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An experimental study of the equation of state of superheated liquid methane
J. Chem. Thermodynamics15439,21, 1009-1016
An experimental study of state of superheated
of the equation liquid methane
V. G. BAIDAKOV and T. A. GURINA Institute of Thermophysics, Ural Division of the U.S.S.R. Academyof Sciences, Sverdlovsk620219,U.S.S.R. (ReceivedI4 March 1989) The results of measurements of densities of liquid methane in stable and metastable phase states are given. The experiments have been carried out in the temperature range from 144 to 185 K. The depth of penetration into the metastable region was 1.5 MPa and 43.6 K. An equation of state of superheated liquid methane is proposed. The spinodal has been approximated. The density of the liquid at the line of attainable superheats has been determined.
1. Introduction As a rule, when the external conditions are altered, a system’s thermodynamic characteristics vary continuously. However, when crossing specific lines on the phase diagram, some of them undergo a discontinuous change. Such a phenomenon is called a first-order phase transition. Two homogeneous macroscopic phases are in equilibrium at the phase-transition Iine (one-component system). In fast processes, the crossing of the line of phase equilibrium may not be accompanied by phase separation. A macroscopically homogeneous and incompletely stable state with properties different from the equilibrium properties under given conditions of the phase is realized in the thermodynamic system. These states are said to be metastable, i.e. not completely stable.“) Upon the expiry of a certain span of time, a metastable phase relaxes to a stable state. In systems that contain no inclusions initiating a phase transition, the relaxation occurs by fluctuation formation and the subsequent growth of the nuclei of a stable phase. When the depth of penetration into the metastable region is small, the time of expectation of a nucleus is usually substantially longer than the time needed for the establishment of local equilibrium in the system, which is a fact of fundamental importance. This allows us to investigate experimentally thermodynamic properties of metastable systems. The most convenient objects to study include superheated liquids, i.e. a particular case of a metastable phase state. The low viscosity of the superheated liquid assures a rapid relaxation of its structure, which is not always the case with, for example, a supercooled liquid. Superheating of liquid gases is a rather common phenomenon. The removal of metastability is accompanied by substantial hydraulic shocks in systems designed to 0021-9614/89/101009+08
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V. G. BAIDAKOV
AND T. A. GURINA
transport and store cryogenic liquids,“) and by vapour explosions arising from an overflow of liquid natural gas on to a water surface. (3) A comprehensive solution of the problems of controlling phase metastability is possible, provided information is available not pnly about the processes of the incipient formation of a vapour phase but also about the thermodynamic properties of a superheated liquid. In the absence of a microscopic theory, experimental investigations of metastable systems are of particular importance. This paper is devoted to experimental investigations of equations of state for superheated liquid methane and is a sequel to our earlier publication(4) where results on ultrasonic speed were reported. The experiments have been carried out in the temperature range from 144 to 185 K and the amount-of-substance density range from 16.95 to 23.10 mol. dme3. In the region of stable states, the (p, &, T) properties of liquid methane were studied earlier. (5m1o)This paper is the first publication to report the density of liquid methane in the metastable region.
2. Experimental The density of liquid methane was measured with a constant-volume piezometer. The apparatus and the procedure for carrying out the experiment have been described.” i) The piezometer was made of glass and had a volume V x 1.6 x 10m6 m3. The temperature of the liquid in the piezometer was measured with a platinum resistance thermometer calibrated according to IPTS-68. The temperature was maintained with an accuracy of f0.005 K. Pressure was created by compressed helium and was transmitted to the liquid through a metal membrane, the position of which was checked with a capacitance null-indicator. The membrane sensor was connected to the piezometer by a thin metal capillary. The temperature of the membrane sensor was maintained at the normal boiling temperature of methane. The methane under investigation had a purity of 99.9 moles per cent. The experiments were carried out along quasi-isochores, i.e. lines of fixed mass of liquid in the (piezometer + detector) system, and along isotherms. The measurements were relative. The mass of liquid in the (piezometer + detector) system was determined from known values of density pS at the phase equilibrium line.@-“) In the temperature range from the triple to the critical point, the orthobaric density and the elasticity curve of methane were approximated by: o, = 1+ 1.521316~“~ +0.812763~~‘~ +0.123765s3,
(1)
Inn, =(T,/T)(-6.005918~+1.193242~3’2-0.844288~3-1.180809~6),
(2)
where E = 1 - T/T,, o = p,Jp,,,, K = pJp,, and T,, p. and p,,,c are the temperature, pressure, and amount-of-substance density at the critical point; T, = 190.54 K, 10.154 mol. dmm3. The general form of equations (1) and p, = 4.598 MPa, and pn,c = (2) was borrowed from Pentermann and Wagner’s paper.“*) Penetration into the metastable region was realized by smoothly lowering the pressure beyond the line of phase equilibrium. The total error in the determination of density taking into account the error of pn,s was 0.1 to 0.15 per cent. The error of relative measurements was 0.05 to 0.07 per cent.
p.(CH,, I, T = 144 TO 185 K, p < 5.2 MPa)
101 I
T/K FIGURE 1. Quasi-isochores of liquid methane: 1, Saturation line; 2, threshold of boiling initiation corresponding to .I, = 6 x lo3 m- )‘s- l;(14) 3, line of attainable superheating with J = 1x10’ m-3.s-1,(14.
15)
3. Results and discussion The density of liquid methane was measured in the temperature range from 144 to 185 K and in the pressure range from 5.0 MPa to pressures close to the boundary of spontaneous boiling of the liquid. Measurements were made along quasi-isochores and isotherms. On each of 10 quasi-isochores, 14 to 18 density values were obtained. The penetration into the metastable region in terms of pressure Ap = {p(K)--p} reached 1.5 MPa, which corresponded to the liquid superheating AT = {T- T(pJ) = 43.6 K. The results are presented in table 1 and figure 1. Figure 2 demonstrates the density dependence of pressure along isotherms. The quasi-isochores are close to straight lines in both the stable region and the metastable region (figure 1). A more detailed analysis of the temperature dependence of pressure along the isochores revealed their curvature. When the amount-ofsubstance density values exceed approximately 2.lp,, c, the isochores are convex with respect to the pressure axis, @‘p/fJT2),. -C 0, and the sign of the curvature changes with decreasing density. Unlike the isochores, the isotherms (figure 2) and isobars are substantially nonlinear. Their curvature increased with approximation to the critical point and with penetration into the metastable region, which is an indication of a decrease in the
V. G. BAIDAKOV
1012
AND T. A. GURINA
TABLE 1. Temperature T, pressure p. and amount-of-substance density pn of liquid methane in stable and metastable states. The values of Ap = {p,(T)-p} correspond to the difference between the saturation pressure at a given temperature and the measured value of pressure. In the region of metastable states Ap > 0 and determines the value of supersaturation of the liquid phase. The last column of the table gives relative deviations of the experimental densities from those calculated by equation (3). For each of the quasi-isochores, the orthobaric amount-of-substance density p., 9 temperature T,, and pressure p, at the saturation line are given T
P
it
MPa
P.
l!!L
mol.dmm3
MPa
6 10’2
T
P
P”
AP
P
K
MPa
mol. dm- 3
MPa
148.76 148.51 148.02 147.16 146.70 146.35 145.80 145.30 145.02
5.046 4.823 4.380 3.606 3.196 2.878 2.383 1.927 1.676
pn.. = 23.051 mol.dmm3 T, = 144.04 K ps = 0.7870 MPa 23.020 -4.259 0.5 144.78 1.459 23.045 23.020 -4.036 -0.3 144.40 1.111 23.045 23.026 -3.598 1.1 144.18 0.916 23.045 23.032 -2.819 1.3 143.99 0.743 23.05 1 23.032 -2.409 -0.2 23.051 143.80 0.567 23.032 -2.091 -1.1 143.67 0.453 23.051 23.039 - 1.596 0.4 143.51 0.306 23.051 23.039 - 1.140 -0.8 143.46 0.264 23.051 23.039 -0.889 - 1.7
155.04 154.04 152.99 152.63 152.22 151.96 151.30 151.07
4.761 3.950 3.102 2.809 2.474 2.261 1.727 1.539
22.241 22.247 22.253 22.253 22.253 22.259 22.266 22.266
159.90 159.13 158.52 158.03 157.77 157.11 156.42 156.06 155.83
4.445 3.877 3.427 3.060 2.870 2.383 1.871 I.601 1.433
165.96 165.16 164.52 163.90 162.90 162.42 161.72 161.32 160.89 169.89 169.20 168.46 168.08 167.64
1042
6 P
-0.672 -0.324 -0.129 0.044 0.220 0.334 0.481 0.523
0.2 -0.6 -1.5 0.7 0.3 -0.2 -0.5 -0.9
= 1.065 MPa 1.448 22.266 1.295 22.266 0.957 22.272 0.703 22.272 0.569 22.272 0.353 22.272 0.242 22.272 0.177 22.272
-0.383 -0.230 0.108 0.362 0.496 0.712 0.823 0.888
0.1 -0.4 1.5 0.7 0.0 -0.4 -0.8 -1.2
p.,$ = 21.586 mol.dm-3 T, = 155.70 K 21.568 -3.109 -0.2 155.65 21.574 -2.541 0.9 155.38 21.574 -2.091 -0.6 155.12 21.574 - 1.724 -1.2 154.98 21.580 - 1.534 0.7 154.63 21.580 - 1.047 -0.9 154.50 21.586 -0.535 0.4 154.32 21.586 - 0.265 -0.2 154.21 21.586 - 0.097 -0.9
p, = 1.336 MPa 1.297 21.586 1.097 21.592 0.905 21.592 0.801 21.592 0.542 21.592 0.447 21.592 0.311 21.592 0.227 21.592
0.039 0.239 0.431 0.535 0.794 0.889 1.025 1.109
-1.1 1.0 0.4 0.1 -0.9 - 1.2 -1.5 -1.5
pn,s = 20.863 mol.dm-3 T, = 160.85 K 20.838 - 3.414 -1.2 160.62 20.844 -2.883 -0.2 160.32 20.844 - 2.455 - 1.3 160.00 20.844 - 2.037 -2.0 159.85 20.85 1 - 1.374 - 1.2 159.61 20.857 - 1.047 1.4 159.42 20.857 -0.581 -0.2 159.00 20.857 -0.317 - 1.3 158.80 20.863 -0.025 0.9 158.61
p,
5.063 4.532 4.104 3.686 3.023 2.696 2.230 1.966 1.674
= 1.649 MPa 1.496 20.863 1.295 20.863 1.083 20.863 0.980 20.863 0.820 20.869 0.694 20.869 0.412 20.869 0.276 20.869 0.146 20.869
0.153 0.354 0.566 0.669 0.829 0.955 1.237 1.373 1.503
0.2 -0.5 -1.4 - 1.4 0.9 0.3 -0.6 -0.7 -0.7
4.908 4.486 4.039 3.804 3.536
pn.p = 20.221 mol.dme3 T, = 165.02 K 20.202 - 2.969 -1.1 167.05 20.202 -2.547 - 1.9 166.73 20.209 -2.100 -0.3 166.06 20.209 - 1.865 -0.6 165.46 20.209 - 1.597 - 1.2 165.25
p. = 1.939 MPa 3.175 20.215 2.980 20.215 2.575 20.215 2.207 20.222 2.081 20.222
- 1.236 -1.041 -0.636 -0.268 -0.142
0.7 0.3 - 1.5 0.6 0.5
mol. dme3 T, = 150.49 K - 3.696 -0.9 150.96 -2.885 -0.7 150.77 - 2.037 -0.8 150.36 - 1.744 -1.7 150.05 - 1.409 -2.6 149.88 - 1.196 -0.5 149.62 -0.662 0.9 149.48 - 0.474 0.3 149.40
pn., = 22.266
p,
p,(CH,,
1, T = 144 TO TABLE
T ii
P MPa
P. mol.dm-3
164.90 164.48 164.16 163.90
1.865 1.607 1.415 1.255
20.222 20.222 20.227 20.227
173.66 172.89 172.43 171.92 171.32 170.86 170.24 169.72 169.10
5.027 4.595 4.342 4.055 3.720 3.465 3.117 2.828 2.480
19.598 19.604 19.604 19.604 19.610 19.610 19.610 19.617 19.617
177.29 176.76 176.13 175.43 174.92 173.92 173.13
5.151 4.883 4.562 4.202 3.944 3.438 3.036
18.968 18.968 18.974 18.874 18.974 18.981 18.981
180.46 179.56 178.93 178.26 177.73 177.02 176.64 176.06
5.176 4.760 4.469 4.156 3.914 3.588 3.415 3.146
18.320 18.320 18.326 18.326 18.326 18.332 18.332 18.332
182.80 182.19 181.85 181.32 180.90 180.23 179.76 179.13
5.060 4.806 4.661 4.437 4.265 3.985 3.784 3.523
17.703 17.703 17.709 17.709 17.709 17.709 17.715 17.715
pm,.=
* 0.074 0.332 0.524 0.684
T P
0.1 -0.3 0.9 0.7
19.617 mol.dmm3 -2.810 - 2.378 -2.125 - 1.838 - 1.503 - 1.248 -0.900 -0.611 - 0.263
4.852 4.602 4.535 4.446 4.342 4.087 3.794 3.510
16.924 16.930 16.930 16.930 16.930 16.930 16.930 16.936
- 1.441 -1.191 - 1.124 - 1.035 -0.931 - 0.676 -0.383 - 0.099
172.84 172.22 171.36 171.14 170.90 170.66 170.32
175.36 175.27 174.96 174.65 174.26 174.04 173.62
T, = 178.08 K -3.3 -4.3 - 3.0 0.0 -1.7 - 2.4 2.0 0.3
P.. I = 16.936mol.dme3
185.02 184.34 184.16 183.92 183.64 182.96 182.16 181.39
168.72 168.55 168.29 168.05 167.88 167.32 167.08 166.95 166.62
T, = 175.32 K -3.6 -4.2 -1.3 -0.8 -‘1.9 0.2 -1.0 -1.0
P.. s = 17.715 mol.dm-3
- 1.975 - 1.721 - 1.576 - 1.352 - 1.180 -0.9 -0.699 -0.438
163.51 163.22 162.99 !62.82
T, = 172.10 K -3.3 -4.4 - 1.7 - 1.7 -2.8 -0.9 -2.0
P.. s = 18.339 mol.dme3
- 2.366 - 1.95 - 1.659 - 1.346 -1.104 -0.778 -0.605 -0.336
K
T, = 168.63 K - 3.4 -1.1 -2.2 -2.7 -0.6 -1.5 -2.2 0.2 -0.5
P.. I = 18.987 mol.dm-3
- 2.639 -2.371 - 2.050 - 1.690 - 1.432 -0.926 -0.524
1013
l-continued
lo4~
MPa
185 K, p < 5.2 MPa)
178.68 178.20 178.02 177.82 177.46 177.29 177.13 176.90 T, = 181.20 K
-2.1 0.1 0.1 0.1 0.1 1.4 -1.1 1.1
181.22 181.09 180.90 180.86 180.65 180.25 180.14 179.93
P
MPa
P. mol.dm-”
1.019 0.840 0.703 0.599
20.227 20.227 20.227 20.234
p, = 2.217
MPa
2.269 2.175 2.030 1.896 1.797 1.489 1.356 1.282 1.093
19.617 19.623 19.623 19.623 19.623 19.623 19.623 19.629 19.629
p. = 2.885 2.573 2.135 2.023 1.905 1.784 1.609
ps= 2.822 2.785 2.641 2.499 2.322 2.217 2.029
p. = 3.337 3.135 3.062 2.976 2.829 2.754 2.686 2.597
ps= 3.449 3.403 3.331 3.315 3.238 3.093 3.050 2.971
L!E
1042s
MPa
P
0.920 1.099 1.236 1.340
-0.5 -0.7 -1.8 1.4
-0.052 0.042 0.187 0.321 0.420 0.728 0.861 0.935 1.124
- 1.4 1.2 0.7 0.3 0.7 -1.3 - 1.8 1.2 1.5
-0.373 -0.063 0.377 0.489 0.607 0.728 0.903
1.7 -4.3 -1.2 1.7 0.1 -0.4 -0.4
-0.012 0.025 0.169 0.311 0.488 0.593 0.781
2.2 0.7 0.7 0.0 -1.5 -0.6 -3.3
- 0.252 - 0.05 0.023 0.109 0.256 0.331 0.399 0.488
-1.1 -1.5 -2.5 2.1 0.0 1.4 1.8 -2.1
-0.038 0.008 0.08 0.096 0.173 0.318 0.361 0.44
-0.6 -1.9 -1.1 -0.2 -1.6 -3.6 -2.3 -1.9
2.512 MPa 18.987 18.987 18.987 18.993 18.993 18.993 18.993 2.810 MPa 18.339 18.339 18.339 18.339 18.339 18.339 18.339
3.085 MPa 17.715 17.715 17.715 17.722 17.722 17.722 17.722 17.722 3.411 MPa 16.936 16.936 16.936 16.936 16.936 16.936 16.936 16.936
V. G. BAIDAKOV
1014
AND T. A. GURINA
‘ ) -’
I? 2 1 0
16
19 p,,/(mol.dm-3)
FIGURE 2. Isotherms of liquid methane: 1, Saturation line; 2, line of attainable superheating with J = 1 x 107m-3.s-l,‘14’
stability of the superheated liquid with respect to the continuous variation of the state parameters, i.e. @p/ap,), decreases. The stability of superheated liquid in relation to fluctuation formation of the nuclei of a stable (vapour) phase is determined by the height W, of the free-energy activation barrier which separates the metastable and stable states. The probability of spontaneous nucleation involves W,. (l) Attainable superheating is determined by prescribing the number of viable nuclei that form in a volume of liquid divided by time, i.e. the nucleation rate J or the mean time (t) of expectation of a nucleus in a volume V which are related by the expression J = ((t)V)-‘.(‘3) As the liquid is superheated, (z) decreases. Under the conditions of our experiment, Yx 1.5 x low6 m3 and the characteristic measurement time F = 100 s. Measurements of thermodynamic parameters in the metastable region are possible if F < (r). Thus the maximum value .I,,,,, of the nucleation rate which may be attained in experiments aimed at measuring (p, p., T) is 6 x lo3 rnm3. s-r. Temperature and pressure dependences of the nucleation rate in liquid methane were investigated,t14’ and it was found that when J < 5 x lo6 rne3. s-r, the boiling of methane was initiated from the natural radiation background.“- r3* r4) The initiatedboiling boundaryo4’ and J, = J,,,., are shown in figure 1 by a dashed line. The closeness of the superheats attained in the investigation of (p, pm, T) to the boundary
p.(CH,, I, T = 144 TO 185 K, p i 5.2 MPa)
145
160
175
190
10
14
18
22
T/K FIGURE 3. 1, Saturation line; 2, line of attainable superheating; 3, liquid methane spinodal.
J, indicates that the radiation background is a major obstacle to a deeper penetration into the region of metastable states. To a first approximation, the equation of state of the liquid near the phaseequilibrium line may be written in the form: n = do) + (ww,,,~~
- a4
(3)
The functions n,(o) and tS(o) in implicit form are given by equations (1) and (2). The values of (a7t/a7),,,at the saturation line have been found by numerical differentiation of (p, p., T) and are approximated by the expression: (an/&),,,
= (drr/dr),{ 1 + 2.1781 l(w - 1)3’2 + 0.954269(w - 1)5’2 +0.226579@ - l)“},
(4)
where (d?r/dr), determines the slope of the elasticity curve at the critical point and, according to equation (2), has a value of 6.005918. Deviations of experimental amount-of-substance densities from the values calculated by the equation of state (3) are presented in table 1. It is assumed that equation (3) also correctly reproduces the thermodynamic properties of metastable methane in the region of state parameters in which measurements cannot be made currently. The results of a calculation of the density on the line of attainable superheats for liquid methane (J = 1 x 10’ me3 *s-1)(141 are given in table 2. In the investigated range of state parameters, the value of p. is about 3 to 4 per cent less than that at the phase-equilibrium line. The value of (ap/ap,J, decreases with penetration into the metastable region. The metastable system loses its reconstructive reaction to arbitrarily small perturbations on the line where (ap/ap,), = 0. The spatial curve on which (ap/ap,), = 0 and
V. G. BAIDAKOV
1016
AND T. A. GURINA
TABLE 2. Temperature T, pressure pn, and amount-of-substance density p... at the line of attainable superheating of liquid methane at the nucleation rate J = 1 x 10’ mm3 .s-t,04’ pressure psr and amountof-substance density pn.Jp at the spinodal. The values of pressure ps and amount-of-substance density p.. J at the line of phase equilibrium have been calculated by equations (1) and (2) T
K 150.00 155.00 160.00 163.00 167.00 170.00 173.00 177.00 180.00 183.00
I P
MPa
1.042 1.297 1.594 1.794 2.088 2.331 2.593 2.975 3.287 3.625
P., J
mol-dme3 22.32 21.67 20.98 20.53 19.89 19.38 18.81 17.97 17.24 16.38
Pn
MPa
- 3.27 -2.10 -0.98 -0.34 0.48 1.07 1.65 2.38 2.91 3.41
P.. n
mol.dm-3 21.54 20.94 20.29 19.88 19.28 18.80 18.29 17.51 16.83 16.03
A!L
MPa
- 8.978 - 6.679 -4.549 -3.351 - 1.849 - 0.793 0.200 1.426 2.267 3.039
P.. rp
mol.dm-3 18.49 18.03 17.53 17.22 16.77
16.41 16.02 15.44 14.94 14.36
TIC,,, = 0, with C,,, being the isobaric heat capacity, is called the spinodal. The results of approximating the spinodal of superheated liquid methane from the equation of state (3) are given in figure 3 and table 2. In the (temperature, density) coordinates the spinodal equation reads (5) +v = 1+ 0.940969~“~ + 0.860512~~‘~ -0.505066~~. At atmospheric pressure, the boundary of spontaneous boiling of liquid methane corresponds to the superheating AT = T,-- T, x 54.4 K. The spinodal point lies 7 K higher. REFERENCES 1. Skripov, V. P. Metastabilhaya Zhiafkost. Nauka: Moscow. 1972. Metastable Liquids. Wiley: New York. Israel Progr. Sci. Transl.: Jerusalem. 1974. 2. Marias, J. L. Fire International 1984, 8, 27. 3. Nakanishi, E.; Reid, R. C. Chem. Eng. Progr. 1971, 67, 36. 4. Baidakov, V. G.; Kaverin, A. M.; Skripov, V. P. J. Chem. Thermodynamics 1982, 14, 1003. 5. Sorokin, V. A.; Blagoy, Y. P. Termodinamicheskie i termokhimicheskie konstanti Nauka: Moscow. 1970, p. 97. 6. Goodwin, R. D.; Prydz, R. J. Res. Nat. Bur. Stand, U.S. 1972, 76A, 81. 7. Straty, G. C.; Goodwin, R. D. Cryogenics 1973, 13, 712. 8. Gammon, B. E.; Douslin, D. R. J. Chem. Phys. 1976, 64, 203. 9. Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Rev. Sci. Instrum. 1976, 47, 1237. 10. McClune, C. R. Cryogenics 1976, 16, 289. 11. Baidakov, V. G.; Gurina, T. A. J. Chem. Thermodynamics 1985, 17, 13 1. 12. Pentermann, W.; Wagner, W. J. Chem. Thermodynamics 1978, 10, 1161. 13. Skripov, V. P.; Sin&sin, E. N.; Pavlov, P. A.; Ermakov, G. V.; Muratov, G. N.; Bulanov, N. V.; Baidakov, V. G. Teplofzicheskie svoistva zhirlkostei v metastabilinom sostoyanii (Thermal Properties of Liquids in the Metastable State) Atomizdat: Moscow. 1980. 14. Baidakov, V. G.; Kaverin, A. M.; Skripov, V. P. Koll. Zh. 1980, 42, 314. 15. Baidakov, V. G. Fluid mechanics-Soviet Research 1987, 16, 62.
An experimental study of state of superheated
of the equation liquid methane
V. G. BAIDAKOV and T. A. GURINA Institute of Thermophysics, Ural Division of the U.S.S.R. Academyof Sciences, Sverdlovsk620219,U.S.S.R. (ReceivedI4 March 1989) The results of measurements of densities of liquid methane in stable and metastable phase states are given. The experiments have been carried out in the temperature range from 144 to 185 K. The depth of penetration into the metastable region was 1.5 MPa and 43.6 K. An equation of state of superheated liquid methane is proposed. The spinodal has been approximated. The density of the liquid at the line of attainable superheats has been determined.
1. Introduction As a rule, when the external conditions are altered, a system’s thermodynamic characteristics vary continuously. However, when crossing specific lines on the phase diagram, some of them undergo a discontinuous change. Such a phenomenon is called a first-order phase transition. Two homogeneous macroscopic phases are in equilibrium at the phase-transition Iine (one-component system). In fast processes, the crossing of the line of phase equilibrium may not be accompanied by phase separation. A macroscopically homogeneous and incompletely stable state with properties different from the equilibrium properties under given conditions of the phase is realized in the thermodynamic system. These states are said to be metastable, i.e. not completely stable.“) Upon the expiry of a certain span of time, a metastable phase relaxes to a stable state. In systems that contain no inclusions initiating a phase transition, the relaxation occurs by fluctuation formation and the subsequent growth of the nuclei of a stable phase. When the depth of penetration into the metastable region is small, the time of expectation of a nucleus is usually substantially longer than the time needed for the establishment of local equilibrium in the system, which is a fact of fundamental importance. This allows us to investigate experimentally thermodynamic properties of metastable systems. The most convenient objects to study include superheated liquids, i.e. a particular case of a metastable phase state. The low viscosity of the superheated liquid assures a rapid relaxation of its structure, which is not always the case with, for example, a supercooled liquid. Superheating of liquid gases is a rather common phenomenon. The removal of metastability is accompanied by substantial hydraulic shocks in systems designed to 0021-9614/89/101009+08
%02.00/O
8 1989 Academic Press Limited
1010
V. G. BAIDAKOV
AND T. A. GURINA
transport and store cryogenic liquids,“) and by vapour explosions arising from an overflow of liquid natural gas on to a water surface. (3) A comprehensive solution of the problems of controlling phase metastability is possible, provided information is available not pnly about the processes of the incipient formation of a vapour phase but also about the thermodynamic properties of a superheated liquid. In the absence of a microscopic theory, experimental investigations of metastable systems are of particular importance. This paper is devoted to experimental investigations of equations of state for superheated liquid methane and is a sequel to our earlier publication(4) where results on ultrasonic speed were reported. The experiments have been carried out in the temperature range from 144 to 185 K and the amount-of-substance density range from 16.95 to 23.10 mol. dme3. In the region of stable states, the (p, &, T) properties of liquid methane were studied earlier. (5m1o)This paper is the first publication to report the density of liquid methane in the metastable region.
2. Experimental The density of liquid methane was measured with a constant-volume piezometer. The apparatus and the procedure for carrying out the experiment have been described.” i) The piezometer was made of glass and had a volume V x 1.6 x 10m6 m3. The temperature of the liquid in the piezometer was measured with a platinum resistance thermometer calibrated according to IPTS-68. The temperature was maintained with an accuracy of f0.005 K. Pressure was created by compressed helium and was transmitted to the liquid through a metal membrane, the position of which was checked with a capacitance null-indicator. The membrane sensor was connected to the piezometer by a thin metal capillary. The temperature of the membrane sensor was maintained at the normal boiling temperature of methane. The methane under investigation had a purity of 99.9 moles per cent. The experiments were carried out along quasi-isochores, i.e. lines of fixed mass of liquid in the (piezometer + detector) system, and along isotherms. The measurements were relative. The mass of liquid in the (piezometer + detector) system was determined from known values of density pS at the phase equilibrium line.@-“) In the temperature range from the triple to the critical point, the orthobaric density and the elasticity curve of methane were approximated by: o, = 1+ 1.521316~“~ +0.812763~~‘~ +0.123765s3,
(1)
Inn, =(T,/T)(-6.005918~+1.193242~3’2-0.844288~3-1.180809~6),
(2)
where E = 1 - T/T,, o = p,Jp,,,, K = pJp,, and T,, p. and p,,,c are the temperature, pressure, and amount-of-substance density at the critical point; T, = 190.54 K, 10.154 mol. dmm3. The general form of equations (1) and p, = 4.598 MPa, and pn,c = (2) was borrowed from Pentermann and Wagner’s paper.“*) Penetration into the metastable region was realized by smoothly lowering the pressure beyond the line of phase equilibrium. The total error in the determination of density taking into account the error of pn,s was 0.1 to 0.15 per cent. The error of relative measurements was 0.05 to 0.07 per cent.
p.(CH,, I, T = 144 TO 185 K, p < 5.2 MPa)
101 I
T/K FIGURE 1. Quasi-isochores of liquid methane: 1, Saturation line; 2, threshold of boiling initiation corresponding to .I, = 6 x lo3 m- )‘s- l;(14) 3, line of attainable superheating with J = 1x10’ m-3.s-1,(14.
15)
3. Results and discussion The density of liquid methane was measured in the temperature range from 144 to 185 K and in the pressure range from 5.0 MPa to pressures close to the boundary of spontaneous boiling of the liquid. Measurements were made along quasi-isochores and isotherms. On each of 10 quasi-isochores, 14 to 18 density values were obtained. The penetration into the metastable region in terms of pressure Ap = {p(K)--p} reached 1.5 MPa, which corresponded to the liquid superheating AT = {T- T(pJ) = 43.6 K. The results are presented in table 1 and figure 1. Figure 2 demonstrates the density dependence of pressure along isotherms. The quasi-isochores are close to straight lines in both the stable region and the metastable region (figure 1). A more detailed analysis of the temperature dependence of pressure along the isochores revealed their curvature. When the amount-ofsubstance density values exceed approximately 2.lp,, c, the isochores are convex with respect to the pressure axis, @‘p/fJT2),. -C 0, and the sign of the curvature changes with decreasing density. Unlike the isochores, the isotherms (figure 2) and isobars are substantially nonlinear. Their curvature increased with approximation to the critical point and with penetration into the metastable region, which is an indication of a decrease in the
V. G. BAIDAKOV
1012
AND T. A. GURINA
TABLE 1. Temperature T, pressure p. and amount-of-substance density pn of liquid methane in stable and metastable states. The values of Ap = {p,(T)-p} correspond to the difference between the saturation pressure at a given temperature and the measured value of pressure. In the region of metastable states Ap > 0 and determines the value of supersaturation of the liquid phase. The last column of the table gives relative deviations of the experimental densities from those calculated by equation (3). For each of the quasi-isochores, the orthobaric amount-of-substance density p., 9 temperature T,, and pressure p, at the saturation line are given T
P
it
MPa
P.
l!!L
mol.dmm3
MPa
6 10’2
T
P
P”
AP
P
K
MPa
mol. dm- 3
MPa
148.76 148.51 148.02 147.16 146.70 146.35 145.80 145.30 145.02
5.046 4.823 4.380 3.606 3.196 2.878 2.383 1.927 1.676
pn.. = 23.051 mol.dmm3 T, = 144.04 K ps = 0.7870 MPa 23.020 -4.259 0.5 144.78 1.459 23.045 23.020 -4.036 -0.3 144.40 1.111 23.045 23.026 -3.598 1.1 144.18 0.916 23.045 23.032 -2.819 1.3 143.99 0.743 23.05 1 23.032 -2.409 -0.2 23.051 143.80 0.567 23.032 -2.091 -1.1 143.67 0.453 23.051 23.039 - 1.596 0.4 143.51 0.306 23.051 23.039 - 1.140 -0.8 143.46 0.264 23.051 23.039 -0.889 - 1.7
155.04 154.04 152.99 152.63 152.22 151.96 151.30 151.07
4.761 3.950 3.102 2.809 2.474 2.261 1.727 1.539
22.241 22.247 22.253 22.253 22.253 22.259 22.266 22.266
159.90 159.13 158.52 158.03 157.77 157.11 156.42 156.06 155.83
4.445 3.877 3.427 3.060 2.870 2.383 1.871 I.601 1.433
165.96 165.16 164.52 163.90 162.90 162.42 161.72 161.32 160.89 169.89 169.20 168.46 168.08 167.64
1042
6 P
-0.672 -0.324 -0.129 0.044 0.220 0.334 0.481 0.523
0.2 -0.6 -1.5 0.7 0.3 -0.2 -0.5 -0.9
= 1.065 MPa 1.448 22.266 1.295 22.266 0.957 22.272 0.703 22.272 0.569 22.272 0.353 22.272 0.242 22.272 0.177 22.272
-0.383 -0.230 0.108 0.362 0.496 0.712 0.823 0.888
0.1 -0.4 1.5 0.7 0.0 -0.4 -0.8 -1.2
p.,$ = 21.586 mol.dm-3 T, = 155.70 K 21.568 -3.109 -0.2 155.65 21.574 -2.541 0.9 155.38 21.574 -2.091 -0.6 155.12 21.574 - 1.724 -1.2 154.98 21.580 - 1.534 0.7 154.63 21.580 - 1.047 -0.9 154.50 21.586 -0.535 0.4 154.32 21.586 - 0.265 -0.2 154.21 21.586 - 0.097 -0.9
p, = 1.336 MPa 1.297 21.586 1.097 21.592 0.905 21.592 0.801 21.592 0.542 21.592 0.447 21.592 0.311 21.592 0.227 21.592
0.039 0.239 0.431 0.535 0.794 0.889 1.025 1.109
-1.1 1.0 0.4 0.1 -0.9 - 1.2 -1.5 -1.5
pn,s = 20.863 mol.dm-3 T, = 160.85 K 20.838 - 3.414 -1.2 160.62 20.844 -2.883 -0.2 160.32 20.844 - 2.455 - 1.3 160.00 20.844 - 2.037 -2.0 159.85 20.85 1 - 1.374 - 1.2 159.61 20.857 - 1.047 1.4 159.42 20.857 -0.581 -0.2 159.00 20.857 -0.317 - 1.3 158.80 20.863 -0.025 0.9 158.61
p,
5.063 4.532 4.104 3.686 3.023 2.696 2.230 1.966 1.674
= 1.649 MPa 1.496 20.863 1.295 20.863 1.083 20.863 0.980 20.863 0.820 20.869 0.694 20.869 0.412 20.869 0.276 20.869 0.146 20.869
0.153 0.354 0.566 0.669 0.829 0.955 1.237 1.373 1.503
0.2 -0.5 -1.4 - 1.4 0.9 0.3 -0.6 -0.7 -0.7
4.908 4.486 4.039 3.804 3.536
pn.p = 20.221 mol.dme3 T, = 165.02 K 20.202 - 2.969 -1.1 167.05 20.202 -2.547 - 1.9 166.73 20.209 -2.100 -0.3 166.06 20.209 - 1.865 -0.6 165.46 20.209 - 1.597 - 1.2 165.25
p. = 1.939 MPa 3.175 20.215 2.980 20.215 2.575 20.215 2.207 20.222 2.081 20.222
- 1.236 -1.041 -0.636 -0.268 -0.142
0.7 0.3 - 1.5 0.6 0.5
mol. dme3 T, = 150.49 K - 3.696 -0.9 150.96 -2.885 -0.7 150.77 - 2.037 -0.8 150.36 - 1.744 -1.7 150.05 - 1.409 -2.6 149.88 - 1.196 -0.5 149.62 -0.662 0.9 149.48 - 0.474 0.3 149.40
pn., = 22.266
p,
p,(CH,,
1, T = 144 TO TABLE
T ii
P MPa
P. mol.dm-3
164.90 164.48 164.16 163.90
1.865 1.607 1.415 1.255
20.222 20.222 20.227 20.227
173.66 172.89 172.43 171.92 171.32 170.86 170.24 169.72 169.10
5.027 4.595 4.342 4.055 3.720 3.465 3.117 2.828 2.480
19.598 19.604 19.604 19.604 19.610 19.610 19.610 19.617 19.617
177.29 176.76 176.13 175.43 174.92 173.92 173.13
5.151 4.883 4.562 4.202 3.944 3.438 3.036
18.968 18.968 18.974 18.874 18.974 18.981 18.981
180.46 179.56 178.93 178.26 177.73 177.02 176.64 176.06
5.176 4.760 4.469 4.156 3.914 3.588 3.415 3.146
18.320 18.320 18.326 18.326 18.326 18.332 18.332 18.332
182.80 182.19 181.85 181.32 180.90 180.23 179.76 179.13
5.060 4.806 4.661 4.437 4.265 3.985 3.784 3.523
17.703 17.703 17.709 17.709 17.709 17.709 17.715 17.715
pm,.=
* 0.074 0.332 0.524 0.684
T P
0.1 -0.3 0.9 0.7
19.617 mol.dmm3 -2.810 - 2.378 -2.125 - 1.838 - 1.503 - 1.248 -0.900 -0.611 - 0.263
4.852 4.602 4.535 4.446 4.342 4.087 3.794 3.510
16.924 16.930 16.930 16.930 16.930 16.930 16.930 16.936
- 1.441 -1.191 - 1.124 - 1.035 -0.931 - 0.676 -0.383 - 0.099
172.84 172.22 171.36 171.14 170.90 170.66 170.32
175.36 175.27 174.96 174.65 174.26 174.04 173.62
T, = 178.08 K -3.3 -4.3 - 3.0 0.0 -1.7 - 2.4 2.0 0.3
P.. I = 16.936mol.dme3
185.02 184.34 184.16 183.92 183.64 182.96 182.16 181.39
168.72 168.55 168.29 168.05 167.88 167.32 167.08 166.95 166.62
T, = 175.32 K -3.6 -4.2 -1.3 -0.8 -‘1.9 0.2 -1.0 -1.0
P.. s = 17.715 mol.dm-3
- 1.975 - 1.721 - 1.576 - 1.352 - 1.180 -0.9 -0.699 -0.438
163.51 163.22 162.99 !62.82
T, = 172.10 K -3.3 -4.4 - 1.7 - 1.7 -2.8 -0.9 -2.0
P.. s = 18.339 mol.dme3
- 2.366 - 1.95 - 1.659 - 1.346 -1.104 -0.778 -0.605 -0.336
K
T, = 168.63 K - 3.4 -1.1 -2.2 -2.7 -0.6 -1.5 -2.2 0.2 -0.5
P.. I = 18.987 mol.dm-3
- 2.639 -2.371 - 2.050 - 1.690 - 1.432 -0.926 -0.524
1013
l-continued
lo4~
MPa
185 K, p < 5.2 MPa)
178.68 178.20 178.02 177.82 177.46 177.29 177.13 176.90 T, = 181.20 K
-2.1 0.1 0.1 0.1 0.1 1.4 -1.1 1.1
181.22 181.09 180.90 180.86 180.65 180.25 180.14 179.93
P
MPa
P. mol.dm-”
1.019 0.840 0.703 0.599
20.227 20.227 20.227 20.234
p, = 2.217
MPa
2.269 2.175 2.030 1.896 1.797 1.489 1.356 1.282 1.093
19.617 19.623 19.623 19.623 19.623 19.623 19.623 19.629 19.629
p. = 2.885 2.573 2.135 2.023 1.905 1.784 1.609
ps= 2.822 2.785 2.641 2.499 2.322 2.217 2.029
p. = 3.337 3.135 3.062 2.976 2.829 2.754 2.686 2.597
ps= 3.449 3.403 3.331 3.315 3.238 3.093 3.050 2.971
L!E
1042s
MPa
P
0.920 1.099 1.236 1.340
-0.5 -0.7 -1.8 1.4
-0.052 0.042 0.187 0.321 0.420 0.728 0.861 0.935 1.124
- 1.4 1.2 0.7 0.3 0.7 -1.3 - 1.8 1.2 1.5
-0.373 -0.063 0.377 0.489 0.607 0.728 0.903
1.7 -4.3 -1.2 1.7 0.1 -0.4 -0.4
-0.012 0.025 0.169 0.311 0.488 0.593 0.781
2.2 0.7 0.7 0.0 -1.5 -0.6 -3.3
- 0.252 - 0.05 0.023 0.109 0.256 0.331 0.399 0.488
-1.1 -1.5 -2.5 2.1 0.0 1.4 1.8 -2.1
-0.038 0.008 0.08 0.096 0.173 0.318 0.361 0.44
-0.6 -1.9 -1.1 -0.2 -1.6 -3.6 -2.3 -1.9
2.512 MPa 18.987 18.987 18.987 18.993 18.993 18.993 18.993 2.810 MPa 18.339 18.339 18.339 18.339 18.339 18.339 18.339
3.085 MPa 17.715 17.715 17.715 17.722 17.722 17.722 17.722 17.722 3.411 MPa 16.936 16.936 16.936 16.936 16.936 16.936 16.936 16.936
V. G. BAIDAKOV
1014
AND T. A. GURINA
‘ ) -’
I? 2 1 0
16
19 p,,/(mol.dm-3)
FIGURE 2. Isotherms of liquid methane: 1, Saturation line; 2, line of attainable superheating with J = 1 x 107m-3.s-l,‘14’
stability of the superheated liquid with respect to the continuous variation of the state parameters, i.e. @p/ap,), decreases. The stability of superheated liquid in relation to fluctuation formation of the nuclei of a stable (vapour) phase is determined by the height W, of the free-energy activation barrier which separates the metastable and stable states. The probability of spontaneous nucleation involves W,. (l) Attainable superheating is determined by prescribing the number of viable nuclei that form in a volume of liquid divided by time, i.e. the nucleation rate J or the mean time (t) of expectation of a nucleus in a volume V which are related by the expression J = ((t)V)-‘.(‘3) As the liquid is superheated, (z) decreases. Under the conditions of our experiment, Yx 1.5 x low6 m3 and the characteristic measurement time F = 100 s. Measurements of thermodynamic parameters in the metastable region are possible if F < (r). Thus the maximum value .I,,,,, of the nucleation rate which may be attained in experiments aimed at measuring (p, p., T) is 6 x lo3 rnm3. s-r. Temperature and pressure dependences of the nucleation rate in liquid methane were investigated,t14’ and it was found that when J < 5 x lo6 rne3. s-r, the boiling of methane was initiated from the natural radiation background.“- r3* r4) The initiatedboiling boundaryo4’ and J, = J,,,., are shown in figure 1 by a dashed line. The closeness of the superheats attained in the investigation of (p, pm, T) to the boundary
p.(CH,, I, T = 144 TO 185 K, p i 5.2 MPa)
145
160
175
190
10
14
18
22
T/K FIGURE 3. 1, Saturation line; 2, line of attainable superheating; 3, liquid methane spinodal.
J, indicates that the radiation background is a major obstacle to a deeper penetration into the region of metastable states. To a first approximation, the equation of state of the liquid near the phaseequilibrium line may be written in the form: n = do) + (ww,,,~~
- a4
(3)
The functions n,(o) and tS(o) in implicit form are given by equations (1) and (2). The values of (a7t/a7),,,at the saturation line have been found by numerical differentiation of (p, p., T) and are approximated by the expression: (an/&),,,
= (drr/dr),{ 1 + 2.1781 l(w - 1)3’2 + 0.954269(w - 1)5’2 +0.226579@ - l)“},
(4)
where (d?r/dr), determines the slope of the elasticity curve at the critical point and, according to equation (2), has a value of 6.005918. Deviations of experimental amount-of-substance densities from the values calculated by the equation of state (3) are presented in table 1. It is assumed that equation (3) also correctly reproduces the thermodynamic properties of metastable methane in the region of state parameters in which measurements cannot be made currently. The results of a calculation of the density on the line of attainable superheats for liquid methane (J = 1 x 10’ me3 *s-1)(141 are given in table 2. In the investigated range of state parameters, the value of p. is about 3 to 4 per cent less than that at the phase-equilibrium line. The value of (ap/ap,J, decreases with penetration into the metastable region. The metastable system loses its reconstructive reaction to arbitrarily small perturbations on the line where (ap/ap,), = 0. The spatial curve on which (ap/ap,), = 0 and
V. G. BAIDAKOV
1016
AND T. A. GURINA
TABLE 2. Temperature T, pressure pn, and amount-of-substance density p... at the line of attainable superheating of liquid methane at the nucleation rate J = 1 x 10’ mm3 .s-t,04’ pressure psr and amountof-substance density pn.Jp at the spinodal. The values of pressure ps and amount-of-substance density p.. J at the line of phase equilibrium have been calculated by equations (1) and (2) T
K 150.00 155.00 160.00 163.00 167.00 170.00 173.00 177.00 180.00 183.00
I P
MPa
1.042 1.297 1.594 1.794 2.088 2.331 2.593 2.975 3.287 3.625
P., J
mol-dme3 22.32 21.67 20.98 20.53 19.89 19.38 18.81 17.97 17.24 16.38
Pn
MPa
- 3.27 -2.10 -0.98 -0.34 0.48 1.07 1.65 2.38 2.91 3.41
P.. n
mol.dm-3 21.54 20.94 20.29 19.88 19.28 18.80 18.29 17.51 16.83 16.03
A!L
MPa
- 8.978 - 6.679 -4.549 -3.351 - 1.849 - 0.793 0.200 1.426 2.267 3.039
P.. rp
mol.dm-3 18.49 18.03 17.53 17.22 16.77
16.41 16.02 15.44 14.94 14.36
TIC,,, = 0, with C,,, being the isobaric heat capacity, is called the spinodal. The results of approximating the spinodal of superheated liquid methane from the equation of state (3) are given in figure 3 and table 2. In the (temperature, density) coordinates the spinodal equation reads (5) +v = 1+ 0.940969~“~ + 0.860512~~‘~ -0.505066~~. At atmospheric pressure, the boundary of spontaneous boiling of liquid methane corresponds to the superheating AT = T,-- T, x 54.4 K. The spinodal point lies 7 K higher. REFERENCES 1. Skripov, V. P. Metastabilhaya Zhiafkost. Nauka: Moscow. 1972. Metastable Liquids. Wiley: New York. Israel Progr. Sci. Transl.: Jerusalem. 1974. 2. Marias, J. L. Fire International 1984, 8, 27. 3. Nakanishi, E.; Reid, R. C. Chem. Eng. Progr. 1971, 67, 36. 4. Baidakov, V. G.; Kaverin, A. M.; Skripov, V. P. J. Chem. Thermodynamics 1982, 14, 1003. 5. Sorokin, V. A.; Blagoy, Y. P. Termodinamicheskie i termokhimicheskie konstanti Nauka: Moscow. 1970, p. 97. 6. Goodwin, R. D.; Prydz, R. J. Res. Nat. Bur. Stand, U.S. 1972, 76A, 81. 7. Straty, G. C.; Goodwin, R. D. Cryogenics 1973, 13, 712. 8. Gammon, B. E.; Douslin, D. R. J. Chem. Phys. 1976, 64, 203. 9. Haynes, W. M.; Hiza, M. J.; Frederick, N. V. Rev. Sci. Instrum. 1976, 47, 1237. 10. McClune, C. R. Cryogenics 1976, 16, 289. 11. Baidakov, V. G.; Gurina, T. A. J. Chem. Thermodynamics 1985, 17, 13 1. 12. Pentermann, W.; Wagner, W. J. Chem. Thermodynamics 1978, 10, 1161. 13. Skripov, V. P.; Sin&sin, E. N.; Pavlov, P. A.; Ermakov, G. V.; Muratov, G. N.; Bulanov, N. V.; Baidakov, V. G. Teplofzicheskie svoistva zhirlkostei v metastabilinom sostoyanii (Thermal Properties of Liquids in the Metastable State) Atomizdat: Moscow. 1980. 14. Baidakov, V. G.; Kaverin, A. M.; Skripov, V. P. Koll. Zh. 1980, 42, 314. 15. Baidakov, V. G. Fluid mechanics-Soviet Research 1987, 16, 62.