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Viscosity measurements of hydrogen at high temperatures up to 573 K by a curved vibrating wire method
J. Chem. Thermodynamics 89 (2015) 22–26
Contents lists available at ScienceDirect
J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct
Viscosity measurements of hydrogen at high temperatures up to 573 K by a curved vibrating wire method N. Sakoda a,b,c,⇑, T. Hisatsugu a, K. Furusato a, K. Shinzato c, M. Kohno a,b,c, Y. Takata a,b,c a
Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka, Japan c Research Center for Hydrogen Industrial Use and Storage (HYDROGENIUS), Kyushu University, Fukuoka, Japan b
a r t i c l e
i n f o
Article history: Received 31 January 2015 Received in revised form 5 April 2015 Accepted 24 April 2015 Available online 2 May 2015 Keywords: Viscosity Vibrating wire Hydrogen High temperature
a b s t r a c t The viscosities of hydrogen were measured at temperatures of (296 to 573) K and at pressures up to 0.7 MPa by the vibrating wire method. In this study, a tungsten wire 50 lm in diameter and 24 mm in length is bent into semicircular form. The direction of the vibrating motion is fixed using the curved wire, and a more compact sample vessel can be used than in a traditional straight vibrating wire method requiring weight for the tension in the wire. Alternating voltages with different frequencies were supplied to the curved wire, which was set between samarium cobalt magnets. The generated induced voltages depending on the supplied frequencies were measured by a lock-in amplifier, and the resonant curve was obtained. The resonant frequency and half-width of the resonant curve were determined by curve fitting. The wire’s effective diameter and internal friction coefficient, which represents the damping from the wire material and the magnetic force, are very important parameters for evaluating the viscosities, and they were precisely calibrated by measuring helium and nitrogen as reference fluids. Finally, the viscosities of hydrogen were obtained with an uncertainty of 1.4% (k = 2). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The thermophysical properties of hydrogen are highly important for the coming hydrogen society, and accurate transport properties in a wide temperature and pressure range are essential for safe and inexpensive design of fuel cell vehicles and hydrogen refueling stations, which are promising key technologies for the use of hydrogen as a clean energy carrier. The viscosities of hydrogen were measured recently at temperatures from (295 to 400) K and at high pressures up to 100 MPa by the capillary tube method [1]. However, at higher temperatures, this method is hard to apply because the sealing part of the capillary tube becomes fragile; thus, the vibrating wire method was adopted in this study. The vibrating wire method is applicable over a wide range of fluid viscosities, including those of gases and liquids [2,3], although a complicated theoretical formula and data analysis are necessary. In the present method, the vibrating wire was bent into semicircular form and set in the sample vessel with magnets. Compared with the traditional straight vibrating wire method, the curved wire method does not require a weight for the tension of the wire, and the direction of the vibrating motion is fixed. In addition, when the curved wire
is used, a more compact sample vessel can be used, and this is desirable for measuring high-temperature hydrogen because careful sample treatment is required. The curved vibrating wire method was previously used for viscosity measurements of liquid helium [4–6] and quite recently applied to hydrogen measurements at room temperature [7,8]. However, the feasibility of the curved wire method at high temperatures has not been well considered; hence, an accurate data acquisition procedure for this method is examined in this study. In the vibrating wire method, the effective diameter and internal friction coefficient of the wire strongly affect the determination of the viscosity. Moreover, at high temperatures, the decrease in the magnetic force should be taken into consideration. Therefore, the effective diameter of the wire and the temperature-dependent internal friction coefficient were precisely calibrated by measuring helium and nitrogen as reference fluids using the present method, and the viscosities of hydrogen were finally obtained from T = (296 to 573) K and at pressures up to 0.7 MPa. 2. Experimental 2.1. Measurement principle
⇑ Corresponding author at: Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan. E-mail address: [email protected] (N. Sakoda). http://dx.doi.org/10.1016/j.jct.2015.04.028 0021-9614/Ó 2015 Elsevier Ltd. All rights reserved.
The fundamental equations for the curved vibrating wire method were described previously [7,8], and are briefly explained
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
Nomenclature B E Hð1Þ n I L P Q 1 vac r T Vi Vmax
magnetic flux density, T Young’s modulus of the wire, Pa first Hankel function driving current of electricity, A length of the wire, m pressure, MPa internal friction coefficient radius of the wire, m temperature, K in-phase component of resonant curve, V maximum induced voltage, V
Voffset Vq
g gCAL gEXP 2k
qS qW x x0
here. The physical model is shown in figure 1(a), and (b) shows the original probe. The motion equation of the vibrating wire is expressed as
! 2 @2y pr 4 @ 4 y @y 0@ y pr qW 2 ¼ E D þc 2 @t 4 @x4 @t @t p 2 @y ; þ BI0 sin x sinðxtÞ Q 1 vac xqW pr @t L
voltage offset of induced voltage, V quadrature component of resonant curve, V viscosity of sample fluid, Pa s viscosity calculated from viscosity correlation, Pa s measured viscosity, Pa s half peak band-width, rad s1 density of sample fluid, kg m3 density of wire, kg m3 frequency, rad s1 resonant frequency, rad s1
2
where Hð1Þ n ðzÞ is the Hankel function. Alternating voltages with different frequencies are supplied to the wire, and the induced voltage is measured by a lock-in amplifier. The induced voltage is separated into two phases, which are the in-phase and quadrature components, and it is given by
VðtÞ ¼ V i cosðxtÞ þ V q sinðxtÞ: ð1Þ
The resonant curve and quadrature curve are given by
with boundary conditions of
yð0; tÞ ¼ yðL; tÞ ¼ 0;
ð2Þ
@ 2 y @x2
ð3Þ
x¼0
@ 2 y ¼ 2 @x
¼ 0;
0
D ¼ pqS r 2 xk ðmÞ;
ð4Þ
c0 ¼ pqS r 2 kðmÞ;
ð5Þ
and
m¼
r ðxqS =gÞ1=2 ; 2 0
k þ ik ¼ 1
z¼
V i ¼ V offset þ
Vq ¼
x¼L
where y is the deflection of the wire, and x is the axial location along the wire, as shown in figure 1(a). The parameters in equation (1) are given by
ð6Þ
pffiffiffi ð1Þ 2ð1 iÞ H1 ðzÞ ; ð1Þ m H0 ðzÞ
ð7Þ
pffiffiffi 2ð1 þ iÞm;
ð8Þ
ð9Þ
4k2 x2 2
ðx x2 Þ þ 4k2 x2 2 0
2kxðx20 x2 Þ 2
ðx x2 Þ þ 4k2 x2 2 0
V 0;
V 0;
0
k ¼
2 2kðpqW r 2 þ pr2 qS kÞ pQ 1 vac x0 qW r : 2 px0 qS r
2.2. Experimental apparatus A schematic diagram of the experimental apparatus is shown in figure 2. A tungsten wire 50 lm in nominal diameter and 23.55 mm in length, which had previously been annealed at
φ 20 mm
Gold-plated samarium cobalt magnet
Curved wire
Ceramic supporter
y x
y=0
I : Alternating current
B : Magnetic field
(a) Schematic diagram
ð12Þ
The viscosity of the sample, g, in equation (6), is finally determined by an iteration calculation to satisfy equations (4–8) and (12).
x=0 x=L
ð11Þ
respectively, where V0 = Vmax Voffset, and Vmax is the peak voltage of the resonant curve. By applying curve fitting to the resonant curve, the resonant frequency x0 and the half peak band-width 2k are obtained. Equation (1) with the boundary conditions leads to
Nickel rod
Magnet
ð10Þ
Curved tungsten wire
(b) Original probe with a ceramic supporter
FIGURE 1. Setup of the probe consisting of the curved tungsten wire and gold-plated samarium cobalt magnets held by a ceramic supporter.
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
supplies alternating voltages to the wire and also measures the resonant and quadrature signals.
3. Experimental results The obtained resonant curves of hydrogen, helium, and nitrogen at T = 298 K and 0.1 MPa are shown in figure 3. The samples were supplied from commercially available gas cylinders, and the purities of hydrogen, helium, and nitrogen were 99.999, 99.99995, and 99.99 vol.%, respectively. Each resonant curve is normalized
100(ηEXP - ηCAL) / ηCAL
T = 800 K, is bent into semicircular form, and the curved wire is welded to nickel rods, as shown in figure 1(b). Gold-plated samarium cobalt magnets of 80 mT are placed on a ceramic supporter. The ceramic supporter is contained in a sample vessel having an inner volume of 40 cm3, and the sample vessel is placed in a thermostatic oven with radiation shield plates. The temperature and pressure of the sample are measured by a platinum thermometer with an uncertainty of 50 mK and by a quartz pressure transducer with an uncertainty of 0.5 kPa, respectively. The lock-in amplifier used in this apparatus is highly sensitive and can measure output signals on the order of a few microvolts. The lock-in amplifier
PC Pressure transducer
Platinum thermometer Pressure T transducer
Vacuum pump
H2 He N2
100(ηEXP - ηCAL) / ηCAL
Lock-in amplifier
Exhaust
Cooling gas
Probe Heating coil
Vacuum pump
(Vi - Voffset) / (Vmax - Voffset)
-5 300
400
600 (b) Nitrogen
0
-5 300
400
500
600
0.3 MPa
0.5 MPa
0.7 MPa
FIGURE 5. Deviations of the helium and nitrogen measurements from the viscosity correlations [9,10] using the calibrated effective diameter and internal friction coefficient of the wire.
1 0.8
ω1
H2 He N2
ω2
0.6
TABLE 1 Results of helium viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a
2λ
0.4
P/MPa
0.2
ω0
0 333
334
335
FIGURE 3. Normalized resonant curves of hydrogen, helium, and nitrogen at 298 K and 0.1 MPa.
[×10-4] 6 5 4 3 2
T/K
gEXP/lPa s
300
400
500
600
T/K Determined by the helium and nitrogen measurements Determined by the resonant curve under vacuum a
FIGURE 4. Temperature dependence of the internal friction coefficient.
gCAL/lPa s [9]
100(gEXP gCAL)/gCAL
0.1003 0.1101 0.1116 0.1115 0.1111
296.26 322.56 373.12 472.08 572.96
19.64 20.84 23.07 27.12 31.07
0.1 MPa 19.76 20.95 23.15 27.25 31.19
0.3321 0.3065 0.3071 0.3137 0.3097
296.26 322.56 373.12 472.08 572.96
19.83 20.96 23.14 27.27 31.19
0.3 MPa 19.77 20.95 23.16 27.26 31.20
0.30 0.05 0.06 0.07 0.03
0.5003 0.5212 0.5042 0.5049 0.5143
296.26 322.56 373.12 472.08 572.96
19.79 21.01 23.18 27.25 31.19
0.5 MPa 19.77 20.96 23.16 27.26 31.20
0.11 0.24 0.06 0.05 0.05
0.7160 0.7230 0.6985 0.7177 0.7217
296.26 322.56 373.12 472.08 572.96
19.82 21.00 23.22 27.32 31.23
0.7 MPa 19.78 20.96 23.17 27.26 31.20
0.18 0.16 0.21 0.20 0.07
1 0
500
5
0.1 MPa
ω (2π)-1 / Hz
-1
0
T/K
FIGURE 2. Schematic diagram of the experimental apparatus.
Qvac
(a) Helium 5
0.59 0.50 0.35 0.50 0.38
Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
by the respective maximum voltage corresponding to the peak voltage at the resonant frequency x0 and corrected with the offset voltages. The maximum voltages range from several to several tens of microvolts, for instance, 8 lV for the nitrogen measurement. The fundamental equation expressed by equation (1) is based on the assumption that the wire is an infinitely long cylinder; thus, the effective diameter of the wire was determined by measuring helium and nitrogen as reference fluids and found to be 50.119 lm at T = 296 K with an uncertainty of 0.125 lm. The TABLE 2 Results of nitrogen viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a P/MPa
gEXP/lPa s
T/K
gCAL/lPa s [10]
100(gEXP gCAL)/gCAL
0.0999 0.1098 0.1112 0.1102 0.1117
296.26 322.56 373.12 472.10 572.98
17.66 18.84 21.04 24.98 28.64
0.1 MPa 17.72 18.91 21.10 25.03 28.66
0.3126 0.3073 0.3006 0.3097 0.3161
296.26 322.56 373.12 472.10 572.98
17.73 18.92 21.09 25.00 28.67
0.3 MPa 17.75 18.94 21.12 25.04 28.67
0.09 0.11 0.16 0.18 0.01
0.5009 0.5007 0.5032 0.5058 0.5020
297.05 322.56 373.12 472.10 572.98
17.83 18.99 21.19 25.09 28.71
0.5 MPa 17.81 18.96 21.14 25.06 28.68
0.13 0.17 0.24 0.14 0.11
0.7009 0.7012 0.6945 0.7034 0.7034
297.05 322.56 373.12 472.10 572.98
17.89 19.04 21.21 25.13 28.68
0.7 MPa 17.84 18.99 21.16 25.07 28.69
0.30 0.31 0.22 0.22 0.05
0.33 0.37 0.30 0.18 0.07
a Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
(a) Temperature dependence
η / μPa ·s
14
internal friction coefficient Q 1 vac in equation (1) represents the damping due to the wire material and magnetic force, and this coefficient is as important as the effective diameter of the wire for evaluating the viscosity. The magnetic force decreases at high temperatures, and the internal friction coefficient was also precisely determined by helium and nitrogen measurements with an uncertainty of 0.5%. The internal friction coefficient can be theoretically determined as 2k/x0 from the resonant curve measured under vacuum, although accurate determination of the coefficient is more difficult. The temperature dependence of the internal friction coefficient is shown in figure 4. As the temperature increases, the internal friction coefficient also increases, and the temperature dependence agrees well with the result obtained from the vacuum resonant curves. With the calibrated effective diameter and internal friction coefficient, the corrected helium and nitrogen viscosities are in good agreement with the viscosity correlations [9,10] within 0.6%, as shown in figure 5. The obtained values of helium and nitrogen are tabulated in tables 1 and 2, respectively. In the data acquisition analysis, the density of the tungsten wire was obtained from a thermophysical properties handbook [11], and the densities of helium and nitrogen were calculated from accurate equations of state [12,13]. Finally, the obtained hydrogen viscosities are shown in figure 6, and their numerical values are summarized in table 3. The uncertainty of the viscosity measurement is estimated as 1.4% (k = 2) by propagation from the measurement uncertainties of the effective diameter and internal friction coefficient. A virial-type equation of state for hydrogen was recently developed on the basis of the latest PVT property measurements up to 100 MPa [14], but the validity range of the equation of state extends only to T = 473 K, and in this analysis an equation of state covering higher temperatures [15] was adopted. At low pressures up to 0.7 MPa, the densities calculated using the two equations of state differ by less than 0.01% from T = (296 to 573) K. In the target temperature and pressure range of the present measurement, the pressure dependence of the viscosity is very small. In the dilute gas region, the viscosity correlation by Yusibani et al. [16] represents existing viscosity measurements within 2%, and comparisons TABLE 3 Results of hydrogen viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a P/MPa
T/K
12 10 8
300
400
500
600
gEXP/lPa s
100( ηEXP - ηCAL) / ηCAL
5
296.26 322.56 373.12 472.13 573.48
8.90 9.41 10.41 12.09 14.16
0.3053 0.3024 0.3012 0.2989 0.2982
296.26 322.56 373.12 472.13 573.48
8.97 9.46 10.47 12.14 14.20
0.3 MPa 8.87 9.39 10.37 12.16 13.89
1.10 0.66 0.97 0.18 2.22
0.5010 0.5020 0.5031 0.5001 0.5030
296.26 322.56 373.12 472.13 573.48
9.00 9.49 10.48 12.15 14.27
0.5 MPa 8.88 9.40 10.37 12.16 13.89
1.39 0.97 1.11 0.13 2.74
0.7006 0.6980 0.7014 0.7011 0.7016
296.26 322.56 373.12 472.13 573.48
9.02 9.51 10.50 12.16 14.25
0.7 MPa 8.88 9.40 10.37 12.17 13.90
1.55 1.10 1.24 0.07 2.53
0
-5 300
400
500
600
T/ K Trautz and Zink [17] Muzny et al. [19] Present data 0.1 MPa
0.3 MPa
Mal'tsev et al. [18]
0.5 MPa
0.7 MPa
FIGURE 6. Measurement results of hydrogen viscosities. (a) Temperature dependence of the viscosity, (b) deviations from a viscosity correlation [16].
a
100(gEXP gCAL)/gCAL
0.1028 0.1110 0.1116 0.1114 0.0989
T/ K (b) Deviation
gCAL/lPa s [16] 0.1 MPa 8.87 9.39 10.36 12.16 13.89
0.42 0.17 0.46 0.54 1.94
Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
with existing experimental data by Trautz and Zink [17] and Mal’tsev et al. [18] are shown in figure 6(b). A viscosity correlation of hydrogen was developed recently by Muzny et al. [19], and the correlation is adopted in REFPROP ver. 9.1 [20]. Figure 6(b) also shows comparison with this correlation at 0.1 MPa. The correlation by Yusibani et al. represents the present data within 2% except at T = 573 K, where the correlation and the present data are in agreement within both of the uncertainties. The viscosities calculated from the latest correlation by Muzny et al. are slightly larger than the values calculated from the correlation by Yusibani et al. above T = 310 K, and the present data agree well with the correlation by Muzny et al. within 2% including temperatures up to 573 K. 4. Summary The viscosities of hydrogen were measured by the curved vibrating wire method at high temperatures up to 573 K and at pressures up to 0.7 MPa. The effective diameter and internal friction coefficient of the wire were determined by measuring helium and nitrogen as reference fluids. With the calibrated effective diameter and internal friction coefficient, the corrected helium and nitrogen viscosities show good agreement with the viscosity correlations within 0.6%, and finally hydrogen viscosities were obtained with an uncertainty of 1.4% (k = 2). Acknowledgments This research was conducted as part of the ‘‘Research and Development of Technical Standards to Prevent Overfilling on Hydrogen Fueling for FCV’’ funded by the New Energy and Industrial Technology Development Organization (NEDO).
References [1] E. Yusibani, Y. Nagahama, M. Kohno, Y. Takata, P.L. Woodfield, K. Shinzato, M. Fujii, Int. J. Thermophys. 32 (2011) 1111–1124. [2] M.J. Assael, C.P. Oliveira, M. Papadaki, W.A. Wakeham, Int. J. Thermophys. 13 (1992) 593–615. [3] J. Wilhelm, E. Vogel, J.K. Lehmann, W.A. Wakeham, Int. J. Thermophys. 19 (1998) 391–401. [4] D.C. Carless, H.E. Hall, J.R. Hook, J. Low Temp. Phys. 50 (1983) 583–603. [5] A.M. Guenault, V. Keith, C.J. Kennedy, S.G. Mussett, G.R. Pickett, J. Low Temp. Phys. 62 (1986) 511–523. [6] M. Morishita, T. Kuroda, A. Sawada, T. Satoh, J. Low Temp. Phys. 76 (1989) 387– 415. [7] E. Yusibani, P.L. Woodfield, K. Shinzato, Y. Takata, M. Kohno, Exp. Therm. Fluid Sci. 47 (2013) 1–5. [8] K. Yoshimura, T. Uehara, K. Shinzato, T. Hisatsugu, N. Sakoda, M. Kohno, Y. Takata, Jpn. J. Thermophys. Prop. 28 (2014) 15–21. [9] V.D. Arp, R.D. McCarty, D.G. Friend, NIST Technical Note 1334 (revised), 1998. [10] E.W. Lemmon, R.T Jacobsen, Int. J. Thermophys. 25 (2004) 21–69. [11] Japan Society of Thermophysical Properties, Thermophysical Properties Handbook, Yokendo, Tokyo, 1990, p. 25. [12] R.D. McCarty, V.D. Arp, Adv. Cryog. Eng. 35 (1990) 1465–1475. [13] R. Span, E.W. Lemmon, R.T Jacobsen, W. Wagner, A. Yokozeki, J. Phys. Chem. Ref. Data 29 (2000) 1361–1433. [14] N. Sakoda, K. Shindo, K. Motomura, K. Shinzato, M. Kohno, Y. Takata, M. Fujii, Int. J. Thermophys. 33 (2012) 381–395. [15] J.W. Leachman, R.T Jacobsen, S.G. Penoncello, E.W. Lemmon, J. Phys. Chem. Ref. Data 38 (2009) 721–748. [16] E. Yusibani, P.L. Woodfield, K. Shinzato, M. Kohno, Y. Takata, M. Fujii, Jpn. J. Thermophys. Prop. 24 (2010) 21–27. [17] M. Trautz, R. Zink, Ann. Phys. (Leipzig) 7 (1930) 427–452. [18] V.A. Mal’tsev, O.A. Nerushev, S.A. Novopashin, V.V. Radchenko, W.R. Licht, E.J. Miller, V.S. Parekh, J. Chem. Eng. Data 49 (2004) 684–687. [19] C.D. Muzny, M.L. Huber, A.F. Kazakov, J. Chem. Eng. Data 58 (2013) 969–979. [20] E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP): Version 9.1, Standard Reference Data, National Institute of Standards and Technology, Gaithersburg, MD, 2013.
JCT 15-62
Contents lists available at ScienceDirect
J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct
Viscosity measurements of hydrogen at high temperatures up to 573 K by a curved vibrating wire method N. Sakoda a,b,c,⇑, T. Hisatsugu a, K. Furusato a, K. Shinzato c, M. Kohno a,b,c, Y. Takata a,b,c a
Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka, Japan c Research Center for Hydrogen Industrial Use and Storage (HYDROGENIUS), Kyushu University, Fukuoka, Japan b
a r t i c l e
i n f o
Article history: Received 31 January 2015 Received in revised form 5 April 2015 Accepted 24 April 2015 Available online 2 May 2015 Keywords: Viscosity Vibrating wire Hydrogen High temperature
a b s t r a c t The viscosities of hydrogen were measured at temperatures of (296 to 573) K and at pressures up to 0.7 MPa by the vibrating wire method. In this study, a tungsten wire 50 lm in diameter and 24 mm in length is bent into semicircular form. The direction of the vibrating motion is fixed using the curved wire, and a more compact sample vessel can be used than in a traditional straight vibrating wire method requiring weight for the tension in the wire. Alternating voltages with different frequencies were supplied to the curved wire, which was set between samarium cobalt magnets. The generated induced voltages depending on the supplied frequencies were measured by a lock-in amplifier, and the resonant curve was obtained. The resonant frequency and half-width of the resonant curve were determined by curve fitting. The wire’s effective diameter and internal friction coefficient, which represents the damping from the wire material and the magnetic force, are very important parameters for evaluating the viscosities, and they were precisely calibrated by measuring helium and nitrogen as reference fluids. Finally, the viscosities of hydrogen were obtained with an uncertainty of 1.4% (k = 2). Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The thermophysical properties of hydrogen are highly important for the coming hydrogen society, and accurate transport properties in a wide temperature and pressure range are essential for safe and inexpensive design of fuel cell vehicles and hydrogen refueling stations, which are promising key technologies for the use of hydrogen as a clean energy carrier. The viscosities of hydrogen were measured recently at temperatures from (295 to 400) K and at high pressures up to 100 MPa by the capillary tube method [1]. However, at higher temperatures, this method is hard to apply because the sealing part of the capillary tube becomes fragile; thus, the vibrating wire method was adopted in this study. The vibrating wire method is applicable over a wide range of fluid viscosities, including those of gases and liquids [2,3], although a complicated theoretical formula and data analysis are necessary. In the present method, the vibrating wire was bent into semicircular form and set in the sample vessel with magnets. Compared with the traditional straight vibrating wire method, the curved wire method does not require a weight for the tension of the wire, and the direction of the vibrating motion is fixed. In addition, when the curved wire
is used, a more compact sample vessel can be used, and this is desirable for measuring high-temperature hydrogen because careful sample treatment is required. The curved vibrating wire method was previously used for viscosity measurements of liquid helium [4–6] and quite recently applied to hydrogen measurements at room temperature [7,8]. However, the feasibility of the curved wire method at high temperatures has not been well considered; hence, an accurate data acquisition procedure for this method is examined in this study. In the vibrating wire method, the effective diameter and internal friction coefficient of the wire strongly affect the determination of the viscosity. Moreover, at high temperatures, the decrease in the magnetic force should be taken into consideration. Therefore, the effective diameter of the wire and the temperature-dependent internal friction coefficient were precisely calibrated by measuring helium and nitrogen as reference fluids using the present method, and the viscosities of hydrogen were finally obtained from T = (296 to 573) K and at pressures up to 0.7 MPa. 2. Experimental 2.1. Measurement principle
⇑ Corresponding author at: Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan. E-mail address: [email protected] (N. Sakoda). http://dx.doi.org/10.1016/j.jct.2015.04.028 0021-9614/Ó 2015 Elsevier Ltd. All rights reserved.
The fundamental equations for the curved vibrating wire method were described previously [7,8], and are briefly explained
23
N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
Nomenclature B E Hð1Þ n I L P Q 1 vac r T Vi Vmax
magnetic flux density, T Young’s modulus of the wire, Pa first Hankel function driving current of electricity, A length of the wire, m pressure, MPa internal friction coefficient radius of the wire, m temperature, K in-phase component of resonant curve, V maximum induced voltage, V
Voffset Vq
g gCAL gEXP 2k
qS qW x x0
here. The physical model is shown in figure 1(a), and (b) shows the original probe. The motion equation of the vibrating wire is expressed as
! 2 @2y pr 4 @ 4 y @y 0@ y pr qW 2 ¼ E D þc 2 @t 4 @x4 @t @t p 2 @y ; þ BI0 sin x sinðxtÞ Q 1 vac xqW pr @t L
voltage offset of induced voltage, V quadrature component of resonant curve, V viscosity of sample fluid, Pa s viscosity calculated from viscosity correlation, Pa s measured viscosity, Pa s half peak band-width, rad s1 density of sample fluid, kg m3 density of wire, kg m3 frequency, rad s1 resonant frequency, rad s1
2
where Hð1Þ n ðzÞ is the Hankel function. Alternating voltages with different frequencies are supplied to the wire, and the induced voltage is measured by a lock-in amplifier. The induced voltage is separated into two phases, which are the in-phase and quadrature components, and it is given by
VðtÞ ¼ V i cosðxtÞ þ V q sinðxtÞ: ð1Þ
The resonant curve and quadrature curve are given by
with boundary conditions of
yð0; tÞ ¼ yðL; tÞ ¼ 0;
ð2Þ
@ 2 y @x2
ð3Þ
x¼0
@ 2 y ¼ 2 @x
¼ 0;
0
D ¼ pqS r 2 xk ðmÞ;
ð4Þ
c0 ¼ pqS r 2 kðmÞ;
ð5Þ
and
m¼
r ðxqS =gÞ1=2 ; 2 0
k þ ik ¼ 1
z¼
V i ¼ V offset þ
Vq ¼
x¼L
where y is the deflection of the wire, and x is the axial location along the wire, as shown in figure 1(a). The parameters in equation (1) are given by
ð6Þ
pffiffiffi ð1Þ 2ð1 iÞ H1 ðzÞ ; ð1Þ m H0 ðzÞ
ð7Þ
pffiffiffi 2ð1 þ iÞm;
ð8Þ
ð9Þ
4k2 x2 2
ðx x2 Þ þ 4k2 x2 2 0
2kxðx20 x2 Þ 2
ðx x2 Þ þ 4k2 x2 2 0
V 0;
V 0;
0
k ¼
2 2kðpqW r 2 þ pr2 qS kÞ pQ 1 vac x0 qW r : 2 px0 qS r
2.2. Experimental apparatus A schematic diagram of the experimental apparatus is shown in figure 2. A tungsten wire 50 lm in nominal diameter and 23.55 mm in length, which had previously been annealed at
φ 20 mm
Gold-plated samarium cobalt magnet
Curved wire
Ceramic supporter
y x
y=0
I : Alternating current
B : Magnetic field
(a) Schematic diagram
ð12Þ
The viscosity of the sample, g, in equation (6), is finally determined by an iteration calculation to satisfy equations (4–8) and (12).
x=0 x=L
ð11Þ
respectively, where V0 = Vmax Voffset, and Vmax is the peak voltage of the resonant curve. By applying curve fitting to the resonant curve, the resonant frequency x0 and the half peak band-width 2k are obtained. Equation (1) with the boundary conditions leads to
Nickel rod
Magnet
ð10Þ
Curved tungsten wire
(b) Original probe with a ceramic supporter
FIGURE 1. Setup of the probe consisting of the curved tungsten wire and gold-plated samarium cobalt magnets held by a ceramic supporter.
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
supplies alternating voltages to the wire and also measures the resonant and quadrature signals.
3. Experimental results The obtained resonant curves of hydrogen, helium, and nitrogen at T = 298 K and 0.1 MPa are shown in figure 3. The samples were supplied from commercially available gas cylinders, and the purities of hydrogen, helium, and nitrogen were 99.999, 99.99995, and 99.99 vol.%, respectively. Each resonant curve is normalized
100(ηEXP - ηCAL) / ηCAL
T = 800 K, is bent into semicircular form, and the curved wire is welded to nickel rods, as shown in figure 1(b). Gold-plated samarium cobalt magnets of 80 mT are placed on a ceramic supporter. The ceramic supporter is contained in a sample vessel having an inner volume of 40 cm3, and the sample vessel is placed in a thermostatic oven with radiation shield plates. The temperature and pressure of the sample are measured by a platinum thermometer with an uncertainty of 50 mK and by a quartz pressure transducer with an uncertainty of 0.5 kPa, respectively. The lock-in amplifier used in this apparatus is highly sensitive and can measure output signals on the order of a few microvolts. The lock-in amplifier
PC Pressure transducer
Platinum thermometer Pressure T transducer
Vacuum pump
H2 He N2
100(ηEXP - ηCAL) / ηCAL
Lock-in amplifier
Exhaust
Cooling gas
Probe Heating coil
Vacuum pump
(Vi - Voffset) / (Vmax - Voffset)
-5 300
400
600 (b) Nitrogen
0
-5 300
400
500
600
0.3 MPa
0.5 MPa
0.7 MPa
FIGURE 5. Deviations of the helium and nitrogen measurements from the viscosity correlations [9,10] using the calibrated effective diameter and internal friction coefficient of the wire.
1 0.8
ω1
H2 He N2
ω2
0.6
TABLE 1 Results of helium viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a
2λ
0.4
P/MPa
0.2
ω0
0 333
334
335
FIGURE 3. Normalized resonant curves of hydrogen, helium, and nitrogen at 298 K and 0.1 MPa.
[×10-4] 6 5 4 3 2
T/K
gEXP/lPa s
300
400
500
600
T/K Determined by the helium and nitrogen measurements Determined by the resonant curve under vacuum a
FIGURE 4. Temperature dependence of the internal friction coefficient.
gCAL/lPa s [9]
100(gEXP gCAL)/gCAL
0.1003 0.1101 0.1116 0.1115 0.1111
296.26 322.56 373.12 472.08 572.96
19.64 20.84 23.07 27.12 31.07
0.1 MPa 19.76 20.95 23.15 27.25 31.19
0.3321 0.3065 0.3071 0.3137 0.3097
296.26 322.56 373.12 472.08 572.96
19.83 20.96 23.14 27.27 31.19
0.3 MPa 19.77 20.95 23.16 27.26 31.20
0.30 0.05 0.06 0.07 0.03
0.5003 0.5212 0.5042 0.5049 0.5143
296.26 322.56 373.12 472.08 572.96
19.79 21.01 23.18 27.25 31.19
0.5 MPa 19.77 20.96 23.16 27.26 31.20
0.11 0.24 0.06 0.05 0.05
0.7160 0.7230 0.6985 0.7177 0.7217
296.26 322.56 373.12 472.08 572.96
19.82 21.00 23.22 27.32 31.23
0.7 MPa 19.78 20.96 23.17 27.26 31.20
0.18 0.16 0.21 0.20 0.07
1 0
500
5
0.1 MPa
ω (2π)-1 / Hz
-1
0
T/K
FIGURE 2. Schematic diagram of the experimental apparatus.
Qvac
(a) Helium 5
0.59 0.50 0.35 0.50 0.38
Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
by the respective maximum voltage corresponding to the peak voltage at the resonant frequency x0 and corrected with the offset voltages. The maximum voltages range from several to several tens of microvolts, for instance, 8 lV for the nitrogen measurement. The fundamental equation expressed by equation (1) is based on the assumption that the wire is an infinitely long cylinder; thus, the effective diameter of the wire was determined by measuring helium and nitrogen as reference fluids and found to be 50.119 lm at T = 296 K with an uncertainty of 0.125 lm. The TABLE 2 Results of nitrogen viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a P/MPa
gEXP/lPa s
T/K
gCAL/lPa s [10]
100(gEXP gCAL)/gCAL
0.0999 0.1098 0.1112 0.1102 0.1117
296.26 322.56 373.12 472.10 572.98
17.66 18.84 21.04 24.98 28.64
0.1 MPa 17.72 18.91 21.10 25.03 28.66
0.3126 0.3073 0.3006 0.3097 0.3161
296.26 322.56 373.12 472.10 572.98
17.73 18.92 21.09 25.00 28.67
0.3 MPa 17.75 18.94 21.12 25.04 28.67
0.09 0.11 0.16 0.18 0.01
0.5009 0.5007 0.5032 0.5058 0.5020
297.05 322.56 373.12 472.10 572.98
17.83 18.99 21.19 25.09 28.71
0.5 MPa 17.81 18.96 21.14 25.06 28.68
0.13 0.17 0.24 0.14 0.11
0.7009 0.7012 0.6945 0.7034 0.7034
297.05 322.56 373.12 472.10 572.98
17.89 19.04 21.21 25.13 28.68
0.7 MPa 17.84 18.99 21.16 25.07 28.69
0.30 0.31 0.22 0.22 0.05
0.33 0.37 0.30 0.18 0.07
a Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
(a) Temperature dependence
η / μPa ·s
14
internal friction coefficient Q 1 vac in equation (1) represents the damping due to the wire material and magnetic force, and this coefficient is as important as the effective diameter of the wire for evaluating the viscosity. The magnetic force decreases at high temperatures, and the internal friction coefficient was also precisely determined by helium and nitrogen measurements with an uncertainty of 0.5%. The internal friction coefficient can be theoretically determined as 2k/x0 from the resonant curve measured under vacuum, although accurate determination of the coefficient is more difficult. The temperature dependence of the internal friction coefficient is shown in figure 4. As the temperature increases, the internal friction coefficient also increases, and the temperature dependence agrees well with the result obtained from the vacuum resonant curves. With the calibrated effective diameter and internal friction coefficient, the corrected helium and nitrogen viscosities are in good agreement with the viscosity correlations [9,10] within 0.6%, as shown in figure 5. The obtained values of helium and nitrogen are tabulated in tables 1 and 2, respectively. In the data acquisition analysis, the density of the tungsten wire was obtained from a thermophysical properties handbook [11], and the densities of helium and nitrogen were calculated from accurate equations of state [12,13]. Finally, the obtained hydrogen viscosities are shown in figure 6, and their numerical values are summarized in table 3. The uncertainty of the viscosity measurement is estimated as 1.4% (k = 2) by propagation from the measurement uncertainties of the effective diameter and internal friction coefficient. A virial-type equation of state for hydrogen was recently developed on the basis of the latest PVT property measurements up to 100 MPa [14], but the validity range of the equation of state extends only to T = 473 K, and in this analysis an equation of state covering higher temperatures [15] was adopted. At low pressures up to 0.7 MPa, the densities calculated using the two equations of state differ by less than 0.01% from T = (296 to 573) K. In the target temperature and pressure range of the present measurement, the pressure dependence of the viscosity is very small. In the dilute gas region, the viscosity correlation by Yusibani et al. [16] represents existing viscosity measurements within 2%, and comparisons TABLE 3 Results of hydrogen viscosity measurement from T = (296 to 573) K and up to 0.7 MPa.a P/MPa
T/K
12 10 8
300
400
500
600
gEXP/lPa s
100( ηEXP - ηCAL) / ηCAL
5
296.26 322.56 373.12 472.13 573.48
8.90 9.41 10.41 12.09 14.16
0.3053 0.3024 0.3012 0.2989 0.2982
296.26 322.56 373.12 472.13 573.48
8.97 9.46 10.47 12.14 14.20
0.3 MPa 8.87 9.39 10.37 12.16 13.89
1.10 0.66 0.97 0.18 2.22
0.5010 0.5020 0.5031 0.5001 0.5030
296.26 322.56 373.12 472.13 573.48
9.00 9.49 10.48 12.15 14.27
0.5 MPa 8.88 9.40 10.37 12.16 13.89
1.39 0.97 1.11 0.13 2.74
0.7006 0.6980 0.7014 0.7011 0.7016
296.26 322.56 373.12 472.13 573.48
9.02 9.51 10.50 12.16 14.25
0.7 MPa 8.88 9.40 10.37 12.17 13.90
1.55 1.10 1.24 0.07 2.53
0
-5 300
400
500
600
T/ K Trautz and Zink [17] Muzny et al. [19] Present data 0.1 MPa
0.3 MPa
Mal'tsev et al. [18]
0.5 MPa
0.7 MPa
FIGURE 6. Measurement results of hydrogen viscosities. (a) Temperature dependence of the viscosity, (b) deviations from a viscosity correlation [16].
a
100(gEXP gCAL)/gCAL
0.1028 0.1110 0.1116 0.1114 0.0989
T/ K (b) Deviation
gCAL/lPa s [16] 0.1 MPa 8.87 9.39 10.36 12.16 13.89
0.42 0.17 0.46 0.54 1.94
Standard uncertainties u are u(T) = 50 mK, u(P) = 0.5 kPa, and the combined expanded uncertainty Ur is Ur(g) = 1.4 % (0.95 level of confidence).
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N. Sakoda et al. / J. Chem. Thermodynamics 89 (2015) 22–26
with existing experimental data by Trautz and Zink [17] and Mal’tsev et al. [18] are shown in figure 6(b). A viscosity correlation of hydrogen was developed recently by Muzny et al. [19], and the correlation is adopted in REFPROP ver. 9.1 [20]. Figure 6(b) also shows comparison with this correlation at 0.1 MPa. The correlation by Yusibani et al. represents the present data within 2% except at T = 573 K, where the correlation and the present data are in agreement within both of the uncertainties. The viscosities calculated from the latest correlation by Muzny et al. are slightly larger than the values calculated from the correlation by Yusibani et al. above T = 310 K, and the present data agree well with the correlation by Muzny et al. within 2% including temperatures up to 573 K. 4. Summary The viscosities of hydrogen were measured by the curved vibrating wire method at high temperatures up to 573 K and at pressures up to 0.7 MPa. The effective diameter and internal friction coefficient of the wire were determined by measuring helium and nitrogen as reference fluids. With the calibrated effective diameter and internal friction coefficient, the corrected helium and nitrogen viscosities show good agreement with the viscosity correlations within 0.6%, and finally hydrogen viscosities were obtained with an uncertainty of 1.4% (k = 2). Acknowledgments This research was conducted as part of the ‘‘Research and Development of Technical Standards to Prevent Overfilling on Hydrogen Fueling for FCV’’ funded by the New Energy and Industrial Technology Development Organization (NEDO).
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JCT 15-62