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Ind. Eng. Chem. Res. 2001, 40, 4610-4614
GENERAL RESEARCH Isoplethic Method to Estimate Critical Lines for Binary Fluid Mixtures from Subcritical Vapor-Liquid Equilibrium: Application to the Azeotropic Mixtures R32 + C3H8 and R125 + C3H8 Lambert J. Van Poolen,* Cynthia D. Holcomb, and James C. Rainwater Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305-3328
A new, isoplethic method is used to determine critical lines of azeotropic binary fluid mixtures using vapor-liquid equilibrium data including saturated vapor and liquid densities below the lowest critical temperature of the pure components. Thermodynamic paths that access mixture critical lines from this region of the vapor-liquid surface are described by the liquid volume fraction (set at 1/2), fixed overall composition, and pressure. Data along these paths behave like those for a pure fluid and are extrapolated to the critical line by means of simple, pure-fluidlike expressions for the density difference, vapor pressure, and rectilinear diameter for vapor and liquid in equilibrium. The method is tested successfully for an azeotropic mixture for which critical-point estimates can be compared to those in the literature [C3H8 (propane) + C4F8 (perfluorocyclobutane)]. The method is then applied to the two azeotropic mixtures, R32 [CH2F2 (diflouromethane)] + C3H8 (propane) and R125 [C2F5H (pentaflouroethane)] + C3H8 (propane), for which no information is available on the critical line. Analytical expressions for the critical temperature, critical pressure, and critical density are given as functions of the overall composition for these mixtures. The combination of relatively easily measured vapor-liquid equilibrium data along with a simple, thermodynamically correct path to the critical line provides an efficient, relatively simple method for characterizing the vapor-liquid coexistence surfaces for binary mixtures. 1. Introduction A new, isoplethic method is proposed to determine critical lines of binary mixtures based on vapor-liquid equilibrium data including saturation vapor and liquid densities at temperatures below the lowest critical temperature of the two pure-fluid components. For the system C3H8 (propane) + C4F8 (perfluorocyclobutane), for example (shown in Figure 1), this region would be below the critical temperature of the propane (near 370 K) along four isotherms at 10-K intervals from about 330 to 360 K. Accurate vapor-liquid thermodynamic data can be obtained in this region in a relatively easy and efficient manner. Similar data in the critical region can be more difficult to obtain. The method presented below allows access to the mixture critical lines from this subcritical region. The method is based on the phase rule developed by Van Poolen.1 This rule accounts for the phase-intensive variables pressure (P); temperature (T); phase compositions of the saturated liquid and vapor (x1l and x1v, respectively), where the subscript 1 is for the component having the lowest critical temperature; and phase * Author to whom correspondence should be addressed. Permanent address: Engineering Department, Calvin College, 3201 Burton St. SE, Grand Rapids, MI 49546. E-mail: vpol@ calvin.edu. Phone: 616-957-6337. Fax: 616-957-6501.
densities of the saturated liquid and vapor (Fl and Fv, respectively), as well as the system-intensive variables overall density (FT), overall composition on a mole fraction basis (X1,T), and liquid volume fraction (LVF). This phase rule yields (C + 1) degrees of freedom, where C is the number of components in the mixture. Note that a specific vapor-liquid coexistence state described in terms of phase-intensive variables by means of the Gibbs phase rule can, in fact, have a range of overall compositions and densities varying from the composition and density of the liquid to those of the vapor. The range of liquid volume fractions is from 0 to 1. There are three degrees of freedom for a binary fluid mixture according to the Van Poolen phase rule.1 Previously,2 paths of fixed overall density, fixed overall composition, and varying temperature were used to obtain critical densities of the binary mixture C2H6 (ethane) + n-C4H10 (normal-butane). Also, for a ternary system (four degrees of freedom), vapor-liquid equilibrium data, together with saturated vapor and liquid densities along paths of constant overall composition, constant temperature, and varying pressure, were used to obtain critical-point values.3 Data used to obtain mixture critical-point information for these paths entail information in the near-critical region. These kinds of data are not always available and often are difficult and very labor-intensive to obtain, as noted above.
10.1021/ie010215x CCC: $20.00 © 2001 American Chemical Society Published on Web 09/15/2001
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2. Method For any vapor-liquid coexistence data point for a binary mixture, the overall composition of the component that has the lowest critical temperature for which the liquid volume fraction is 1/2 can be calculated from the following equation based on mass balances1
XT,1 )
Figure 1. Pressure versus temperature for the binary fluid mixture C3H8 + C4F8.
The thermodynamic paths mentioned in previous work (see above) can accurately determine critical points where there are data in the critical region. However, to utilize data away from the critical region (below the lowest critical temperature of the two pure components), a different thermodynamic path is needed to ensure accessibility to the critical line from this region. The thermodynamic path chosen for this work is that for which the overall composition is constant and the liquid volume fraction is constant at a value of exactly 1/ . This means that the overall density is always the 2 average of the saturated vapor and liquid densities and that this overall density is the critical value as the critical line is approached; see the discussion in Van Poolen et al.2 If this is so, then the thermodynamic path will access the critical point as the pressure increases. The thermodynamic path with three degrees of freedom is then described by the liquid volume fraction (set at 1/ ), a constant overall composition, and a varying 2 pressure. The constant overall composition along this thermodynamic path accessing the critical point is what gives the method its namesan isoplethic method. The method, outlined below, is applied successfully to the vapor-liquid equilibrium data of Barber et al.4,5 for the azeotropic mixture C3H8 (propane) + C4F8 (perfluorocyclobutane). The critical properties derived agree well with the published values. The procedure is then utilized to determine the hitherto unknown critical lines of two azeotropic, refrigerant + propane mixtures: R32 [CH2F2 (diflouromethane)] + C3H8 (propane) and R125 [C2F5H (pentaflouroethane)] + C3H8 (propane). These mixtures are being studied as possible alternative refrigerants in various commercial applications. (For convenience below, the refrigerants are designated as R32 and R125, propane as C3H8, and perflourocyclobutane as C4H8.)
[
]
LVF(Flx1l - Fvx1v) + Fvx1v FT
(1)
The coexisting densities and compositions in eq 1 must be known. In addition, a liquid volume fraction (LVF) of 1/2 entails FT ) (Fl + Fv)/2. The overall composition from eq 1 is calculated for discrete data points on a series of isotherms for which the liquid and vapor compositions vary from one pure component through the azeotrope to the other pure component. The P, Fl, and Fv values along each isotherm can be fitted separately to simple polynomials as functions of XT,1 (the calculated overall composition) at LVF ) 1/2. This procedure is done separately for those data points between the least volatile component (for a given isotherm) and the azeotrope and for those in the region between the azeotrope and the most volatile component. The percent deviations for these fits are within experimental uncertainties of the original data. A specific overall composition for an isopleth is then selected and inserted into this set of polynomial equations to obtain P, Fl, and Fv along an isopleth for which the liquid volume fraction is 1/2. This is done for the data along the several isotherms available below the lowest critical temperature of the two components. The result is an interpolated set of data (P, T, Fl, and Fv) at several points along a line of constant overall composition (for which the LVF is 1/2). This calculated line of constant overall composition at LVF ) 1/2 is an isopleth that can be extrapolated to the critical line. In essence, this procedure transforms two-phase binary fluid mixture data into data that behave similarly to those for a pure fluid. For example, a simple, pure-fluid-like vapor pressure curve (P versus T) results for each isopleth (see Figures 1-3). Data along these paths can then be extrapolated to the critical line using equations such as those for a pure fluid to obtain the critical point for each isopleth. To determine the critical temperature (Tc) for a given isopleth, the following equation is used
Fl - Fv ) C1
[
]
Tc - T Tc
β
(2)
where C1 and Tc are fitted constants, the nonclassical scaling exponent value β ) 0.325 is used, and the densities are determined from the fitted polynomials at the same overall composition with LVF ) 1/2. To determine the critical pressure (Pc) for a given isopleth, the pressures along the isopleth are fitted by the equation
ln P ) C2 +
[ ] C3 T
(3)
where C2 and C3 are fitted constants. Tc is then inserted into eq 3 to obtain Pc. Most likely, the expression for ln P, over a wide temperature range, needs more than the terms shown in eq 3 to accurately represent data.
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overall composition (as found by means of eq 1 for LVF at 1/2) as it varies from one pure component to the other. The pressure versus these overall compositions is fitted to a simple polynomial that has a maximum at the azeotrope. From this maximum, the overall composition and the associated pressure can be found for the azeotrope. Saturated densities along the isotherm are also fitted to simple polynomials over this composition range. By inserting the overall composition (found from the pressure maximum) into these polynomials, the saturated densities for the given pressure and temperature can also be found to provide a specific point on the azeotrope. To obtain Tc, Pc, and Fc of the azeotrope, eqs 2-4 are used. To obtain the composition of the azeotrope at its critical point (Xc,1), the overall composition (for which LVF is 1/2) for each calculated point along the azeotrope is plotted versus pressure. A linear relationship was sufficient to represent these calculated values to within 1%
XT,1 ) C6 + C7P Figure 2. Pressure versus temperature for the binary fluid mixture R32 + C3H8.
(5)
where C6 and C7 are fitted constants. The critical pressure (Pc) is inserted into eq 5 to obtain Xc.1 for the azeotrope. 3. Test of Method
Figure 3. Pressure versus temperature for the binary fluid mixture R125 + C3H8.
However, the few terms shown in eq 3 are utilized to avoid over-fitting, which, in turn, might create unnecessary errors in the extrapolated critical pressure. Care must be taken to appropriately limit the range of application of eq 3 as given. To obtain the critical density (Fc) for the isopleth, its rectilinear diameter is fitted by the equation
Fl + Fv ) C4 + C5T 2
(4)
where C4 and C5 are fitted constants. Tc is then inserted into eq 4 to obtain Fc. The average absolute percent deviations for the fits of the derived isoplethic data to eqs 2-4 are in the range of 1%. Values of Tc, Pc, and Fc are also obtained for the azeotrope. As a first step, the pressures for each twophase point on an isotherm are plotted against the
Binary mixture vapor-liquid equilibrium data (dew and bubble points, including density information) for an azeotropic mixture of C3H8 and C4F8 were available in the literature5 from Kay’s group at The Ohio State University, Columbus, OH. They also provided estimates for critical line values.4 The critical temperature and pressure values were obtained by closing the dewand bubble-point curves by means of a graphical technique. Their method is accurate, especially as it is based on large amounts of high-quality data available in the near-critical region. Critical densities were also estimated.4 No information is given as to how they were determined. Judging from a comment in another paper by the same group,6 it is likely that the critical densities were obtained by extrapolating an average of dew- and bubble-point densities. The authors called this a rectilinear diameter. However, the densities in the C3H8 and C4F8 paper are not in thermodynamic equilibrium with each other. Hence, the equation for the rectilinear diameter cannot be used to calculate the critical density. To utilize the isoplethic method, the dew- and bubblepoint data (below the critical temperature of C3H8) had to be transformed to vapor-liquid coexistence data. To do this, first, the isothermal dew- and bubble-point data (density and composition) were represented by simple polynomials in pressure. Then, at a given pressure, chosen to be constant in both phases, phase densities and compositions were found, to yield a vapor-liquid equilibrium data point (P, T, x1l, x1v, Fl, Fv) along an isotherm. The isoplethic method was then applied to these data to estimate the critical temperatures and pressures to be compared with the literature values. The results for the critical pressure and temperature are shown graphically in Figure 1. The points shown are all derived, isoplethic points including the critical point; hence, the same symbol is used for an entire isopleth. (This is true also for Figures 2 and 3.) Numerical comparisons including percent differences between the mean values are given in Table 1. The uncertainties
Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4613
in the derived critical temperatures and pressures are given for this work. The uncertainty in the critical temperature is based on eq 2 and experimental data uncertainties in the temperature and saturated densities ((0.1 K and (0.5 kg/m3, respectively). The uncertainty in the critical pressure is based on the propagation of uncertainty of the critical temperature in eq 3. The literature4 did not provide estimates for the uncertainty in the critical temperatures given. A precision estimate for the critical pressure is given as (0.017 Mpa.4 Estimates for the azeotrope critical-point values are also given in Table 1 for comparison. The criticalpoint values agree within reasonable uncertainties for critical points of binary mixtures. For comparison, all temperatures were referred to the temperature scale appropriate to the literature4,5 publication date. 4. Application of Method The method was then applied to the azeotropic mixture R32 + C3H8. Vapor-liquid equilibrium data,7 including saturation vapor and liquid densities recorded in our laboratory, are available for three isotherms (310.29, 324.82, and 340.26 K) below the critical temperature, 351.33 K, of R32. The method described above was used to obtain the isopleths shown in Figure 2. The azeotrope is also shown, along with the derived critical line. Values for the critical temperature, pressure, and density for the isopleths and azeotrope (shown in Figure 2) are given in Table 2, along with the critical values of the pure components. The uncertainties in the critical temperatures listed are based on eq 2 and the experimental data uncertainties given above. The uncertainties in the critical pressure and density are estimated by propagating the uncertainty in the critical temperature through eqs 3 and 4, respectively. The uncertainties in the derived azeotropic critical compositions (not given) are very small in that the pressure term in eq 5 had little effect on the result. The discrete critical temperature, pressure, and density values were fitted to the following equations similar to Redlich-Kister8 expansions, where XT,1 is denoted by X for convenience
Tc,mix ) XTc,1 + (1 - X)Tc,2 + GT1X(1 - X) + 3
GTiX(1 - X)(2X - 1)i-1 ∑ i)2
(6)
Pc,mix ) XPc,1 + (1 - X)Pc,2 + GP1X(1 - X) + 3
GPiX(1 - X)(2X - 1)i-1 ∑ i)2
(7)
Fc,mix ) XFc,1 + (1 - X)Fc,2 + GF1X(1 - X) + 3
GFiX(1 - X)(2X - 1)i-1 ∑ i)2
(8)
where the subscript 2 refers to the component having the highest critical temperature and GT1, GT2, GT3, GP1, GP2, GP3, GF1, GF2, and GF3 are fitted constants. The critical line values in Table 2 are fitted to eqs 6-8 with maximum deviations of 0.05, 0.5, and 1.0%, respectively. The values for the fitted constants for eqs 6-8 are given in Table 3. The method was also applied to the azeotropic mixture R125 + C3H8. Vapor-liquid equilibrium data7
Table 1. Comparison of Critical Temperatures and Pressures for C3H8 + C4F8 XT,1
Tc (K)a
0.2483 0.4861 0.6359 0.7457
375.78 ( 0.13 366.56 ( 0.13 363.30 ( 0.13 361.92 ( 0.13
Tc (K)b % diff
Pc (MPa)a
Pc (MPa)b % diff
376.69 -0.24 3.095 ( 0.007 366.36 0.05 3.328 ( 0.007 362.63 0.18 3.481 ( 0.008 361.81 0.03 3.597 ( 0.009
azeotrope comparison 0.8948 364.71 ( 0.16 3.901 ( 0.013 365.09c 0.8960c a
b
3.132 3.354 3.472 3.596
-1.15 -0.78 0.26 0.03
3.936c
c
This work. Reference 4. Reference 5.
Table 2. Critical Temperature, Pressure, and Density for Pure Components and Mixtures XT,1
Fc (kg/m3)
Tc (K)
Pc (MPa)
0.0000 0.1862 0.2275 0.3006 0.6859 0.8500 1.0000
369.82a 354.78 ( 0.21 352.81 ( 0.20 349.86 ( 0.19 342.24 ( 0.14 343.01 ( 0.13 351.33
R32 + C3H8 4.2475 4.6681 ( 0.0176 4.7729 ( 0.0158 4.9426 ( 0.0165 5.3400 ( 0.0136 5.5162 ( 0.0141 5.7745b
comment
0.0000 0.1174 0.1974 0.3615 0.6609 1.0000
369.82a 357.39 ( 0.21 350.89 ( 0.18 343.22 ( 0.14 337.27 ( 0.15 339.33
R125 + C3H8 4.2475 220.600 C3H89 4.0514 ( 0.0155 259.816 ( 0.144 3.9415 ( 0.0128 294.435 ( 0.106 3.9025 ( 0.0101 373.727 ( 0.006 3.8166 ( 0.0107 483.618 ( 0.064 azeotrope 3.6290 571.290 R-12512
220.600 C3H89 258.261 ( 0.078 265.922 ( 0.069 279.283 ( 0.067 355.571 ( 0.011 azeotrope 380.752 ( 0.202 429.040 R-3211
a ITS-90 value. b Obtained at 351.33 K from vapor pressure data of Holcomb et al.10
Table 3. Constants for Equations 6-8 GT1 (K) GT2 (K) GT3 (K) GP1 (MPa) GP2 (MPa) GP3 (MPa) GF1 (kg/m3) GF2 (kg/m3) GF3 (kg/m3)
R32 + C3H8
R125 + C3H8
-0.622 40 × -0.371 23 × 101 -0.436 88 × 102 0.771 70 -0.964 31 -0.908 49 -0.448 57 × 101 -0.612 96 × 102 -0.144 61 × 103
-0.628 04 × 102 0.266 34 × 102 0 -0.376 30 0.102 19 × 101 0 0.115 723 × 103 0.114 167 × 103 0
102
including saturated vapor and liquid densities, also recorded in our laboratory, are available for two isotherms (310.97 and 326.21 K) below the critical temperature, 339.33 K, of the component R125. As an exception to data being below the lowest pure-component critical temperature, data along a third isotherm (340.47 K) just above the critical-point temperature of R125 were also used. The resultant critical line is shown in Figure 3, along with the isopleths and the azeotrope. Discrete values of the critical temperature, critical pressure, and critical density for the mixtures and pure components are given in Table 2, along with the critical parameters for the azeotrope. The uncertainties shown are estimated in the manner given above. The critical values in Table 2 are fitted to eqs 6-8 to within less than 0.2, 0.8, and 2.0%, respectively. Table 3 gives the constants for the critical-line representations (eqs 6-8) for this mixture. 5. Summary and Future Work An isoplethic method has been developed to access (and to determine) the critical lines of binary fluid mixtures from the area of the two-phase surface away from the critical region. The method relies on thermodynamic paths having three degrees of freedom accord-
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ing to the phase rule of Van Poolen.1 The three variables chosen to specify the path are the liquid volume fraction at a constant value of 1/2, a constant overall composition, and the pressure. The liquid volume fraction of 1/2 entails a total or overall density equal to the average of the coexisting saturated liquid and vapor densities, that is, the rectilinear diameter. The method was tested for azeotropic mixtures by estimating the critical pressure and temperature of the binary fluid mixture C3H8 + C4F8. The results seen in Figure 1 and Table 1 warranted application of the method to other azeotropic mixtures for which no critical line information is available. Critical lines for the two mixtures R32 + C3H8 and R125 + C3H8 were developed using the proposed method. The shapes of the critical lines in pressure-temperature space (see Figures 2 and 3) are typical of measured azeotropic mixture critical lines. Planned future work is to use this method to obtain critical line values that, in turn, can be used to correlate and predict vapor-liquid surface tensions throughout the two-phase region. Either the fugacity fraction corresponding states method of Nadler et al.13 or a modification of that method along constant overall composition lines (isopleths) with liquid volume fractions equal to 1/2 is to be pursued. Acknowledgment This work was supported in part by a grant from the U.S. Department of Energy, Office of Building Technology. Nomenclature C ) number of mixture components in the phase rule, degrees of freedom ) (C + 1) C1-C7 ) fitted constants in eqs 2, 3, 4, and 5 GT1-GF3 ) fitted constants in eqs 6,7, and 8 LVF ) liquid volume fraction P ) pressure T ) temperature x ) phase composition on a mole fraction basis X ) overall composition on a mole fraction basis Greek Symbols F ) density Subscripts c ) critical point mix ) mixture T ) total or overall system property 1 ) component having the lowest critical temperature 2 ) component having the highest critical temperature Superscripts l ) saturated liquid v ) saturated vapor
Literature Cited (1) Van Poolen, L. J. Extended Phase Rule for Non-Reactive, Multiphase, Multicomponent Chemical Systems. Fluid Phase Equilib. 1990, 58, 133. (2) Van Poolen, L. J.; Niesen, V. G.; Rainwater, J. C. Experimental Method for Obtaining Critical Densities of Binary Mixtures: Application to Ethane + n-Butane. Fluid Phase Equilib. 1991, 66, 161. (3) Van Poolen, L. J.; Rainwater, J. C. Vapor-Liquid Equilibria of Ternary Mixtures in the Critical Region on Paths of Constant Temperature and Overall Composition. Int. J. Thermophys. 1995, 16, 473. (4) Barber, J. R.; Kay, W. B.; Teja, A. S. A Study of the Volumetric and Phase Behavior of Binary Systems, Part I: Critical Properties of Propane-Perfluorocyclobutane Mixtures. AIChE J. 1982, 28, 134. (5) Barber, J. R.; Kay, W. B.; Teja, A. S. A Study of the Volumetric and Phase Behavior of Binary Systems, Part II: Vapor-Liquid Equilibria and Azeotropic States of PropanePerfluorocyclobutane Mixtures. AIChE J. 1982, 28, 138. (6) Genco, J. M.; Teja, A. S.; Kay, W. B. Study of the Critical and Azeotropic Behavior of Binary Mixtures. 1. Critical States of Perfluoromethylcyclohexane-Isomeric Hexane Systems. AIChE J. 1980, 25, 350. (7) Holcomb, C. D.; Magee, J. W.; Scott, J. L.; Outcalt, S. L.; Haynes, W. M. Selected Thermodynamic Properties for the Mixtures of R-32 (Difluoromethane), R-125 (Pentafluoroethane), R-134A (1,1,1,2-Tetrafluoroethane), R143A (1,1,1-Trifluoroethane), R-41 (Fluoromethane), R-290 (Propane), and R-744 (Carbon Dioxide); NIST Technical Note 1397; National Institute of Standards and Technology: Boulder, CO, Dec 1997. (8) Redlich, O.; Kister, A. T.; Turnquist, C. E. Thermodynamics of solutions. Analysis of vapor-liquid equilibria. Chem. Eng. Prog. Symp. Ser. 1952, 48 (2), 49. (9) Goodwin, R. D.; Haynes, W. M. Thermophysical Properties of Propane from 85 to 700 K at Pressures to 70 MPa; NBS (NIST) Monograph 170; National Institute of Standards and Technology: Boulder, CO, Apr 1982. (10) Holcomb, C. D.; Niesen, V. G.; Van Poolen, L. J.; Outcalt, S. L. Coexisting Densities, Vapor Pressures and Critical Densities of Refrigerants R-32 and R152a at 300-385 K. Fluid Phase Equilib. 1993, 91, 145. (11) Van Poolen, L. J.; Holcomb, C. D.; Niesen, V. G. Critical Temperature and Density from Vapor-Liquid Coexistence Data: Application to Refrigerants R32, R124, and R152a. Fluid Phase Equilib. 1997, 129, 105. (12) Outcalt, S. L.; McLinden, M. O. Equations of State for the Thermodynamic Properties of R32 (Difluoromethane) and R125 (Pentafluoroethane). Int. J. Thermophys. 1995, 16, 79. (13) Nadler, K. C.; Zollweg, J. A.; Streett, W. B. Global Representation of Interfacial Tension in the System (Ar + Kr). Int. J. Thermophys. 1989, 10, 333.
Received for review March 5, 2001 Revised manuscript received July 26, 2001 Accepted July 30, 2001 IE010215X
