Improved Methods for Gas Mixture Viscometry ... - ACS Publications

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Improved Methods for Gas Mixture Viscometry Using a Vibrating Wire Clamped at Both Ends Clayton R. Locke,† Paul L. Stanwix,† Thomas J. Hughes,† Austin Kisselev,† 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 ‡ Schlumberger Technology Corporation, Sugar Land, Texas 77478, United States ABSTRACT: We present a clamped vibrating-wire instrument and the associated methods of measurement and analysis that enabled gas-mixture viscosity measurements at densities up to 110 kg·m−3 with a standard uncertainty of 0.09 μPa·s, which is a relative uncertainty of 0.60 %. The vibrating-wire was clamped at both ends and operated in the steady-state mode to make the apparatus more compact and allow operation over a broad range of conditions. New modifications to the method include an interleaved measurement protocol to minimize errors arising from fluctuations in temperature and pressure, and optimization of the signal-tonoise while ensuring that the driven wire’s response remained in the linear regime. The wire’s radius was determined from calibration measurements with He, and the viscometer’s performance was verified with N2, CO2, and CH4. The discrepancies between the measured pure fluid viscosities and those predicted with models implemented in the software REFPROP 9.1 were smaller than 1 %; literature data for these fluids exhibit similar deviations. Viscosities of (1 − x)C3H8 + xCH4 with x = 0.9452, and (1 − x)CO2 + xCH4 with x = 0.57 were also determined at pressures between (1.5 and 6.5) MPa and temperatures of (280, 303 and 328) K. The largest rms deviation of 3.6 % of the measured viscosities relative to those calculated with the extended corresponding states model implemented in REFPROP occurred for CH4 + CO2 at a temperature of 328 K.

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INTRODUCTION The vibrating-wire viscometer is an attractive technique for the measurement of gas and liquid viscosities.1−4 The viscometer has complete working equations firmly based in the principles of physics and requires knowledge of both the average wire radius over the length of the wire and the internal-damping of the wire in vacuum. Since all coefficients can be determined independently, the vibrating-wire viscometer has the capacity for absolute viscosity measurements. Vibrating-wire viscometers are designed to operate in one of two modes: steady-state, in which the wire is actively and continuously driven at a frequency near the fundamental vibrational mode of the wire; or transient, in which the wire’s fundamental mode of vibration is excited by an impulse and allowed to decay. The steady-state mode of operation is favored when the chemical composition, temperature, and pressure are constant, since the attainable operating viscosity range is larger for a given wire diameter. However, while the method permits measurements with both increasing and decreasing frequency and thus provides a measure of the thermal hydrostatic and chemical stability, it does require a measurement time on the order of 100 s. The transient method, which requires a measurement time from between 1 ms and 1 s, is preferred for a flowing fluid where the temperature, pressure, and chemical © 2014 American Chemical Society

composition might vary sufficiently to introduce a significant systematic error into the measured viscosity. However, the transient method limits the operating range of the viscometer. In addition to their mode of operation, vibrating-wire viscometers are also characterized by their wire clamping arrangement (Table 1 of Kandil et al.5): either a wire clamped at both ends or, more commonly, a wire clamped at one end and the other end weighed with a buoyant sinker. The latter is able to simultaneously determine both viscosity and density of a fluid, but has disadvantages in terms of vessel size, alignment, and instrument robustness. In this work we have developed a steady-state vibrating-wire viscometer with the wire clamped at both ends, designed to take advantage of its broad range of operation and robust and compact attributes. In doing so, several technical challenges were observed that resulted in large uncertainties when measurements were performed at moderate gas-densities between (2 and 150) kg·m−3. The steady-state method is highly dependent on the stability of the temperature and pressure; the time required to acquire the measurement makes Received: January 24, 2014 Accepted: April 7, 2014 Published: April 22, 2014 1619

dx.doi.org/10.1021/je500083n | J. Chem. Eng. Data 2014, 59, 1619−1628

Journal of Chemical & Engineering Data

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perpendicular modes of oscillation of the wire. The modes were observed by adjusting the magnetic flux orientation, preferentially exciting each mode in turn. For our apparatus, the two modes occurred at frequencies of (2743 and 2745) Hz. In operation, the magnetic field was aligned to excite solely one of these modes; however, cross-coupling between the two modes cannot be eliminated completely. This cross-coupling created a precession about the axis of the wire, which could have corrupted the measured amplitude and frequency of the resonance, leading to a source of systematic error (as shown later) in the viscosity obtained from the measurements. The pressure vessel was placed in a temperature-controlled bath with a temperature variation about the set-point of 0.03 K for integration times of about 60 s and temperature variation about the set-point lower than 0.004 K at integration times greater than 0.3 h. Temperature variations of 0.06 K were observed for the thermostat for times on the order of days and these manifested themselves as variations in the average temperature over extended measurement times. Platinum resistance thermometers with a nominal resistance of 100 Ω at T = 273 K were placed on wire mounts within the pressure vessel, in the bath, and outside the bath. The thermometers were sampled continuously throughout the time required to acquire the voltages from which the viscosity was determined. The pressure of the gas was determined with an oscillating quartz-crystal transducer (Paroscientific model 745) having a relative uncertainty of 0.008 % of its 7 MPa fullscale. Typically, the pressure remained constant during the determination of the resonance curve with a relative variation of less than 0.03 %. The uncertainties associated with measurements of viscosity made with the apparatus are summarized in Table 1.

it particularly susceptible to variations of these quantities that occur on time scales less than that of the acquisition. This work introduces data acquisition and processing strategies to mitigate the deleterious effect of these factors. The performance of the system was verified by measuring the viscosity of methane, carbon dioxide, and nitrogen. The instrument was then used to measure binary mixture viscosities of the systems C3H8 + CH4 and CO2 + CH4.

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APPARATUS

A vibrating-wire viscometer reported previously5 was modified to include a revision of the wire-holder and reduction of the diameter of the wire. For the apparatus used in this work, the wire-holder was fabricated from nonmagnetic 316 stainless steel to increase the robustness of the device compared to the previous holder, which was machined from MACOR. The wire was electrically isolated from the holder by cups machined from polyimide that were inserted into the stainless steel holder and served to electrically isolate the wire clamps and retaining screws from the holder. Polyimide was chosen as it can be used over a wide range of temperatures (from cryogenic to about 520 K) and is also compatible with the gases to be studied. The vibrating wire was formed from a centerless ground tungsten rod of length ≈ 40 mm which was clamped at both ends under tension to achieve a fundamental resonance frequency of about 2 kHz. The wire had a nominal radius of about 25.4 μm, chosen on the basis of simulations6 which suggested this would give the viscometer the desired operational range. The wire and wire-holder were annealed at a temperature of 473 K, repeatedly, until the mechanical resonance frequency at a temperature of 298 K varied upon thermal cycling by less than 10 % of the resonant frequency. The viscometer was located within a pressure vessel filled with the fluid under test. An alternating current was passed through the wire, which was located in the field of two samarium cobalt permanent magnets producing a field of 0.3 T, to thus drive the wire motion. The wire motion in turn induced a potential difference proportional to the oscillation’s amplitude, which was enhanced when the drive frequency corresponded to a mechanical resonance of the wire. This motional potential difference was detected as the difference between the two inputs (A − B) of a lock-in amplifier, shown in Figure 1. The two quadratures (that are equivalent to the phase and amplitude of the induced voltage, with some arbitrary phase offset relative to the drive signal) were acquired as a function of drive frequency over the resonance. As the cross-section of the wire was elliptical there existed two families of resonant frequencies corresponding to the two

Table 1. Contribution to Uncertainties of the Viscosities, u(η), Determined with the Doubly-Clamped Vibrating-Wire Viscometer source

100·u(η)/η

nonlinear motion/out of plane vibration wire radius calibration density of wire material pressure sensor temperature sensor vacuum damping subtotal standard deviation of 20 measurements of η at each p and T total

0.44 0.24 0.05 0.008 0.005 0.002 0.50 0.33 0.60

To drive the wire optimally (maximizing signal while avoiding nonlinearity in the response), the potential difference was set to maintain the maximum displacement of the wire at less than 1 % of the wire radius. To do so, the resistance of the tungsten wire was estimated from material datasheets,7 and the electrical current that would give rise to a potential difference in the presence of a magnetic flux was calculated. The Catenary curve equation8 was used to determine the displacement, which was enhanced by the pressure dependent Q factor when at resonance. For further verification, the potential difference induced by a wire moving in the magnetic field was estimated and compared to the value obtained experimentally to ensure an optimal drive current was applied. This relationship between the amplitude (obtained from the appropriate phase balance between channels X and Y of the lock-in amplifier) allowed an

Figure 1. Schematic of the vibrating wire viscometer and associated interrogation electronics. 1620



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calculations of viscosity used have a relative uncertainty of less than 10−4.12,13 As shown in Figure 2, helium measurements at mass densities between (2.4 and 12.9) kg·m−3, were used to

immediate estimate of the maximum displacement of the wire; in our case a relative displacement of 1 % of the wire radius corresponded to a measured voltage of 32 μV. When calculating the fluid viscosity from the measured resonance, the vacuum damping parameter, Δ0, of eq 2 (see later) must be known. This term quantifies all losses, including those of the wire itself and the supporting structure, other than the loss arising from motion within a fluid, and is determined from measurements under vacuum. The pressure inside the chamber containing the viscometer was reduced until no further decrease in loss was observed (in this case, pressures of less than 2.5 Pa were sufficient), and the value of the vacuum damping parameter was calculated as Δ0 = (2.148 ± 0.012)· 10−5. In measurements of viscosity of binary mixtures presented here, the contribution of vacuum damping to total loss is most significant (approximately 10 %) at low densities, reducing to less than 3 % at high densities. The wire in our apparatus was formed from tungsten with a mass fraction purity of 0.9995 and was centerless ground to give a radius cited by the manufacturer8 of (25.400 ± 0.635) μm. The density and radius of the wire are required in the analyses to determine viscosity; in this case, the density of tungsten was taken from Lassner et al.7 as ρ(298 K) = (19 256 ± 10) kg·m−3. When thin wires are employed, which is favorable for measurements of fluids with low viscosity,6 the radius of the wire can prove technically difficult to measure with the required uncertainty (approximately < 0.5 %). For example, Goodwin and Marsh3 determined the radius R by contact micrometry combined with laser interferometry. It is therefore common practice to determine the radius R with a reference fluid for which the viscosity is known. In this work, a series of measurements were made with the wire immersed in helium at pressures between (0.5 to 11) MPa at a temperature of 280 K. The source and purity of the helium, and all the pure fluids used in this work, are listed in Table 2. We determined the

Figure 2. Radius of the wire R as a function of mass density ρ at T = 280 K: ●, R determined from ab initio calculations of viscosity having relative uncertainty of less than 10−4;12,13 black line, ⟨R⟩; dash-dotted line, R provided by the manufacturer; dotted line, the standard uncertainty of R provided by the manufacturer. The error bars are one standard deviation.

determine the mean radius over the 40 mm wire length with the result ⟨R⟩ = (25.518 ± 0.065) μm, which is within the bounds of the value reported by the manufacturer of the wire. The measurements at pressures below 1.4 MPa (ρ ≈ 2.4 kg· m−3) were limited by molecular slip and were not used to determine ⟨R⟩; in a future publication we will further discuss molecular slip in vibrating wire viscometry. The viscometer was automated to continuously acquire the viscosity of the high-pressure fluid. Each determination of viscosity at a temperature and pressure listed in the following sections was typically an average of 20 determinations of the resonance frequency that were fit to eq 2, and which resulted in a statistical contribution to the standard uncertainty in the determination of viscosity of 0.05 μPa·s.

Table 2. Chemical Sample Sources. All Gases Were Used without Further Purification chemical name

source

manufacturer’s purity

carbon dioxide helium nitrogen methane propane

Coregas Coregas Coregas BOC Air Liquide

0.99999 0.99999 0.99999 0.99995 0.99995

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RESONANCE CURVE ANALYSIS The standard approach for determining fluid viscosity is through a fit of the measured wire resonance to the hydrodynamic response function (Vhydro). The hydrodynamic response1,2 measured by the two quadratures of voltage u( f) and v( f) is given by

viscosity η(He, 280 K, p) from ab initio calculations combined with corresponding states theory10,11 to estimate its small viscosity virial coefficient. For this purpose, the pressure was determined with an uncertainty of 0.001 MPa and temperatures with an uncertainty of 0.01 K so that the ab initio values of the density and viscosity were determined with an expanded uncertainty of 0.02 kg·m−3 and 0.001 μPa·s, respectively. Calculations of viscosity for a range of pressures at T = 280 K were anchored by the calculated ab initio value at T = 298 K and zero density of

Vhydro(f ) = u(f ) − iv(f ) =

(A1 + iA 2 )f 2

i[f (1 + β) − f02 ] + f 2 (β′ + 2Δ0)

(2)

where β is the dimensionless added mass owing to the fluid surrounding the wire, β′ is the dimensionless viscous damping of the fluid surrounding the wire, and Δ0 is the logarithmic decrement in vacuum (vacuum damping). The parameters β and β′ contain terms5 involving modified complex Bessel functions K0 and K1 that include the ratio of viscosity η to mass density ρ of the fluid, the density of the wire ρs, a dimensionless quantity closely related to the Reynolds number Ω, and the radius of the wire R:

η(He, T = 298 K, p = 0) = (19.8253 ± 0.0002) μPa· s (1)

obtained in Cencek et al.10 The viscosity virial coefficient of helium for (T > 130 K) and (p < 25.3 MPa) is small and was estimated from a corresponding states model.11 The ab initio 1621


Journal of Chemical & Engineering Data β=

ρ k ρS

and

β′ =

k = −1 + 2Im(()

ρ k′ , ρS

and

k′ = 2Re((),

⎡ 2K1(iΩ)1/2 ⎤ ⎥, ( = i⎢1 + ⎢⎣ (iΩ)1/2 K 0(iΩ)1/2 ⎥⎦ Ω=

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and

2πfρR2 η

(3)

This approach is an effective and precise means of determining viscosity. However, it requires knowledge of the phase between the voltage quadratures and, if the data need to be reanalyzed (using, for example, a revised density or vacuum damping), the entire resonance curve must be regressed again for each data point. These issues can be avoided if a generic resonance function (Vgeneric) is regressed to the measured resonance curve and the viscosity is then extracted directly from the best-fit parameters (frequency, f r; and full width at halfmaximum amplitude, γ). A generic Lorentzian in the presence of a background of a drive frequency swept over a resonance14,15 is given by

Figure 3. Fractional differences between the calculations of viscosity using the generic model (ηgeneric) and the hydrodynamic model (ηhydro) as a function of detuning (fdetuning = f − f r). Internal damping (Δ0) determined from experiment is varied: bold black line, 0.985Δ0; black line, 0.995Δ0; gray line, 1.005Δ0; light gray line, 1.015Δ0. Consecutive curves are displaced by 5 Hz about that of Δ0 for clarity.

experimental data fit the generic resonance function, and thus can be used as an indicator of the data quality. Moreover, the variation of ηgeneric with f can be used to test the values of the quantities ρ, ρs, R, and Δ0 used in eq 5. The variation of ηgeneric with f is shown in Figure 3 for several different values of the vacuum-damping parameter Δ0. Figure 4 shows how an

Vgeneric(f ) = u(f ) + iv(f ) =

(a1 + ia 2)f 2γfr + i(f 2 − fr2 + γ 2)

+ b1 + ib2 + (c1 + ic 2)f (4)

The terms b1 and b2 account for a frequency independent background, and c1 and c2 account for frequency dependent background (owing possibly to tails from other resonances). A fit using the resonance function of eq 4 to experimental data yields γ, phase θ (where we have defined a2 = a1 tan θ) and resonance frequency f r. To obtain viscosity from these fit parameters, an algebraic relation between the hydrodynamic resonance function (Vhydro) and the generic resonance (Vgeneric) must be established. Removing background terms, equating the real and imaginary components of eq 2 and eq 4, and considering that a1 of eq 4 is not necessarily equal to A1 of eq 2 and likewise a2 is not necessarily equal to A2, we obtain

Figure 4. Background slope of the generic model viscosity calculation vs stimulus frequency curve (dηgeneric/df) at a frequency far from resonance ( f = f r + 3γ) as a function of the phase θ, where θ, f r, and γ are as defined in Vgeneric of eq 4

2a1γfr + a 2(f 2 − fr2 + γ 2) (a12 + a 22)1/2 [4γ 2fr2 + (f 2 − fr2 + γ 2)2 ]1/2 =

incorrect assignment of the phase θ, which can occur as discussed in the method section below, can cause the background slope, dηgeneric/df, far from resonance to deviate from zero. Excluding values of ηgeneric obtained for stimulus frequencies within ± 2γ of f r, a viscosity calculated using eq 4 and eq 5 was found to generally deviate by less than 0.02 % from the viscosity calculated in the conventional manner (i.e., regression of the data to eq 2.)

f 2 (β′ + 2Δ0) + [f 2 (1 + β) − f02 ]tan θ {f 4 (β′ + 2Δ0)2 + [f 2 (1 + β) − f02 ]2 }1/2(1 + tan 2 θ)1/2 (5)

By substituting the best-fit values of the parameters f r, γ, a1, and a2 determined by regression of Vgeneric to the measured resonance curve, eq 5 can be solved (numerically) for the viscosity η, which is contained within the terms β and β′ (as shown in eq 3), by also using the known values of the other quantities θ, ρ, ρs, R, and Δ0. The remaining variable f in eq 5 is the stimulus frequency, which is driven across the resonance. In principle the viscosities should be valid at any value of f; in practice, however, the viscosities calculated at different stimulus frequencies f across resonance do vary slightly. Figure 3 shows the variation with stimulus frequency of viscosities calculated from Vgeneric using eq 5, ηgeneric, relative to the (single) viscosity, ηhydro determined by regression of eq 2. The degree to which ηgeneric varies with f provides information as to how well the

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METHOD IMPROVEMENTS There are a number of challenges in the application of doubly clamped vibrating-wire viscometers to the measurement of gases at moderate densities, as follows: (1) correctly setting the parameters of the measurement, specifically the vibrating-wire drive amplitude and thus the amplitude of the wire motion; (2) implementing strategies to mitigate environmental fluctuations that prove difficult to control over the time scales of approximately 100 s for the steady-state measurement; and 1622



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(3) optimizing the data analysis to minimize the statistical uncertainty and computational load of the regression from which the viscosity is determined. We turn to a description of each of these items in the order listed. The hydrodynamic model that is used to interpret the vibrating-wire measurements assumes that the amplitude of oscillation is much less than the wire radius, typically less than 1 %. Furthermore, if the resonant wire is driven to relatively large amplitudes, there is a corresponding increase in nonplanar vibration and also a jump phenomena. These jumps occur when the excitation amplitude increases over the resonance and causes the restoring force to become nonlinear, so that the amplitude response depends on the direction of the drive signal (a downward jump when the driving frequency is increasing and an upward jump when the driving force is decreasing). This effect can be modeled and incorporated into the solutions of the equations of motion, giving nonlinear regions in which the amplitude increases as the driving force decreases. This effect was observed and analyzed in detail by Murthy and Ramakrishna.16 Second, a forced vibration will acquire a component of motion normal to the driving force (a “whirling tubular motion.”). Both of these systematic effects are present even at small driving forces and become more significant as the driving force increases. On the other hand, larger amplitudes of oscillation will induce a greater electric potential, thereby improving the signalto-noise ratio (SNR) of the device. The amplitude of oscillation was set to be 1 % of the wire radius, corresponding to a measured voltage amplitude of about 32 μV as derived from the Catenary equation. Then, in processing the data from a frequency sweep, any deviations of the measured resonance (for both increasing and decreasing drive frequencies) from the hydrodynamic model were analyzed to ensure that nonlinear jump and nonplanar vibration phenomena were unobservable within the experimental limits. From analysis of response data acquired, it was estimated that nonlinear effects and nonplanar vibration accounted for less than a 0.25 % change of the bandwidth, which corresponded to a systematic shift in the derived viscosity of less than 0.44 %. Sullivan et al.17 claimed that it is possible to increase the SNR by overdriving the amplitude of motion (up to 60 % of the wire radius) and, by using a nonlinear interpretation, correctly measure the viscosity; this approach was not used in this work as the SNR achieved with small drive amplitudes of motion did not impose significant uncertainty on the viscosity calculations. In the experiment, it was also important to determine the optimal settings for the lock-in amplifier used for demodulation and amplification of the measured voltage signal. If the drive frequency scan across the resonance was performed too fast there was insufficient time for the mechanical oscillation of the system to equilibrate, which precluded a reliable determination of the fluid viscosity. In addition, the integration time constant must be sufficient to filter high frequency noise. Therefore, to determine the optimal values for the lock-in operation, both of these parameters (time constant and scan rate) were systematically varied, as illustrated in Figure 5. The lock-in time constant was chosen to be 1 s, since this reduced the high frequency noise sufficiently without significantly increasing the total measurement time. The wait time, twait, is defined as the period during which the wire is driven at each frequency step before the next measurement is triggered; this allows time for the mechanical resonator to respond. As demonstrated in Figure 5a it was found that, in general, twait should be a factor of

Figure 5. (a) Percentage change in apparent viscosity ηapparent from viscosity determined at long wait times η0 as a function of wait time twait between measurements at lock-in amplifier integration times. ■, 1 s; ○, 300 ms; ●, 100 ms. (b) The effect of filtering on the measured voltage V as a function of frequency f over the resonance as a function of lock-in amplifier integration times. bold line, 1 s; black line, 300 ms; dotted line, 100 ms. Consecutive curves are offset by 1 μV for the sake of clarity.

5 larger than the time constant of the lock-in to eliminate any dependence of the apparent viscosity on twait. For example, when using a time constant of 1 s, the apparent viscosity with a wait time of 1 s will be 16 % larger than that measured using a wait time of greater than 10 s. Similarly, a wait time of 2 s will give an apparent viscosity 4.4 % larger, whereas a wait time of 5 s will give a difference of less than 0.1 %. Therefore, the wait time was chosen to be 5 s. Acquiring low noise data (i.e., reducing statistical uncertainty to a level below other noise contributions) required long scan times of greater than 0.3 h, during which time the cell’s temperature and pressure could vary because of drifts in the sections of the apparatus not under active thermal control and exposed to variations of room temperature. In addition, fluctuations in the bath temperature over this measurement time also gave rise to fluctuations in the cell pressure and temperature. The background voltage upon which the resonance is located is strongly correlated with these variations in cell temperature and pressure: the observed variation in the measured background level of the resonance curve was sufficiently large that, if uncorrected, it would have led to uncertainties in viscosity of about 0.3 μPa·s. The conventional approach of measuring background signal before and/or after the resonance sweep (e.g., using the lock-in offset feature) was not sufficient to allow adequate mitigation of the effects of these fluctuations and drifts, and similarly they could not be 1623



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Figure 6. (a) Real component u and imaginary component v of the measured resonance V as a function of detuning frequency fdetuning from the resonant frequency when background is not corrected, with atop the residuals of ΔV from the fit to eq 2. The measured offset in the background levels of u and v is shown. (b) Real component u and imaginary component v of V when the background is corrected (removed) using the interleaved technique, with residuals ΔV of the results from eq 2: filled circles, u; open circles, v; dashed line, Δv; solid line, Δu.

removed during the postprocessing of the data with any degree of certainty. To address this problem, we adopted an interleaved data collection technique, which has been used previously to improve resonance curve measurements18 but that has not, to our knowledge, been previously applied to vibrating wire viscometer measurements. In between every frequency step used to scan across the wire’s resonance, a voltage measurement was also made at a frequency 3.5γ from the resonance. In postprocessing the measurement at successive drive frequencies could then be subtracted from that signal obtained at (almost) the same time but at a frequency far from resonance. Continuous measurement and subsequent subtraction of this background allowed any drifts or jumps that occurred during a sweep to be accounted for, improving the performance of the viscosity determination by minimizing the fit residuals. Figure 6 shows a comparison of the resulting resonance curves measured with and without the interleaved technique. In Figure 6a, where the interleaved technique was not used, the deviations of the data from eq 2 are greater than 1 % and structured, while those shown in Figure 6b appear random and have a root mean squared uncertainty almost an order of magnitude lower. The interleaved measurement and data processing also removes any offsets from zero in the measured background, which can simplify the regression. Longer-term drifts in the vibrating-wire resonant frequency cannot be completely eliminated (unless temperature and pressure drifts are removed), and these drifts become especially obvious with gases at low pressure where the loss and measured resonance line-width are a factor of 100 smaller relative to those obtained at elevated pressures. Figure 7 shows the apparent viscosity (ηapparent) derived from the measured resonance curve as a function of drift rate ( fdrift), which we define as the change in resonant frequency over the time taken to complete one scan. If the scan direction is in the same direction as the drift rate, the measured half width increases compared to the actual value and thus the viscosity calculated is systematically greater than the correct value, while if the drift is opposite the scan direction, the measured half width will be smaller resulting in a systematically lower viscosity. For the purposes of obtaining reliable measurements of viscosity, it was first ensured the drift rate was not excessive; by this we mean less than 0.1 Hz over the time of one scan (about

Figure 7. The effect of drift rate, fdrift, (defined as change in resonant frequency over the time taken to complete one scan across resonance) on viscosity measurements: ■, scan increasing in frequency; ●, scan decreasing in frequency; ○, the average of a pair of consecutive increasing and decreasing scans.

0.3 h) giving 90 μHz·s−1. Additionally, scans were made in both directions and the average half width was taken to determine the viscosity. As expressed in eq 2, the measured signal is complex with a small phase offset between the wire’s response and the lock-in amplifier’s reference. This phase offset is dependent on experimental conditions (set by the length of the electrical wiring) and it is common practice to account for this phase offset in the lock-in’s setting before the start of a series of measurements. However, as the measurements progress this phase offset may change (predominately as temperature affects permittivity in the electrical circuit) and it is difficult to predict the offset precisely for each measurement in advance. When both A1 and A2 of eq 2 (or a1 and a2 of eq 5) are treated as adjustable parameters in the data regression, this phase imbalance is accounted for as part of the fitting process. However, it is common practice in vibrating wire viscometry to set A2 to zero19,20 to simplify the regression. The result of an incorrectly set phase will show up in the residuals of the fit, as illustrated in Figure 8. However, when both A1 and A2 were allowed to be free parameters in the fit, the rms residual was reduced to the equivalent of a statistical uncertainty in the viscosity of 0.04 μPa·s. We note that the structure of the residuals as a function of stimulus frequency apparent in Figure 8 when the phase was incorrectly set is similar to that observed 1624



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Figure 8. (a) Imaginary component v(f) of Vmeas(f) where f is the stimulus frequency. × , when phase is incorrectly (A2 set to zero) tuned; and ●, correctly tuned (A2 a free parameter); with residuals, dotted line; and solid line, respectively. (b) Real component u( f) of Vmeas( f): × , when phase is incorrectly tuned; and ●, correctly tuned; with residuals, dotted line; and solid line, respectively.

in several literature publications,19,20 suggesting that the results of those works might have been improved with a refined phase offset.

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RESULTS The performance of the doubly clamped vibrating wire was tested by conducting measurements of the viscosity of pure CO2, N2, and CH4. The measured viscosities were compared with values calculated from the default (reference) equations of state21−23 and viscosity correlations24−26 for these pure fluids implemented within the software REFPROP 9.127 The relative deviations between the measured and calculated viscosities are all less than ± 1 % and are shown in Figure 9; in most cases they are smaller than the estimated experimental uncertainty and, since the relative uncertainty of each pure fluid viscosity correlation is larger than 0.3 %,27 the deviations are all well within the combined uncertainty. Results were compared with those reported in literature; the measurements of Kestin et al.28 and Evers et al.29 were obtained using different measurement techniques with quite different sources of error to those that might plausibly arise with a vibrating-wire viscometer, while the results of Schley et al.4 were determined with an instrument for which the principle is similar to ours. As shown in Figure 9, the differences between the results of this work and these literature values are less than 0.5 %. As discussed by Berg et al.13,30 the viscosity ratio of two gases can be measured with a lower uncertainty than the absolute viscosity of either gas because of the suppression of systematic errors that afflict a particular technique. Comparison of viscosity ratios provides a stringent test of an apparatus and we do so here using the reference viscosity ratios for 11 gases reported by Berg et al.13 at a temperature of 298.15 K and in the limit of zero density. The reference value for the ratio of the viscosity of N2 to that of He at this condition is (0.89529 ± 0.00027). Similarly, the ratio of the viscosities of CH4 and He is (0.55803 ± 0.00031). No reference ratio was given for CO2 and He. To compare our measured viscosity ratios for these gases with these reference values, the viscosities of each pure fluid were regressed to a second-order polynomial in density to allow extrapolation to zero density. Equivalent values of these extrapolated zero density viscosities, measured at a temperature of 303.20 K, were then calculated for the reference temperature of 298.15 K using

Figure 9. Fractional differences between the measured pure gas viscosity (ηmeas) and viscosity calculated (ηcalc) using the default pure fluid equations of state21−23 and viscosity correlations24−26 implemented in REFPROP 9.127 for N2 (top), CO2 (middle), and CH4 (bottom). This work: ⧫, T = 303 K. Viscosity data of Evers et al.,29 □, T = 293 K; △, T = 333 K; Kestin et. al,28 +, T = 305 K; Schley et al.,4 ○, T = 300 K.

η(g, T = 303.2 K, p = 0) ⎡ 303.20 K ⎤b = η(g, T = 298.15 K, p = 0)⎢ ⎣ 298.15 K ⎥⎦

(6)

where g denotes the gas and the exponent b = 0.69 for He, 0.88 for CH4, and 0.77 for N2.13,30 The ratio η (N2, T = 298.15 K, p 1625



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= 0)/η (He, T = 298.15 K, p = 0) measured in this work is 0.26 % higher than the reference ratio reported by Berg et al.13 Similarly, the measured ratio η (CH4, T = 298.15 K, p = 0)/η (He, T = 298.15 K, p = 0) is 0.39 % larger than that reported by Berg et al.13 These deviations are consistent with the uncertainty estimates listed in Table 1. The viscosities of two binary gas mixtures were also measured. A mixture of (1 − x)CH4 + xC3H8 with x = (0.0548 ± 0.0002) was prepared gravimetrically, where the mole fraction uncertainty was determined from the contributions due to mass measurement, mole fraction purity of the component gases, and dead volume in the cylinder valve. The measured viscosities for the mixture are listed in Table 3. The Figure 10. Fractional differences between the measured viscosity (ηmeas) and viscosity calculated (ηcalc) using the REFPROP ECS model27,32 for the (C3H8 + CH4) system as a function of density ρEOS: ●, T = 280 K; and, ■, T = 298 K. The error bars are one standard deviation of repeat measurements (Table 3).

Table 3. Viscosity η and Statistical Uncertainty u(η) as a Function of Temperature T and Pressure p for (1 − x)CH4 + xC3H8, with x = 0.0548. Density ρ Calculated from the GERG-2008 EOS Implemented in REFPROP 9.127,31 T/K

p/MPa

ρEOS/(kg·m−3)

η/(μPa·s)

u(η)/(μPa·s)

280.16 280.16 280.17 280.17 280.18 280.20 280.21 298.18 298.18 298.20 298.19 298.20 298.22 298.22

0.61 1.55 3.15 4.97 5.88 6.56 6.97 0.87 1.45 2.65 4.11 5.43 6.34 6.70

4.679 12.199 25.944 43.198 52.517 59.783 64.289 6.285 10.608 19.902 31.883 43.364 51.633 54.984

10.394 10.510 10.832 11.435 11.795 12.111 12.296 10.831 10.965 11.235 11.633 12.107 12.436 12.588

0.087 0.067 0.080 0.094 0.078 0.081 0.077 0.089 0.093 0.120 0.073 0.081 0.080 0.078

Table 4. Viscosity η and Statistical Uncertainty u(η) as a Function of Temperature T and Pressure p for (1 − x)CO2 + xCH4 with x = 0.57a

data were compared with values calculated using the extended corresponding states model in the software REFPROP 9.1,27 making use of the GERG-2008 equation of state31 to obtain from the measured temperature and pressure the densities needed for scaling the mixture relative to the reference fluid. This extended corresponding states (ECS) model also uses interaction parameters32 to improve viscosity predictions of mixtures for which literature data are available. The differences between the calculated viscosities and viscosities obtained in this work, as shown in Figure 10, are almost within the experimental uncertainty. The interaction parameters used in the ECS model for the CH4 + C3H8 binary system were derived from the ambient pressure measurements of Abe et al.33 at temperatures between (298.15 and 468.15) K, and agreement at higher pressures is taken to indicate the robustness of the model. Other data sets for this binary mixture are few: the data of Bicher and Katz34 measured using a rolling ball viscometer with stated average uncertainty of 3 % have an rms relative deviation from the REFPROP27,32 ECS model of 4.2 % and maximum relative deviation of −11.5 %. Table 4 lists the viscosities measured for the second mixture studied: (1 − x)CO2 + xCH4 with mole fraction x = (0.57 ± 0.01). The composition of this mixture was determined using a gas chromatograph. The results of the viscosity measurements and their deviations from the values calculated using the REFPROP27,31,32 ECS model are shown in Figure 11. Compared to the C3H8 + CH4 mixture, the CO2 + CH4 mixture is a more stringent test of the ECS model contained

T/K

p/MPa

ρEOS/(kg·m−3)

η/(μPa·s)

u(η)/(μPa·s)

280.32 280.31 280.24 280.22 280.20 280.17 303.20 303.20 303.19 303.19 303.18 303.18 328.16 328.15 328.14 328.14 328.14 328.13

1.51 2.51 3.50 4.48 5.49 6.41 1.50 2.50 3.50 4.50 5.44 6.33 1.51 2.50 3.49 4.49 5.50 6.45

19.188 33.156 48.196 64.461 82.930 101.53 17.387 29.806 42.969 56.946 70.895 84.873 16.012 27.054 38.553 50.651 63.381 75.834

13.031 13.097 13.381 13.644 14.077 14.566 13.811 13.872 14.109 14.405 14.765 15.227 14.470 14.533 14.770 15.041 15.380 15.732

0.117 0.161 0.091 0.057 0.028 0.025 0.060 0.072 0.034 0.046 0.042 0.051 0.060 0.072 0.034 0.047 0.042 0.051

a The mixture density ρEOS used to determine the measured viscosity from the resonance curve was calculated from the GERG-2008 EOS implemented in REFPROP 9.1.13,31 The composition uncertainty corresponds to an uncertainty in the calculated viscosity of 0.017 μPa·s (that is a relative uncertainty of 0.13 %).

within REFPROP,27 because it is almost equimolar and the molecular similarity is less. At temperatures of 280 K and 303 K, the viscosity values measured in this work are within 1.6 % of those estimated from the REFPROP ECS model,27,32 which is close to the experimental uncertainty of 0.6 %, particularly if effects of composition uncertainty (corresponding to an uncertainty in the viscosity of 0.13 %) are considered. At a temperature of 328 K, however, our data deviate by a statistically significant margin of between (−3 and −4) %. Available viscosities for (CO2 + CH4) in the archival literature are limited and as seen from Figure 12, the most extensive data set of deWitt and Thodos35 exhibits significant systematic deviations from the REFPROP ECS27 model. The deviations become particularly large at higher densities, reaching a maximum value of about 8 % near carbon dioxide’s 1626



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CH4, and CO2, as well as mixtures of (CO2 + CH4) and (C3H8 + CH4). Given the robust and compact nature of the doubly clamped geometry, this represents an important step toward the application of this type of apparatus to measurements requiring operation over a wide range of viscosity and density. When compared with literature data and models for mixture viscosities, the data measured here for the (CO2 + CH4) binary system indicate that further measurements are needed to improve the reliability of viscosity predictions for such mixtures.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

Figure 11. Fractional differences between the measured viscosity (ηmeas) and calculated (ηcalc) using the REFPROP ECS model27,32 for the (CO2 + CH4) system as a function of density ρ. This work: ◆, T = 280 K; ■, T = 303 K; ●, T = 328 K. The error bars are one standard deviation of repeat measurements (Table 4). Viscosity data of Kestin and Yata:36 +, T = 293 K; × , T = 303 K; and DeWitt and Thodos:35 ◇, T = 323 K; ○, T = 373 K; △, T = 423 K; □, T = 473 K.

E.F.M. acknowledges Chevron for their support of the research through the Gas Process Engineering endowment. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Craig Grimm for assisting with the apparatus setup. A.R.H.G. wishes to acknowledge Schlumberger for permission to collaborate on this project.

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REFERENCES

(1) Ciotta, F.; Trusler, J. M. Improved Understanding of VibratingWire Viscometer−Densimeters. J. Chem. Eng. Data 2010, 55, 2195− 2201. (2) Retsina, T.; Richardson, S.; Wakeham, W. The Theory of a Vibrating-Rod Densimeter. Appl. Sci. Res. 1986, 43, 127−158. (3) Goodwin, A. R.; Marsh, K. N. An Absolute Viscometer for Liquids: Measurement of the Viscosity of Water at T = 298.15 K and p = 0.1 MPa. J. Chem. Eng. Data 2011, 56, 167−170. (4) Schley, P.; Jaeschke, M.; Küchenmeister, C.; Vogel, E. Viscosity Measurements and Predictions for Natural Gas. Int. J. Thermophys. 2004, 25, 1623−1652. (5) Kandil, M. E.; Marsh, K. N.; Goodwin, A. R. Vibrating Wire Viscometer with Wire Diameters of (0.05 and 0.15) mm: Results for Methylbenzene and Two Fluids with Nominal Viscosities at T = 298 K and p = 0.01 MPa of (14 and 232) mPa·s at Temperatures between (298 and 373) K and Pressures below 40 MPa. J. Chem. Eng. Data 2005, 50, 647−655. (6) Harrison, C.; Sullivan, M.; Godefroy, S.; Matsumoto, N.; Goodwin, A. R.; Hsu, K. Operation of a Vibrating Wire Viscometer at Viscosities Greater than 0.2 Pa·s: Results for a Certified Reference Fluid with Nominal Viscosity at T = 273 K and p = 0.1 MPa of 0.652 Pa·s while Stagnant and a Fluid of Nominal Viscosity of 0.037 Pa·s while Flowing. J. Chem. Eng. Data 2007, 52, 774−782. (7) Lassner, E.; Schubert, W. D. Tungsten: Properties, Chemistry, Technology of the Elements, Alloys, and Chemical Compounds; Kluwer Academic/Plenum: New York, 1999. (8) Lockwood, E. H. A Book of Curves; Cambridge University Press: England,1971. (9) Metalcutting Corporation, 89 Commerce Rd, Cedar Grove, NJ, USA. www.metalcutting.com (accessed 02/03/2014). (10) Cencek, W.; Przybytek, M.; Komasa, J.; Mehl, J. B.; Jeziorski, B.; Szalewicz, K. Effects of Adiabatic, Relativistic, and Quantum Electrodynamics Interactions on the Pair Potential and Thermophysical Properties of Helium. J. Chem. Phys. 2012, 136, 4303−4334. (11) Najafi, B.; Ghayeb, Y.; Rainwater, J.; Alavi, S.; Snider, R. Improved Initial Density Dependence of the Viscosity and a Corresponding States Function for High Pressures. Physica A 1998, 260, 31−48. (12) Moldover, M. R.; Gillis, K. A.; May, E. F., Stringent Tests of the Ab Initio Equation of State of Helium-4. In 18th Symposium on Thermophysical Properties, Boulder, CO., June 24-29, 2012.

Figure 12. Fractional differences between the measured viscosity (ηmeas) and viscosity calculated (ηcalc) using the REFPROP ECS model27,32 for (CO2 + CH4) system as a function of density ρ. This work: ◆, T = 280 K; ■, T = 303 K; ●, T = 328 K. Viscosity data of DeWitt and Thodos:35 ◇, T = 323 K; ○, T = 373 K; △, T = 423 K; □, T = 473 K.

critical density of 473 kg·m−3. The data of Kestin et al.36 measured at temperatures of (293 and 303) K, show an rms deviation of 0.17 % and a maximum deviation of less than 0.5 %, which reflects the fact that the ECS model was tuned to the data of Kestin for this binary system. We note that the deviations of this work, as well as the data of deWitt and Thodos,35 from the ECS model exhibit consistent deviation at higher temperatures than the data of Kestin et al.,36 which indicate that the temperature dependence in the REFPROP ECS model27 for this system may require adjustment. Furthermore, these results demonstrate the need for further measurements at high pressures for these binary mixtures.

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CONCLUSION We have introduced new techniques for the measurement and analysis of vibrating-wire viscometry of gases and gas mixtures at moderate densities using a doubly clamped vibrating-wire operated in steady-state mode. The methods introduced have been validated by measuring the viscosity of pure fluids He, N2, 1627



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(36) Kestin, J.; Yata, J. Viscosity and Diffusion Coefficient of Six Binary Mixtures. J. Chem. Phys. 1968, 49, 4780−4791.

(13) Berg, R. F.; May, E. F.; Moldover, M. R. Viscosity Ratio Measurements with Capillary Viscometers. J. Chem. Eng. Data 2014, 59, 116−124. (14) Mehl, J. B. Analysis of Resonance Standing-Wave Measurements. J. Acoust. Soc. Am. 1978, 64, 1523−1525. (15) Morse, P. M.; Ingard, K. U.; Shankland, R. Theoretical Acoustics. Phys. Today 1969, 22, 98. (16) Murthy, G. S.; Ramakrishna, B. Nonlinear Character of Resonance in Stretched Strings. J. Acoust. Soc. Am. 1965, 38, 461−471. (17) Sullivan, M.; Harrison, C.; Goodwin, A. R.; Hsu, K.; Godefroy, S. On the Nonlinear Interpretation of a Vibrating Wire Viscometer Operated at Large Amplitude. Fluid Phase Equilib. 2009, 276, 99−107. (18) Wynands, R.; Weyers, S. Atomic Fountain Clocks. Metrologia 2005, 42, S64. (19) Audonnet, F.; Pádua, A. A. Simultaneous Measurement of Density and Viscosity of n-Pentane from 298 to 383 K and up to 100 MPa using a Vibrating-Wire Instrument. Fluid Phase Equilib. 2001, 181, 147−161. (20) Caudwell, D. R. Viscosity of Dense Fluid Mixtures. Ph.D. Dissertation, Imperial College London: UK, 2004. (21) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1509−1596. (22) Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.; Yokozeki, A. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa. J. Phys. Chem. Ref. Data 2000, 29, 1361−1433. (23) Setzmann, U.; Wagner, W. A New Equation of State and Tables of Thermodynamic Properties for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 100 MPa. J. Phys. Chem. Ref. Data 1991, 20, 1061−1151. (24) Fenghour, A.; Wakeham, W. A.; Vesovic, V. The Viscosity of Carbon Dioxide. J. Phys. Chem. Ref. Data 1998, 27, 31−44. (25) Lemmon, E.; Jacobsen, R. Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air. Int. J. Thermophys. 2004, 25, 21−69. (26) Vogel, E.; Wilhelm, J.; Küechenmeister, C.; Jaeschke, M. HighPrecision Viscosity Measurements on Methane. High Temp. High Pressure 2000, 32, 73−81. (27) Lemmon, E. W.; McLinden, M. O.; Huber, M. L. Reference Fluid Properties Database 23, version 9.1, Physical and Chemical Properties Division; National Institute of Standards and Technology: Gaithersburg, VA, 2010. (28) Kestin, J.; Korfali, Ö .; Sengers, J. Density Expansion of the Viscosity of Carbon Dioxide near the Critical Temperature. Physica A 1980, 100, 335−348. (29) Evers, C.; Lösch, H.; Wagner, W. An Absolute ViscometerDensimeter and Measurements of the Viscosity of Nitrogen, Methane, Helium, Neon, Argon, and Krypton over a Wide Range of Density and Temperature. Int. J. Thermophys. 2002, 23, 1411−1439. (30) Berg, R. F.; Moldover, M. R. Recommended Viscosities of 11 Dilute Gases at 25 °C. J. Phys. Chem. Ref. Data 2012, 41, 043104. (31) Kunz, O.; Klimeck, R.; Wagner, W.; Jaeschke, M. The GERG2004 Wide-Range Equation of State for Natural Gases and Other Mixtures; de Recherches Gazières, Groupe Européen. VDI Verlag: Brussels, Belgium, 2007. (32) Chichester, J. C.; Huber, M. L. Documentation and Assessment of the Transport Property Model for Mixtures Implemented in NIST REFPROP (Version 8.0); National Institute of Standards and Technology: Gaithersburg, VA, 2008. (33) Abe, Y.; Kestin, J.; Khalifa, H.; Wakeham, W. The Viscosity and Diffusion Coefficients of the Mixtures of Four Light Hydrocarbon Gases. Physica A 1978, 93, 155−170. (34) Bicher, L. B., Jr.; Katz, D. L. Viscosities of the Methane− Propane System. Ind. Eng. Chem. 1943, 35, 754−761. (35) Dewitt, K. J.; Thodos, G. Viscosities of Binary Mixtures in the Dense Gaseous State: The Methane−Carbon Dioxide System. Can. J. Chem. Eng. 1966, 44, 148−151. 1628


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