherms, the ECS model overestimates slightly the mixture viscosity, with the average relative deviation of the entire data set from the ECS model being −1.0 %. Dolan et al.3 reported that viscosities were measured for x = 0.25, 0.5, 0.7, and 0.9 at temperatures between (294 and 444) K, which extend in density from (13 to 533) kg·m−3. The relative uncertainty in the viscosity claimed by Dolan et al.3 was between “± 0.5 %” for gas phase measurements and “± 1.5 %” for liquid phase measurements. These uncertainties were attributed primarily to the uncertainty in the mixture composition, which was stated to be “± 3 %”. Such a composition uncertainty, however, gives rise to a relative uncertainty in both density and viscosity that can be up 8 % at (x = 0.9, 294 K, 235 kg·m−3). When compared with the ECS correlation, the data of Dolan et al.3 have relative deviations with magnitudes of up to 19.4 % from the ECS predictions, which is between two and three times that which could be expected from the estimated composition uncertainty. At densities below 100 kg·m−3, where the relative viscosity sensitivity to the stated composition uncertainty is less than 1.5 %, the viscosity data of Dolan et al.3 have relative deviations from the ECS correlation from (−3.8 to +5.4) %. Some of the larger magnitude viscosity deviations (e.g., −19 % at x = 0.7, 311 K, 13.1 MPa) are associated with large deviations between the measured density (282 kg·m−3) and either of the EOSpredicted21 saturated phase densities at that condition (184 kg· m−3 for the saturated vapor and 363 kg·m−3 for the saturated liquid phase). In other cases, (e.g., at x = 0.7, 344 K, 20.7 MPa), the measured density was within 1 % of the GERG-EOS density, but the measured viscosity was 17.4 % larger than that predicted with the ECS model.

Figure 2. Measured viscosity, η, for {xCH4 + (1 − x)C4H10} with x = 0.9484 as a function of (a) mass density, ρEOS, and (b) pressure, p. ●, T = 204 K; ■, T = 233 K; ▲, T = 270 K; ▼, T = 313 K; ◀, T = 351 K; ▶, T = 390 K ; ◆, T = 423 K. Viscosities estimated from the ECS model within REFPROP are shown as solid curves.

The fractional differences between the measured viscosities and the values estimated from the ECS model implemented in REFPROP are shown as a function of (EOS) mass density in Figure 3. Viscosities for methane + butane mixtures reported in the literature3−6 are also shown in Figure 3 as relative differences from the estimates obtained using the ECS model. The low density (< 7 kg·m−3), ambient temperature data of Kestin and Yata4 and Abe et al.5 to which the CH4−C4H10 binary interaction parameter in the ECS model was tuned,22 have a relative root-mean-square (r.m.s.) deviation of 0.5 % and 0.6 %, respectively. The new viscosity data measured in this work, which extend to densities of 371 kg·m−3 at temperatures

Figure 3. Fractional differences between the measured viscosity, η, and viscosity calculated, ηcalc, using the REFPROP ECS model22 for {xCH4 + (1 − x)C4H10} with x = 0.9484 as a function of density, ρEOS, with vertical error bars showing experimental combined standard uncertainty. ●, T = 204 K; ■, T = 233 K; ▲, T = 270 K; ▼, T = 313 K; ◀, T = 351 K; ▶, T = 390 K ; ◆, T = 423 K; +, Dolan et al.,3 0.25 ≤ x ≤ 0.9, 294 ≤ T/K ≤ 444; ◊, Kestin and Yata,4 0.3553 ≤ x ≤ 0.8432, 293 ≤ T/K ≤ 303; ×, Abe et al.,5 0.1614 ≤ x ≤ 0.7367, 298 ≤ T/K ≤ 468; ○, Gozalpour et al.,6 0.515 ≤ x ≤ 0.8550, T = 310.95 K. D



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Figure 4. Fractional differences between the viscosity, η, of {xCH4 + (1 − x)C4H10} with x = 0.9484 from the viscosity calculated, ηcalc, using the REFPROP ECS model22 at the pressure and temperature of the measurements. Vertical error bars denote the combined experimental uncertainty. Symbols: This work, ●, T = 204 K; ■, T = 233 K; ▲, T = 270 K; ▼, T = 313 K; ◀, T = 351 K; ▶, T = 390 K; ◆, T = 423 K; ◊, Kestin and Yata;4 ×, Abe et al.5 Solid curves show the relative differences between predicted viscosities, ηMD, made using a Lennard-Jones correlation based on molecular dynamics simulations24 at the experimental temperature and pressure and those calculated using the ECS model in REFPROP.

Gozalpour et al.6 reported eight high-pressure viscosity measurements at 310.95 K for mixtures of methane + butane with a claimed relative uncertainty of “± 3 %” using a capillary tube viscometer connected between two equilibrium cells. Two of the measurements were made for a gravimetrically prepared single-phase mixture with x = 0.736, while the other measurements were made for saturated vapor and liquid phases obtained by decreasing the pressure in the equilibrium cells below the upper dew point of the single phase mixture. The compositions of the saturated vapor and liquid phases were determined by sampling and analysis with a gas chromatograph. The single-phase viscosities measured at (GERG-2008 EOS) densities of 325 kg·m−3 and 265 kg·m−3 have relative deviations from the ECS model of 6.4 % and 1.2 %, respectively. The saturated vapor phase viscosities measured at (EOS) densities between (124 and 168) kg·m−3 have fractional deviations from the ECS model ranging between + 3 % and + 18 %, while the saturated liquid phase viscosities measured at (EOS) densities between (379 and 424) kg·m−3 have fractional deviations from the ECS model ranging between −4 % and −18 %. While the agreement between the viscosity data measured in this work and the ECS predictions is quite good, the predictive estimates of viscosity obtained from the model of Galliéro et al.24 are even more impressive. This model, which does not contain any binary interaction parameters, consists of a Lennard-Jones (LJ) correlation based on molecular dynamic simulations and uses a corresponding states scheme combined with a van der Waals one-fluid approximation. Figure 4 shows the relative deviation of these predictions, ηMD, from the viscosities calculated using the ECS model in REFPROP, ηcalc, together with the data measured in this work and by Kestin and co-workers.4,5 The agreement between the ηMD values and the measurements made here at all temperatures except the lowest (204 K) is excellent; for the isotherms between (233 and 423) K the consistency between the density dependence of ηMD and that of the new data suggests that the structure apparent in the residuals is caused by the functional form of the ECS model rather any experimental systematic. If the measurements at 204

K are excluded, the average relative deviation of the data from ηMD is (−0.08 ± 1.03) %, where the error bound, which corresponds to the standard deviation of the deviations, is essentially equivalent to the combined experimental uncertainty. The average relative deviation of the data from ηMD increases in magnitude to (−0.50 ± 1.51) % if the 204 K data are included. By comparison, excluding the data at 204 K, the average relative deviation of the data from the ECS model, ηcalc, is (−0.87 ± 1.59) %, which increases in magnitude only to (−0.97 ± 1.51) % if the 204 K data are included. It is unclear why the data at 204 K do not agree as well with ηMD as at the higher temperatures, although the average relative difference for this isotherm of (−3.17 ± 1.33) % still constitutes a reasonable level of agreement given the simplifying assumptions inherent in the model.

4. RESULTS AND DISCUSSION: DEW-PRESSURE Dew-pressure measurements of the same binary mixture were performed using the VWV technique to establish a proof-ofconcept for this approach. Figure 1 shows the T, p conditions at which the dew-pressures were measured, namely along isothermal pathways at 263 K and 273 K for both the upper and lower dew pressures, where at this composition the mixture is in the retrograde region. Since the calculation of viscosity from eq 1 requires knowledge of the fluid density, the detection of a discontinuity indicative of the phase boundary directly from the viscosity is complicated by the requirement to know the density of the fluid as the phase boundary is crossed. To avoid this, the halfwidth γ of the wire resonance was monitored instead by measuring the wire response using the steady-state approach and the generic resonance function discussed by Locke et al.17 To measure the upper dew-pressures at 263 K or 273 K (Points 1 and 3 in Figure 1) the experimental cell was first charged with a single-phase gas mixture at 313 K. The pressure of the mixture was controlled at 12 MPa while the cell was cooled to 263 K or 273 K. To approach the upper dewpressures, the pressure in the cell was decreased incrementally, E



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Table 3. Dew Point Pressures, pdew,meas, Determined by Monitoring the Resonance Half-Width of the Vibrating Wire and Its Estimated Measurement Uncertainty u(pdew,meas), together with the Dew Pressure Calculated Using the GERG2008 EOS,21 pdew,EOSb

allowed to stabilize, and then the wire resonance was measured; this was repeated until the gas mixture was unambiguously in the two-phase range. The lower dew-pressure measurements (points 2 and 4 in Figure 1) followed a similar procedure, except that the measurement proceeded from the single-phase gas to the two-phase region by increasing the pressure along each isotherm. Prior to each dew pressure measurement the mixture was warmed to above 310 K and fluid was cycled in and out of the cell to help return the mixture to a single-phase condition. Figure 5 presents the measured half-width versus pressure for the upper dew-point determination at 263 K. The approach

point no.

T/K

pdew, EOS/MPa

pdew,meas/MPa

u(pdew,meas)/MPa

1 2 3 4

262.96 262.92 272.67 272.59

10.1 1.74 8.91 3.02

10.1 1.63 8.93 3.33

0.2 0.1 0.2 0.1

b

Standard uncertainty u for temperature T is u(T) = 0.05 K.

shown can be in error by up to 10 % of the saturation pressure for the methane + butane binary. In addition, the experimental approach applied here was not optimized for dew point measurements, with problems including (a) no control or knowledge of where the condensation has occurred in the apparatus, and (b) no ability to actively mix the fluid to ensure its homogeneity. Improvements to the design and operation of the apparatus are currently being implemented and these may allow more stringent tests of the vibrating wire’s ability to determine dew points in hydrocarbon mixtures in the future. To this end, a technique to measure the viscosity of a fluid using a modification to the ring-down technique was investigated. Current experimental practice for transient measurements is, for example,27,28 to rapidly sample the voltage-response of the wire in time and fit an exponentially decaying sinusoid. The sampling rate required for such a measurement is at least twice the wire’s resonant frequency to avoid aliasing effects, and the instrumentation required to make such a measurement is quite different to that needed for measurements in the frequency domain using the steady state technique. In this work, we explored the use of a lock-in amplifier to measure the resonance ring-down time, eliminating the need for rapid sampling and exploiting the filtering and amplification provided by such instruments. This has the advantage of the measurement system being the same as that used for the steady-state approach, eliminating the need for additional infrastructure and allowing the vibrating wire’s mode of operation to be switched as required. A high speed single-pole double-throw radio frequency switch, as shown in Figure 6 and with a switching time of less than 2 μs, was used to connect the wire to, first, the driving source for excitation near the wire’s resonant frequency and, second, the lock-in amplifier for readout. Upon switching from the drive source, the free oscillation and decay of the wire was recorded by monitoring the voltage induced by the wire through the applied magnetic field. This signal was demodulated using the lock-in amplifier, allowing the exponential decay envelope of the wire’s oscillation to be measured. The drive frequency must be tuned to within the bandwidth of the resonant frequency of the vibrating wire for adequate excitation, and similarly, the detected amplitude of the ringdown depends on the demodulation frequency supplied to the reference input of the lock-in (i.e., the drive frequency), with a larger detuning resulting in greater signal attenuation. An appropriate drive/demodulation frequency to use can be determined in advance using the steady-state technique. An example measurement is shown in Figure 7, for which a lock-in time constant (integration time) of 30 ms was used, which effectively sets the sampling rate at which the transient

Figure 5. Measured half-width, γ, as pressure is reduced, corresponding to Point 1 in Figure 1: ●, measured half-width (lower panel) and deviation, ◆, (upper panel) from linear fit with pressure, − (lower panel). The residual plot containing the deviations from the linear fit in the upper panel also shows the statistical threshold used to identify the dew-pressure represented as dashed lines.

used to analyze the results is as follows. The measurement was started at a pressure which was at least 1 MPa higher than the dew point pressure predicted by the GERG-2008 EOS. It was observed that, when in the single-phase region, the measured half-width varied linearly with pressure over a small pressure range. Accordingly, a linear fit was applied to the first three data points that were measured well-within the single-phase region. This line of best fit was then extrapolated down in pressure to predict the resonance half-width that would be expected if the fluid remained single-phase. A statistical analysis of repeat measurements at each single-phase pressure was performed to determine the standard deviation, which then served as a threshold value (± 0.005 Hz for T = 263 K, p ≥ 10.5 MPa) to help identify a statistically significant deviation from the singlephase trend. The pressure at which the deviation of the measured half width exceeded this threshold was identified as the dew-point pressure with an uncertainty bound set by the size of the pressure step (0.2 MPa or 0.1 MPa). The dewpressures measured at all four conditions are presented in Table 3 together with the dew-pressures estimated using the GERG2008 EOS implemented in REFPROP.23 It is apparent from Table 3 that the predicted and measured dew point pressures are consistent within the experimental uncertainty, except for the lower dew point at 273 K where the measured pressure is higher by about 3 times the estimated experimental uncertainty. Some of this discrepancy may be attributable to the EOS prediction, which May et al.26 have F



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Figure 6. Layout of the transient response apparatus and behavior of single-pole double-throw (SPDT) high speed switch in drive (excitation) and readout (decay) configuration.

Initial proof-of-concept measurements demonstrating the applicability of the VWV technique to dew pressure determinations have also been presented, in which the steady-state technique was used to track changes in the resonance half-width as the fluid mixture’s phase boundary was crossed. To increase the speed with which dew points could be located in poorly characterized hydrocarbon mixtures using the vibrating wire, a modification to the transient technique was demonstrated that allows the ring-down to be measured in the time domain using the same instrumentation as used in the steady-state technique. The narrow-bandwidth and highamplification of a lock-in amplifier can be used to monitor the decay envelope of the transient signal without the rapid sampling rate normally required for measurements in the time domain. These results further demonstrate that vibrating wire instruments have the potential to enable rapid and reliable measurements of fluid-mixture dew-points and viscosity over a wide range of temperature and pressure.

Figure 7. Transient response of the vibrating wire, measured with a lock-in amplifier: red oscillating curve, experimental transient data; solid black line, fit to data.

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signal’s envelope is monitored. We found that the same ringdown time was measured even with the drive/demodulation frequency detuned from resonance by up to 10 Hz. Using the steady-state technique, the half-width γ calculated by fitting the measured resonance curve to a function of the form 1/[( f − f 0)2 + γ] was 2.06 Hz. Using the lock-in amplifier based transient method, the ring-down time τ was calculated by fitting the measured decay curve to a function of the form e−τ·t and found to be 0.155 s. Comparing the ring-down time to the steady-state result via γ = 1/πτ, shows that the two measures of the half-width were consistent within 1 %, which is comparable with the instrument’s uncertainty.

AUTHOR INFORMATION

Corresponding Author

*Tel: +61 8 6488 2954. Fax.: +61 8 6488 1024. E-mail: eric. [email protected]. Funding

This work was supported by the Australian Research Council’s Linkage Program (LP130101018). E.F.M. acknowledges Chevron for their support of the research through the Gas Process Engineering endowment. P.L.S. is supported by a DECRA fellowship (DE140101094). D.F. acknowledges financial support from the China Scholarship Council during her visit to the University of Western Australia.

5. CONCLUSIONS A doubly clamped vibrating-wire viscometer operated in steadystate mode was used to measure the viscosity of a CH4 + C4H10 mixture at temperatures from (204 to 423) K and pressures in the range (2 to 30) MPa. Very few high-pressure viscosity data are available in the literature for this important binary mixture, and those data exhibit relative deviations from viscosity predictions made with the ECS model in REFPROP of up to 17.4 % (Dolan et al.3) and 18 % (Gozalpour et al.6). In contrast, for densities up to 371 kg·m−3 the new viscosity data reported here have relative deviations of (−3.6 to +2.3) % from the ECS model in REFPROP. Furthermore, the viscosity data measured between (233 and 423) K had an average relative deviation of (−0.08 ± 1.03) % from the model of Galliéro et al.,24 which is based on molecular dynamics simulations and contains no binary interaction parameters. This deviation is comparable with the estimated combined experimental uncertainty.

Notes

The authors declare no competing financial interest.

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for x = 0.9484 with a Vibrating-Wire Viscometer - ACS Publications

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