A recently developed dual vibrating-wire technique has been used to perform viscosity measurements of liquid toluene in the temperature range 213 K ≤ T ≤ 298 K, and at pressures up to approximately 20 MPa. The results were obtained by operating the vibrating-wire sensor in both forced and free decay modes. The estimated precision of the viscosity measurements, in either mode of
operation, is ±0.5%, for temperatures above or equal to 273 K, increasing with
decreasing temperature up to ±1% at 213 K. The corresponding overall uncertainty
is estimated to be within ±1% and ±1.5%, respectively

Caetano, Fernando J. P.

Fareleira, João M. N. A.

Mata, José L. Correia da

Oliveira, Carla Padrel de

Wakeham, William A.

International Journal of Thermophysics

Repositório Aberto da Universidade Aberta

10195-928X/04/0100-0001/0 © 2004 Plenum Publishing CorporationInternational Journal of Thermophysics, Vol. 25, No. 1, January 2004 (© 2004)Viscosity Measurements of Liquid Toluene at LowTemperatures Using a Dual Vibrating-Wire TechniqueF. J. P. Caetano,1, 2 J. L. Correia da Mata,1, 3 J. M. N. A. Fareleira,1, 4C. M. B. P. Oliveira,1, 2 and W. A. Wakeham51 Centro de Química Estrutural, Instituto Superior Técnico, Universidade Técnica de Lisboa,Av. Rovisco Pais, 1049-001 Lisbon, Portugal.2Universidade Aberta, Rua da Escola Politécnica, 147, 1269-001 Lisbon, Portugal.3 Academia Militar, Paço da Rainha, 29, 1150-244 Lisbon, Portugal.4 To whom correspondence should be addressed. E-mail: j.fareleira@ist.utl.pt5University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom.Received April 28, 2003A recently developed dual vibrating-wire technique has been used to performviscosity measurements of liquid toluene in the temperature range 213 K [T [ 298 K, and at pressures up to approximately 20 MPa. The results wereobtained by operating the vibrating-wire sensor in both forced and free decaymodes. The estimated precision of the viscosity measurements, in either mode ofoperation, is ±0.5%, for temperatures above or equal to 273 K, increasing withdecreasing temperature up to ±1% at 213 K. The corresponding overall uncer-tainty is estimated to be within ±1% and ±1.5%, respectively.KEY WORDS: dual vibrating-wire technique; low temperature; toluene;viscosity.1. INTRODUCTIONWe have previously described a vibrating-wire instrument, operated in aforced mode, which has been used for the simultaneous measurement of thedensity and the viscosity of fluids, over wide ranges of temperature andpressure [1–4]. Recently [5], we have described some preliminary tests ofa novel instrument, showing that it is possible to perform mutually consis-tent viscosity measurements by operating a vibrating-wire sensor both inforced and free decay modes. The present work reports viscosity mea-surements of toluene, obtained by the latter instrument using both modesof operation, at low temperatures and pressures up to 20 MPa. New elec-tronic equipment has been purposely built that enables the monitoring ofeither the frequency or the transient responses of the vibrating-wire sensor.Both modes of operation are described by an exact theory, so that the vis-cosity can be obtained from an analysis of the resonance curve, or from thetime response of the sensor [5, 6].The consistency of the two approaches, under the same working con-ditions, as well as the quality of the measuring system, have been assessedduring the measurements. It is concluded that the new equipment improvesthe forced mode of operation, adds new flexibility to the vibrating-wiremethod, and provides further evidence that the instrument is workingaccording to its theoretical description.2. THEORETICAL MODELThe theoretical model for the vibrating-wire viscometer developed byRetsina et al. [7, 8] is based on the characteristics of the oscillation of asolid body surrounded by a fluid. The theoretical model for the forcedmode of operation has been published by Pádua et al. [9], and, for the freedecay mode of operation, the model equations are essentially analogousto the ones previously described by Oliveira [10] and Assael et al. [11].Hence, only a brief description of the working equations for both modes isgiven here.For the forced mode, the wire’s velocity resonant frequency, wr, is thesolution of the equation [7, 9]:ddw˜3 w˜2(bŒ+2D0)2+5w˜(1+b)− w˜2Bw˜624=0 (1)and the half-width, of a resonance curve, Dw=w+−w− , is related to thecell parameters through the equation:w˜± (bŒ+2D0)w˜2± (bŒ+2D0)2+5w˜± (1+b)− w˜2Bw˜±62=12w˜2r (bŒ+2D0)w˜2r (bŒ+2D0)2+5w˜r(1+b)− w˜2Bw˜r62 (2)2 Caetano et al.The tilde above a symbol indicates a corresponding dimensionlessquantity, as defined in [7], such that the relation between a dimensionlessfrequency and the corresponding angular frequency, w, is given by:w˜=w 14rsL4ER221/2 (3)where rs is the wire material density, E is its Young’s modulus, L is the half-length of the wire, and R is its radius.In the free decay mode, the wire’s oscillation frequency, w, and therate of decay, D, are described in terms of the cell parameters, by Eqs. (4)and (5) [8, 10, 11]:D=bŒ+2D02(b+1)(4)w2=(1+D20) w2B(1+b)(1−D2)+D(bŒ+2D0)(5)In the above equations, D0 is the internal damping of the wire and wBrepresents a resonant frequency of the wire, in the hypothetical situationwhere the wire is under vacuum and the weights tensioning it vertically areimmersed in the fluid.The oscillation characteristics of the vibrating-wire sensor, both inforced and free decay modes of operation, are dependent on the density, r,and viscosity, g, of the surrounding fluid, as well as on rs, through thefunctions:b=(r/rs) k and bŒ=(r/rs) kŒ (6)where, in the forced mode:k=−1+2 Im(A) and kŒ=2Re(A)with A=i 51+ 2K1(`iW)`iW K0(`iW)6 (7)In the free decay mode a corresponding set of equations applies:k=−1+2 Im(A) and kŒ=2Re(A)+2D Im(A)with A=(i−D) 11+ 2K1(`(i−D) W)`(i−D) W K0(`(i−D) W)2 (8)Viscosity of Liquid Toluene at Low Temperatures 3K0 and K1 are modified complex argument Bessel functions, and W isa dimensionless frequency related to the Reynolds number for the fluidmotion around the wire:W=rwR2g(9)3. EXPERIMENTAL SETUPThe current viscosity measurements of toluene at temperatures213 K [ T [ 298 K, and pressures up to 20 MPa, have been carried outwith a vibrating-wire viscometer described in detail elsewhere [5, 6].The vibrating-wire sensor consists of a tungsten wire with a nominaldiameter of 100 mm. The wire is suspended from a stainless steel chucksupported from the cap of a pressure vessel. At its lower end an assemblycomposed of two weights, through a pulley system, tensions the wire verti-cally. The main aim of this assembly is to inhibit any motion of the wirelower end, as discussed previously [11].The measuring cell was placed inside a stainless steel pressure vessel.For the present range of temperature the thermostatic system was based ona circulation cryostat (Colora KT 190S), using ethanol as the circulatingfluid. The temperature stability during a typical run was within ±0.005 K.Temperature measurements were performed using a platinum resistancethermometer with an uncertainty of ±0.01 K (ITS-90). Pressure readingswere carried out using a pressure transducer, supplied by Setra SystemsInc, with an uncertainty of ±0.13% (FS).3.1. Electrical Measuring Device/Signal DetectionThe new electronic equipment, which is described in detail elsewhere[6], was purposely developed to enable the use of both forced and freedecay modes of operation. In the forced mode a resonance curve isobtained by measuring the impedance of the vibrating-wire sensor, using afour-wire technique, over an adequate frequency range about the resonantfrequency. The forced mode oscillations are induced by passing a sinusoi-dal current through the wire, placed in the gap of a magnetic circuit. Theresponse of the wire can thus be interpreted as a complex impedance, Z¯,Z¯=Rs+iwLs+1Gw+iwC+1iwLp(10)of an equivalent electrical circuit, as shown in Fig. 1.4 Caetano et al.RsGω CLpLsFig. 1. Complete equivalent electrical circuitmodel of the oscillating wire in forced mode.The three parallel elements: the capacitor C, the inductor Lp, and thefrequency dependent conductance Gw, describe the resonance characteris-tics of the wire. The series resistor, Rs, models the circuit resistance, andthe inductance, Ls, is caused mainly by the high-pressure feed-through. Ina typical experiment these parameters are obtained by fitting the experi-mental data to the modulus of Eq. (10).In the free decay mode of operation a symmetrical current pulse isemployed to deflect the wire from its rest position. The subsequent decay-ing oscillations are detected by the induced voltage at the wire ends, in anelectro-magnetic coupling device, with a 12-bit A/D converter at a rateof 105 samples per second. The corresponding transient decay signal isexpected to conform to the following equation:V=Ae−Dwt sin(wt+f) (11)where the measured voltage, V, and the time interval, t, are the experimen-tal parameters. The amplitude A, the logarithmic decrement D, thefrequency of oscillation w, and the phase shift f, are obtained throughnonlinear fitting.3.2. Cell ParametersThe working equations (Eqs. (1)–(5)) incorporate several parametersthat are independent of the fluid, namely, D0, L, R, and rs. Once these andthe density of the fluid sample are known, the viscosity of the fluid can beobtained from the resonance frequency and the width of the resonancecurve—forced mode—or from the measured frequency and the decay of themotion—free decay mode.Viscosity of Liquid Toluene at Low Temperatures 5Table I. Cell Parameters at 298 KLength of the wire 2L 0.04077 mRadius of the wire R 48.965×10−6mDensity of the wire [12] rs 19230 kg ·m−3The length of the wire is measured, using a cathetometer, with anuncertainty of ±0.05%, and the density of the wire is taken from theliterature [12]. The accuracy target in the viscosity measurements isincompatible with a direct measurement of the wire radius. Therefore, thisparameter is advantageously determined in a single experimental run,under thermodynamic conditions where the viscosity of a reference fluid iswell known. Hence, a single experimental run in the forced mode, withtoluene as a reference fluid was performed near 298 K and 0.1 MPa inorder to obtain an effective average radius for the wire sensor. The viscos-ity and density reference data for toluene were taken from Gonçalves et al.[13] and Kashiwagi et al. [14], respectively.The actual cell parameters for the present setup, at 298 K, are shownin Table I. Given that the current measurements are concerned solely withviscosity, the only calibration parameter is the wire radius, which is essen-tially determined from the viscosity of the calibration fluid.4. EXPERIMENTAL RESULTS4.1. SampleThe sample of toluene (Sigma Aldrich, HPLC grade, with a nominalpurity of 99.8%) was dried using molecular sieves (Riedel–deHaën, 0.4 nm),over a long period of exposure. The final water content was less than10 ppm as shown by a Karl–Fischer titration analysis. The toluene wasintroduced into the vessel, under vacuum.4.2. Density DataA new correlation for the density of toluene has been developed byAssael et al. [15], based on experimental data from 198 to 380 K and up to280 MPa. This Tait-type correlation, with a claimed uncertainty of ±0.1%,within 95% confidence limits, has been used in the present work in order toprovide the density of liquid toluene over the temperature and pressureranges of the viscosity measurements. It should be noted that a previously6 Caetano et al.performed sensitivity analysis has indicated that an uncertainty of ±0.1%in the density yields an uncertainty of approximately ±0.1% in the viscos-ity for the present measurements.4.3. Experimental MeasurementsThe estimated precision of the present viscosity measurements, eitherin forced or free decay modes of operation, is ±0.5% for temperaturesabove 273 K, increasing with decreasing temperature up to 1% at 213 K.The mutual consistency of the results obtained using the two modes wasverified for the entire temperature range at several pressures, as shown inFig. 2. It should be noted that each point represents the average of two orthree experimental values. It can be seen that the differences between theresults obtained by the two modes are mostly within ±0.4%, whichadequately confirms the internal consistency of the two modes of operationfor viscosity measurements. The corresponding overall uncertainty is esti-mated to be within ±1% and ±1.5%, at temperatures above, and below,273 K, respectively.In Table II the results of the present measurements for the viscosity oftoluene as a function of pressure are shown, together with the correspond-ing density values, for the temperature range of 213 to 298 K, and pressuresup to approximately 20 MPa. These data are the result of an average oftwo or three experimental values obtained in sets of measurements per-formed both in increasing and decreasing cycles of pressure. No hysteresis-0.6-0.4-0.200.20.40.60 5 10 15 20 25p , MPa100( ηforced- ηfree)/ηforced, %Fig. 2. Differences of the viscosity measurements for toluene, using forcedand free decay modes of operation: (J) 298.31 K, (I) 273.81 K, (G) 248.47 K,(n) 222.57 K, (g) 213.45 K.Viscosity of Liquid Toluene at Low Temperatures 7Table II. Experimental Viscosity Measurements of Toluene, as a Function of Temperatureand Pressure. (The density data were evaluated according to the correlation equation givenin [15])p (MPa) r (kg ·m−3) g (mPa · s) p (MPa) r (kg ·m−3) g (mPa · s)T=298.31 K T=273.15 K0.12 862.4 0.5543 0.13 885.0 0.76370.13 862.4 0.5550a 0.14 885.0 0.76270.13 862.4 0.5539 0.14 885.0 0.7652a2.57 864.3 0.5637a 2.51 886.6 0.77902.57 864.3 0.5684 2.54 886.6 0.7777a4.96 866.1 0.5743 2.54 886.6 0.77944.97 866.1 0.5724a 5.00 888.3 0.79334.97 866.1 0.5753 5.07 888.3 0.79365.04 866.1 0.5756 9.98 891.5 0.82467.60 868.0 0.5867a 10.03 891.5 0.82657.60 868.0 0.58659.12 869.2 0.594010.01 869.8 0.597910.05 869.8 0.6006a10.05 869.8 0.5986T=248.47 K T=222.57 K0.12 908.4 1.178 0.11 932.5 2.0990.13 908.4 1.178 1.00 932.9 2.1150.13 908.4 1.185a 1.00 932.9 2.118a2.49 909.8 1.200 2.52 933.7 2.1482.49 909.8 1.197a 2.52 933.7 2.155a4.96 911.2 1.226 4.97 934.9 2.1954.96 911.2 1.225a 5.00 935.0 2.217a7.52 912.7 1.252 5.00 935.0 2.2047.52 912.7 1.248a 7.52 936.2 2.266a9.98 914.1 1.278 7.52 936.2 2.23510.00 914.1 1.278 10.02 937.5 2.31110.00 914.1 1.284a 10.02 937.5 2.316a14.91 939.8 2.429a14.91 939.8 2.42620.04 942.3 2.533a20.04 942.3 2.545a Values obtained by using free decay mode.8 Caetano et al.Table II. (Continued)p (MPa) r (kg ·m−3) g (mPa · s) p (MPa) r (kg ·m−3) g (mPa · s)T=213.45 K T=298.41 K0.11 941.1 2.723 0.11 862.3 0.55150.11 941.1 2.734a 0.11 862.3 0.5536a1.13 941.6 2.752a 0.13 862.3 0.55172.49 942.2 2.792 2.52 864.1 0.56202.49 942.2 2.799a 2.52 864.1 0.5635a5.02 943.4 2.864 14.94 873.2 0.61955.02 943.4 2.852a 14.94 873.2 0.6205a9.98 945.8 3.043 19.94 876.7 0.641110.02 945.8 3.027 19.94 876.7 0.6424a10.02 945.8 3.039a 20.02 876.7 0.641815.11 948.2 3.18715.11 948.2 3.203a15.23 948.2 3.19915.23 948.2 3.201a20.05 950.4 3.346a20.05 950.4 3.345-2.0-1.5-1.0-0.50.00.51.01.52.00.0 5.0 10.0 15.0 20.0 25.0p , MPa100 (ηexp- ηcorr)/ηcorr, %Fig. 3. Deviations of the present measurements from the correlation of Assaelet al. [15]. Forced mode: (J) 298.31 K, (G) 273.81 K, (N) 248.47 K, (I) 222.57 K,(w| ) 213.45 K, (J) 298.41 K; Free decay mode: (j) 298.31 K, (g) 273.81 K, (n)248.47 K, (i) 222.57 K, (×| ) 213.45 K, (j) 298.41 K.Viscosity of Liquid Toluene at Low Temperatures 9has been detected within the experimental precision of the measurements.Furthermore, it should be noted that, for the same experimental condi-tions, measurements were performed for both modes of operation and thevalues are appropriately identified in Table II.For comparison purposes we used the viscosity correlation developedby Assael et al. [15], which has an estimated uncertainty of ±2.7% withina 95% confidence limit. This correlation, based on the most recent exper-imental data, covers a temperature range from 213 to 373 K and a pressurerange from atmospheric up to 250 MPa. However, at low temperatures andhigh densities the uncertainty in the correlation is the greatest [15] (up to4% for density values above 920 kg ·m−3) because of the scarcity of theexperimental data and their dispersion. Indeed, it should be pointed outthat the present work provides the first measurements below 217 K.The experimental values obtained by both modes of operation arecompared with the above-mentioned correlation in Fig. 3. The comparisonshows that the actual measurements are well within the overall uncertaintyof the correlation over the entire range of conditions.5. CONCLUSIONSThe results of new measurements of the viscosity of liquid toluenehave been presented, which expand the available published experimentaldata in the very low temperature region (down to 213 K) and pressures upto 20 MPa. Our results are intended to supply data for these thermo-dynamic regions, and thus reinforce the establishment of toluene as astandard reference fluid for viscosity.As a final note it should be stressed that the verification of the internalconsistency between the two different modes of operation in the presentmeasurements constitutes a stringent validation of the correct operation ofthe present viscometer.ACKNOWLEDGMENTSThe present work has been supported by the Fundação para a Ciênciae a Tecnologia (Portugal) under PRAXIS and POCTI programmes, withEuropean Community funds (FEDER) and national funds.REFERENCES1. A. A. H. Pádua, J. M. N. A. Fareleira, J. C. G. Calado, and W. A. Wakeham, Int. J.Thermophys. 17:781 (1996).10 Caetano et al.2. A. A. H. Pádua, J. M. N. A. Fareleira, J. C. G. Calado, and W. A. Wakeham, J. Chem.Eng. Data 41:731 (1996).3. A. A. H. Pádua, J. M. N. A. Fareleira, J. C. G. Calado, and W. A. Wakeham, J. Chem.Eng. Data 41:1488 (1996).4. H. M. T. Avelino, J. M. N. A. Fareleira, and W. A. Wakeham, Int. J. Thermophys. 24:323(2003).5. J. L. Correia da Mata, J. M. N. A. Fareleira, C. M. B. P. Oliveira, F. J. P. Caetano, andW. A. Wakeham, High Temp.-High Press. 33:669 (2001).6. J. L. Correia da Mata, J. M. N. A. Fareleira, P. M. B. S. Gira˜o, and W. A. Wakeham, tobe published.7. T. Retsina, S. M. Richardson, and W. A. Wakeham, Appl. Sci. Res. 43:127 (1986).8. T. Retsina, S. M. Richardson, and W. A. Wakeham, Appl. Sci. Res. 43:325 (1987).9. A. A. H. Pádua, J. M. N. A. Fareleira, J. C. G. Calado, and W. A. Wakeham, Rev. Sci.Instrum. 69:2392 (1998).10. C. M. B. P. Oliveira, Viscosity of Liquid Hydrocarbons at High Pressure, Ph.D. thesis(Imperial College of Science, Technology and Medicine, London, United Kingdom, 1991).11. M. J. Assael, C. M. B. P. Oliveira, M. Papadaki, and W. A. Wakeham, Int. J. Ther-mophys. 13:593 (1992).12. C. S. Barrett, Structure of Materials (McGraw–Hill, New York, 1952).13. F. A. Gonçalves, K. Hamano, J. V. Sengers, and J. Kestin, Int. J. Thermophys. 6:641(1987).14. H. Kashiwagi, T. Hashimoto, Y. Tanaka, H. Kubota, and T. Makita, Int. J. Thermophys.3:201 (1982).15. M. J. Assael, H. M. T. Avelino, N. K. Dalaouti, J. M. N. A. Fareleira, and K. R. Harris,Int. J. Thermophys. 22:789 (2001).Viscosity of Liquid Toluene at Low Temperatures 11

Viscosity measurements of liquid toluene at low temperatures using a dual vibrating-wire technique

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