Vapour—liquid equilibrium of nitrogen—oxygen mixtures and air at high pressure

Vapour-liquid equilibrium of nitrogen-oxygen mixtures and air at high pressure* J.C. R a i n w a t e r

a n d R.T. J a c o b s e n t

Thermophysics Division, National Bureau of Standards, Boulder, CO 80303, USA

Received 10 August 1987 The vapour-liquid equilibrium surface of the binary mixture nitrogen-oxygen is correlated over an extended critical region with the Leung-Griffiths model as modified by Rainwater and Moldover. No single comprehensive experimental measurement of the coexistence surface is available. However, several different experiments along isopleths, isotherms and isobars collectively provide enough data to make possible the development of a reasonable correlation. The model is optimized to modern data and is shown to be consistent with pioneering measurements done before 1930, to within experimental and temperature-scale uncertainties. It is shown that air in the critical region can be accurately modelled as a nitrogen-oxygen binary mixture by including the small argon component with oxygen. Ancillary equations for the saturation properties of air as functions of temperature are also constructed.

Keywords: air; binary mixtures; nitrogen; oxygen; vapour-liquid equilibrium

Recently, significant progress has been made in the correlation of vapour-liquid equilibrium (VLE) of binary mixtures within the critical region. The model of Leung and Griffiths1, as modified by Rainwater, Moldover and co-workers 2-6, has been successfully applied to more than 20 binary mixtures. Conventional approaches based on classical equations of state and standard phase equilibrium algorithms typically fail to converge or are inaccurate near the critical locus. The modified Leung-Griffiths method requires as input the saturation properties of the two pure fluids and (at least) a reasonable estimate of the critical locus in pressure, temperature and density as a function of composition. From this information and a small number of adjustable parameters, the coexistence surface in pressure, temperature, density and composition (P-T-px) is accurately represented over an extended critical region. In past applications, the range of applicability has been from the mixture critical pressure to about half of that pressure. Unlike classical equations of state, the model is consistent with scaling-law critical behaviour and correctly incorporates non-classical critical exponents. Successful application of the model requires adequate VLE measurements over the entire extended critical region. For previous tests of the model, highest priority has been assigned to mixtures with the most extensive critical region data. The guidelines have been that experimental studies of (P, 7) points along constant-composi*Contribution of the National Bureau of Standards, not subject to copyright tpermanent address: Center for Applied Thermodynamic Studies, University of Idaho, Moscow, ID 83843, USA 0011-2275/88/010022-10 $03.00 © 1988 Butterworth & Co (Publishers) Ltd

22 Cryogenics 1988 Vol 28 January

tion lines should include at least four isopleths, and measurements of (P, x) points along lines of constant temperature should include at least four isotherms. Also, the data should fully span the thermodynamic region from one pure substance to the other. The inclusion of coexisting density measurements is also highly desirable. Rainwater has located more than 70 mixtures with data, including density measurements, that conform to the above criteria. Surprisingly, nitrogen + oxygen is not one of them, in spite of its obvious importance. Air is conventionally modelled as a ternary mixture of nitrogen, oxygen and argon. As we shall show, because argon and oxygen have nearly the same (P, T, p) critical point, the small amount of argon in air may be included with oxygen in the critical region with negligible error. Applications to date of the modified Leung Griffiths method demonstrate that a single isopleth of a mixture should not be studied in isolation, but rather that a thermodynamic description should be constructed for the complete range of mixture compositions. Although several different workers have reported measurements on high pressure nitrogen + oxygen 7- 10 as well as air 11 13 there is no single source that provides a thorough determination of the mixture coexistence surface. Also, many of the results are from pioneering experiments, conducted before the establishment of modern experimental techniquesS,9,11. The objective of this Paper is to construct a modified Leung-Griffiths model for the high pressure VLE of nitrogen + oxygen that is consistent with scaling-law critical behaviour and the somewhat limited experimental data. A precedent for such a study is the model of isobutane + isopentane t4 based on limited data near the -

Vapour-fiquid equilibrium. J.C. Rainwater and E.T. Jacobsen critical line and results of a classical equation of state ~5 0.5 MPa below critical, and constructed for a particular geothermal application. We begin in the following section with a review of experimental measurements of nitrogen + oxygen and air in the critical region. Following this, the model is fitted to the data, and we then show for nitrogen + oxygen that it is successful over a somewhat larger region than has been used for previous mixtures. Specific applications to air, and comparisons with ancillary temperature-density and pressure-temperature equations, are presented in the section on Results for air. A summary is provided in the Conclusions section, and the most important equations of the model are presented in Appendix A. Appendix B gives ancillary equations for the calculation of dew line and bubble line properties of air, and some detailed analysis of the data of Michels et al~ 3 and related correlations is provided in Appendix C.

Available data in the critical region The critical point of oxygen is Tc = 154.581 K, Pc = 5.043 MPa and Pc = 13.63 kmol m - a, and that of nitrogen is Tc = 126.24 K, Pc = 3.398 MPa and Pc = 11.21 kmol m-3. As previously defined, the extended critical region includes data for nitrogen-rich mixtures if P > 1.7 MPa and for oxygen-rich mixtures if P > 2.5 MPa. Measurements on air are also considered, despite the small argon component. The composition of air given by Waxman and Davis 16 is 0.7814 N z + 0.2093 O 2 + 0.0093 At. The first measurements of high pressure VLE for this system were by Kuenen and Clark 11 (1917) for air. Subsequently, Kuenen et al. 8 (1922) measured dewbubble curves of the binary mixture for mole fractions of x = 0.25 and x = 0.5, where x = 1 is pure nitrogen. Their unit of molar volume, as specified by Verschaffelt ~7 in an earlier communication, is that of the specified mixture at 0°C and 0.101325 MPa (1 arm) assuming ideal gas behaviour, and they used the conversion 0°C = 273.09 K. Thus, their unit of molar density is 0.0446 kmol m -3.

Kuenen et al. s tabulated their data points in pressure and volume along isotherms and listed the liquid volume fraction for the two-phase points. Dew and bubble points were identified by a liquid volume fraction of 0 or 1, respectively. In some cases the phase boundary was not identified, but could be determined from the discontinuity of (~?P/~P)r according to their isothermal measurements. In view of the uncertainties in such a procedure, we restrict our analysis to the data points explicitly identified by the authors at liquid volume fractions of 0 or 1. Shortly thereafter, Dodge and Dunbar 9 (1927) measured (P, x) points for nitrogen + oxygen along isotherms below the nitrogen critical point. Their data is extensive at rower temperatures and pressures, but only one of their isotherms, at 124.95K, extends above P = 1.7MPa. Dodge and Dunbar did not study isotherms between the critical temperatures of pure nitrogen and pure oxygen. More modern experimental measurements include an investigation of the critical line by Jones and Rowlinson 7. Critical temperatures, but not critical pressures or critical densities, were reported as functions of composition for a variety of cryogenic binary mixtures including nitrogen + oxygen. Blanke~ 2 and Michels et al.l 3 have measured

pressure, temperature and density of the dew-bubble curve for air to the critical point. Finally, Wilson et al, 1° have reported extensive VLE measurements on the binary mixtures of nitrogen, oxygen and argon. Wilson et al. measured (T, x) points along isobars; their highest pressures were 2.027, 2.331 and 2.635 MPa (20, 23 and 26 atm). Although the x = 0.25 dew-bubble curve was measured by Kuenen et alY from the critical pressure down to 4.089 MPa, there are evidently no VLE data for oxygen-rich mixtures (x < 0.5) between 2.7 and 4.0 MPa.

Thermodynamic model The Leung Griffiths model, as modified by Rainwater and Moldover, has been described at length elsewhere 1'4'6. Here we present only a qualitative description of the model. A summary of the most important equations is provided in Appendix A. Leung and Griffiths 1 introduced the variable ~, defined as

exp(/~I/RT)

(1)

= Kexp(l~2/RT) + e x p ( / q / R T )

where: R is the gas constant; l~i is the chemical potential for fluid i (here i = 1 for oxygen and i = 2 for nitrogen); and K is an arbitrary constant, Since #1 approaches - oo in the limit of pure fluid 2 and vice versa, and in our convention x = 0 for pure fluid 1, it follows that ~ = 1 when x = 0 and ~ = 0 when x = 1. Coexisting liquid and vapour states have the same chemical potentials, so each (P, T) point on the coexistence surface corresponds to a single value of ~ for non-azeotropic mixtures. A second important variable within the modified Leung -Griffiths model 3 is t, where

t = T - L(0 L(0

(2)

and, on the coexistence surface, for non-azeotropic mixtures each (P, T) point corresponds to a unique (~, t) point. The chemical potentials, /~i, in Equation (1) are not directly measurable. It is assumed in the model that, because of the freedom to choose K and the zeros of the chemical potentials, a definition of ~ exists with some very simple properties in the critical region. In particular, the assumptions of the model are that x = 1 - ~ on the critical line, and that lines of constant ~ on the coexistence surface in P T space form a parallel set of curves bounded by the pure vapour pressure curves. The latter condition is expressed mathematically by Equations (AI) and (A3); see also Figure 1 of Reference 4. Because of the steepness of P T vapour pressure curves, and thus on the VLE surface of lines of constant ~, t = 0.1 corresponds to P ~ 0.5Pc(~). In past applications of the model, its range of validity was taken to be -0.1 < t < 0. However, to compare better with the limited amount of high pressure data for nitrogen + oxygen, in the present work the lower limit of the range is extended to t = -0.14. Within the model, the critical line, Equations (A4)-(A7), and the pure fluid saturation curves, Equations (AI) and (A2) with ~ = 0 or 1, are fit to polynomials, where some non-analytic terms appear to account for scaling-law behaviour. Given the critical line and saturation curves,

Cryogenics 1988 Vol 28 January

23

Vapour-liquid equilibrium: J.C. Rainwater and E. T. Jacobsen -~

140 -

I

I

~

I

I

I

I

0.6

0.8

\\N

t30

L 120t 0

I

I

0.2

0.4

7 1.0

X Figure 1 Temperature-composition diagram with VLE isobars for nitrogen + oxygen: a, 2.027 MPa; b, 2.331 MPa; c, 2.635 MPa; d, 3,0 MPa; e, 3.5 MPa; f, 4.0 MPa; g, 4.5 MPa. Also shown are data of Wilson etal.l°: r~, 2.027 MPa; IS],2.331 MPa; O, 2.635 M P a . - - - - , Critical locus

points. According to modern terminology, the terms 'plait point' and 'critical point of contact', as used by Kuenen and co-workers 8, denote the critical and maxcondentherm points, respectively. Since dPc/d T~(dP/d T along the critical line) is positive, the critical pressure and temperature are slightly below those of the maxcondenbar point. The critical line has been represented in this work as follows. Equations (A4) and (A6) were solved given the three critical points reported by Kuenen and coworkers 8'1~. From this solution, critical pressures were calculated for the four additional critical temperatures measured by Jones and Rowlinson 7. Equations (A4) and (A6) were then fitted to these seven (Pc, To, xc) points by standard linear regression. The resulting coefficients are given in Table 2. In view of the lack of modern data for the critical density locus, we adopt the simplest possible assumption for Equation (A7), i.e. ~1 = P2 = P3 = 0. Our policy is to assign greater weight to the recent VLE data than to the data taken before 1960. Accordingly, optimization of CH and Cx was based on the measurements of Blanke 12 and of Wilson et al. ~°. The resulting model was compared with the data used in the optimization, and with the earlier data of Kuenen and co-workers TM of Michels et al. 13, and of Dodge and Dunbar 9.

Results for nitrogen + oxygen the VLE surface in P, T, p and x is completely determined by two adjustable parameters, Cn and Cx. Increasing (decreasing) C H moves the dew-bubble curves to lower (higher) temperatures, and increasing (decreasing) Cx narrows (widens) the dew-bubble curves. Since the parameters C n and Cx enter the model in a highly non-linear manner, we have found it best to optimize them by plotting the VLE data using computer graphics and by varying C H and Cx until a best fit is obtained by visual criteria. The mixture nitrogen + oxygen has a convenient relative volatility difference for application of the model. Geometrically, the dew-bubble curves are wide enough that the liquid and vapour sides are easily distinguished graphically, yet narrow enough that only two adjustable parameters are needed. Additional parameters are required for mixtures of greater relative volatility, e.g. ethane + n-butane 5. For pure oxygen, Equations (A1) and (A2) are fitted to the saturation data of Weber TM. The corresponding fits for pure nitrogen are those previously used for nitrogen + methane by Rainwater and Moldover 4, and are based on the data of Jacobsen et al.~ 9 The coefficients of the pure fluid fits are listed in Table 1. Fitting the critical line presents some difficulty, since only Kuenen and Clark 11 and Michels et al. 13 for air, and Kuenen et al. 8 for x = 0.5 and 0.25, have reported measurements at pressures near the mixture critical Table 1

From a graphical optimization, we find that nitrogen + oxygen is best described with Cn = - 6 and Cx = 0.7. This value of Cx is somewhat large compared with normal alkane mixtures of similar relative volatility TM, but mixtures of members of a homologous series generally allow a more simple description than mixtures of dissimilar molecules. In Figure 1 the model is compared with the three highest isobars of Wilson et al. ~° at 2.027, 2.331 and 2.635 MPa (20, 23 and 26 atm). Also shown are the theoretical isobars for 3.0, 3.5, 4.0 and 4.5 MPa. Wilson et al. present one duplicate measurement (at the same temperature and pressure) on their middle isobar, and several duplicate and triplicate measurements on their lowest isobar of Figure I. For the single measurements, the model in all cases agrees with the data to within 0.008 in composition. For the multiple measurements, the model agrees with at least one of the measurements to within 0.0065, whereas the measurements disagree among themselves by as much as 0.0252. The model thus is judged to fit the data well within their precision. Loci of constant composition on a P - T diagram are shown in Figure 2. The data points for air due to Blanke ~2 and Michels et al.13 are discussed in the following section. Also displayed are the data of Kuenen et al. s for x = 0.25 and x = 0.5, and the predicted dew-bubble curves from

Coefficients of Equations (A1) and (A2) Table 2

C1 C2 C3 C4 C5 C6

24

Nitrogen

Oxygen

1.90 -0.670 25.0 5.10 22.54 - 7.20

1.846 -0.7112 20.6 5.07 - 15.16 8.01

Cryogenics 1988 Vol 28 January

71 72 73 Pl P2

Coefficients of Equations (A4) and (A6)

--4.2522 x 10 2kmol (MPam 3) 1 - 8 . 1 9 x 10 4kmol (MPa m 3) 1 - 1 . 2 0 4 x 1 0 3kmol (MPa m 3) 1 4.3346 x 10-2kmol m 3 -0.116513 kmol m -3 5.4401 x 1 0 - 2 k m o l m 3

Vapou~fiquid equilibrium: J.C. Rainwater and E.T. Jacobsen

5

I

I

I

Figure 4 is a plot of temperature versus density. The Blanke 12 and Michels et al. 13 air data are discussed in

I

//

/ /

_

////¸

v (3_ 3

/ 2

110

120

I

130

I

140

160

150

T (K) Figure 2 Pressure temperature diagram s h o w i n g , left to right, nitrogen v a p o u r pressure curve, d e w - b u b b l e curves for x = 0.7814, 0.5 and 0.25, and o x y g e n v a p o u r pressure curve. , Critical locus. Data are as f o l l o w s : E], air, from Blanke12; [~, x - 0.5, from Kuenen etal.8; © , x = 0.25, from Kuenen etal.8; A , air, from Michels eta/. 13

the model at these compositions. As previously discussed, only experimental points explicitly identified as dew or bubble points, as well as their maxcondentherm and critical points, are included in Figure 2. We assume that the temperature for their bubble point VI. 4 at x = 0.5, is misprinted and should be -133.00°C, not -132.00°C. The model agrees with the data of Kuenen et al. for x = 0.5 and P < 3.7 MPa to within 0.25 K in temperature and 0.03 MPa in pressure. For P > 3.7 MPa, there is evidence that the data are systematically low in temperature. A detailed view of the region near the critical point for x = 0.25 is shown in Figure 3. The experimental and theoretical loci have essentially the same shape and curvature, but are displaced in temperature from each other by 0.4K. A similar description applies to the neighbourhood of the critical point for x = 0.5. 4.70

I

the following section. For liquid and vapour densities somewhat removed from critical, i.e. p < 10.5 kmol m-3 or p > 14.55 kmol m-3, the model predictions agree with the density measurements of Kuenen et al. a to within 0.35 kmol m - 3. Furthermore, the simple assumption that /3i = 0 in Equation (A7) is consistent with their critical densities to within 0.17 kmol m - 3. In the near-critical region along the coexistence curves, ((?T/?p)x is very small so small errors in temperature lead to large errors in density and the above comparison is not very meaningful. However, as seen from the figure, a small shift of the experimental points in temperature would lead to consistency with the model density predictions. In comparison to the fit to Equation (A4), the critical point of Jones and Rowlinson 7 (x = 0.202, T~ = 149.4 K) differs by + 0.24 K, whereas that of Kuenen et al. s (x = 0.25, T~= 147.49 K) differs by - 0 . 3 3 K. We expect that, for a mixture as well-behaved as nitrogen + oxygen, the fifth order polynomial of Equation (A4) should fit the actual critical temperature locus to within 0.1 K, so the modern results of Jones and Rowlinson also suggest that the Kuenen et al. s measurements are low in temperature by ~0.4 K. A pressure--composition diagram is constructed in Figure 5, and the isothermal data ( T = 124,95K) of Dodge and Dunbar 9 for P > 2.0 MPa are shown. According to Dodge and Dunbar, the temperature along the approximate isotherm varies from 124.93 to 125.06K. The model is in reasonable agreement with experiment but is systematically high in composition, and the data are most compatible with a model isotherm of 125.1 K. According to Din 2°, the temperature scale of Dodge and Dunbar is in error by 0.5 K at the methane point, and Dodge and Dunbar themselves state an uncertainty of 0.1 K, so the model predictions are well within the experimental uncertainty. Also shown on Figure 5 are the model isotherms at 130, 135, 140, 145 and 150K. 160

I

I

I

I

I 10

[I

I

I

I

150

j~ 4.65 140

%J /J

j )--

jjJ'~

Q.

4.60

fJ

130

J J /J f

0_

120

4.55

110 4.50 146.5

147.0

147.5

148.0

T (K) F i g u r e 3 Expanded view of critical region for x-0.25. - --, Critical locus; , model dew-bubble curve; ©, data points of Kuenen et al. 8

/ 0

p (kmol/m

I

20

\\

I] 30

3)

Figure 4 Temperature-density diagram s h o w i n g , t o p to bottom, coexistence curves for pure oxygen, x = 0.25, 0.5 and 0 . 7 8 1 4 and pure nitrogen. Data symbols as for Figure 2

Cryogenics 1988 Vol 28 J a n u a r y

25

Vapour-fiquid equilibrium." J.C. Rainwater and E.T. Jacobsen I

I

I 0.2

0.4

I

I

5.0~,,~,

4.0 ft. v

0.. 3.0

2.0 I

I

0.6

I

0.8

.0

Figure 5 Pressure-composition diagram with VLE isotherms for nitrogen+oxygen: a, 124.95K; b, 130K; c, 135K; d, 140K; e, 145 K; f, 150 K. Data are from Dodge and Dunbar9 at ~124.95 K

in part reflect the differences between the representation of air as a binary or a ternary mixture. Further analysis of this data is presented in Appendix C. The dew and bubble points for air from the model are listed in Table B-3. According to our numerical routines for calculating isopleths, points of(~, t) are located starting from an essentially geometric series in t. The resulting intervals in P, T and p are thus irregular, as shown. The model dew-bubble curve may be compared with ancillary equations for pressure and density as functions of temperature. The four ancillary equations and coefficients are given in Appendix B. Figures 6 - 1 3 illustrate the results from Blanke 12 and from this work used as input to construct the ancillary equations, and the deviations between the ancillary equations and the input. We can also make a rough estimate of the error in approximating air to be a binary, rather than a ternary, mixture. Many of the ternary effects would depend on the model parameters for the binary mixtures nitrogen + argon and argon + oxygen, and sufficient data is not available to determine such parameters. However, perhaps the dominant ternary effect would be a shift in the critical point of the mixture that would propagate along the dew-bubble curve. For an estimate of the shift, we consider the crude approximation for the ternary critical temperature

Results for air

T~ = T~(O2)x(O2) + T~(Ar)x(Ar) + T~(N2)x(Nz)

Air is conventionally modelled as a ternary mixture of 78.14 mole per cent nitrogen, 20.93 mole percent oxygen and 0.93 mole per cent argon ~6. The modified LeungGriffiths model can, in principle, be applied to ternary mixtures TM, but would require as input the critical surface of the nitrogen + oxygen + argon system, which is not known. However, the critical point of argon is very close to that of oxygen. Specifically, the critical temperatures and pressures of argon and oxygen differ by 3.92 K and 0.18 MPa, respectively, whereas the corresponding differences for nitrogen and oxygen are 28.34K and 1.65 MPa, respectively. Since the amount of argon is small, in the critical region it should be an excellent approximation to include the argon with the oxygen, i.e. to model air as a binary mixture of nitrogen + oxygen with x = 0.7814. In the present analysis, we use the dew-bubble data for air of Blanke ~2 and of Michels et al. 13 in preference to those of Kuenen and Clark ~1 which display considerable scatter. The data for air are plotted in Figures 2 and 4, together with the x = 0.7814 isopleth from the model. Agreement between theory and the data of Blanke ~2 is excellent, within 0.15 K in temperature, 0.02 MPa in pressure and 0.2 kmol m - 3 in density, and with no clear systematic deviations. According to the model, the critical point of air is To= 132.56K, Pc = 3.789MPa and p c = 11.85 kmol m -3. The estimated uncertainties (from all sources) of these values as an estimate of the actual critical point of air are 0.20 K, 0.03 MPa and 0.3 kmol m-3. The data of Michels et al. ~3 needs to be treated with some care, as discussed in Appendix C. Except for the density of the bubble point closest to critical, the selected data of Michels et al. agree with the model as well as do the Blanke points. There is a small systematic trend on the dew curve in that temperatures for the data are lower than those of the model, which, as discussed below, may

and similarly for Pc and Pc. This approximation is sufficient to estimate the difference in critical points between the binary mixture 0.7814 N 2 + 0.2186 0 2 and the ternary mixture 0.7814 N 2 + 0.2093 O z + 0.0093 Ar that represents air. The estimated shift from binary to ternary is - 0.037 K for T~, -0.0017 MPa for Pc and -0.0026 kmol m -3 for Pc. These shifts are much smaller than the uncertainties in the model due to other sources. We also note that, while the ternary critical point is slightly lower in pressure and temperature than the binary critical point, the dew-bubble curve for the ternary should be slightly wider than that of the binary. These two small effects would thus tend to cancel each other at temperatures and pressures below critical.

26

Cryogenics 1988 Vol 28 January

(3)

Conclusions A modified Leung-Griffiths model has been constructed for the binary mixture nitrogen + oxygen and has been shown to apply to air if the small argon component is included with oxygen. The agreement between the model and a diverse but limited body of high pressure coexistence data is comparable to the magnitude of the experimental uncertainties. Unlike classical equations of state, the model is consistent with scaling-law critical behaviour and non-classical critical exponents. Further verification of the model would require additional VLE data. It is surprising that more than 70 binary mixtures have been more thoroughly measured in the critical region than the fluid mixture which is, essentially, our atmosphere. Measurements of oxygen-rich mixtures within the pressure range 2.7 MPa < P < 4.0 MPa would be particularly valuable. The data situation for nitrogen + argon and argon + oxygen is even less satisfactory, although Wilson et al. ~°

Vapou~fiquid equilibrium. J.C. Rainwater and E. T Jacobsen have made extensive isobaric measurements for P = 2.635 MPa and below, and Narinskii 22 has measured isotherms for T <~ 120 K. Similarly, for the nitrogen + argon + oxygen ternary system, Israel et al, 23 have reported isobaric measurements for P ~<2.635 M P a and Narinskii 24 has reported limited measurements up to 120 K and 2.32 MPa. With more extensive high pressure VLE data, the nitrogen + argon + oxygen ternary system would be an excellent candidate mixture for application of the modified Leung-Griffiths model, since the components are completely miscible, the volatility differences are relatively small and there is no azeotrope

Acknowledgement The authors thank J . M H . Levelt Sengers for valuable discussions concerning the data of Reference 13

29 Pominov, A.M. Russ J Phys Chem (1985) 59 578 30 Rainwater, J.C. Nat Bur Stand (US) Tech Note 1061 (1983) 83

Appendix model

A.

Modified

Leung-Griffiths

In this Appendix, the basic equations of the Leung Griffiths model 1 as modified by Moldover, Rainwater and co-workers 2 - 6 are presented with minimal comment. Their derivation and thermodynamic justification are presented at length elsewhere 4"6. The independent variables of the model are ~ and t, Equations (1) and (2). On the coexistence surface P T

P¢(~) [1 + C 3 ( ~ ) ( - t ) 1'9 + C4(~)t + Cs(~)t 2 + C6(~)t 3] T~(~)

(AI) and

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22 23

24 25 26

27 28

Leung, S.S. and Griffiths, R.B. Phys Rev (1973) A8 2670 Moldover, M.R. and Gallagher, J.S. ACS Symp Ser (1977) 60 498 Moldover, M.R. and Gailagher, J.S. A1ChE J (1978) 24 267 Rainwater, J.C. and Moldover, M.R. in: Chemical Engineering at Supercritical Fluid Conditions (Eds Paulaitis, M.E. et al.) Ann Arbor Science, USA (1983) 199 Rainwater, J.C. and Williamson, F.R. Int J Thermophys (1986) 7 65 Rainwater, J.C. to be published Jones, I.W. and Rowlinson, J.S. Trans Far Soc (1963) 59 1702 Kuenen, J.P., Verschoyle, J. and van Urk, A.T. Commun Phys Lab Leiden (1922) 161 reprinted in Proc Acad Sci Amsterdam (1923) 26 49 Dodge, B.F. and Dunbar, A.K. J Am Chem Soc (1927) 49 591 Wilson, G.M., Silverberg, P.M. and Zellner, M.G. Adv Cryog Eng Plenum, New York, USA (1965) 10 192 Kuenen, J.P. and Clark, A.L. Commun Phys Lab Leiden (1917) 150b Blanke, W. in: Proc Seventh Syrup Thermophysical Properties (Ed Cezairliyan, A.) ASME, New York, USA (1977) 461 Michels, A., Wassenaar, T., Levelt, J.M. and DeGraaff, W. Appl Sci Res (1954) A4 381 DiRer, D.E., Gallagher, J.S., Kamgar-Parsi, B., Morrison, G., Rainwater, J.C., Lever Sengers, J.M.H., Sengers, J.V., Van Poolen, L.J. and Waxman, M. Nat Bur Stand (US) Interagency Report NBSIR 85-3124 (May 1985) Gailagber, J.S. lnt J Thermophys (1986) 7 923 Waxman, M. and Davis, H. J Res Nat Bur Stand (US) (1978) 83(5) 415 Verschaffelt, J.E. Commun Phys Lab Leiden (1898) 45 Weber, L.A. J Res Nat Bur Stand (US) (1970) 74A 93 Jacobsen, R.T., Stewart, R.B, MeCarty, R.D. and Hanley, H.J.M. Nat Bur Stand (US) Tech Note 648 (1973) Din, F. Thermodynamic Functions of Gases Vol 2, Butterworth Scientific Publications, London, UK (1956) 4649 Balbuena, P. MSc Thesis Univ. of Pennsylvania, USA (1983) Narinskii, G.B. Russ J Phys Chem (1966) 40 1093 Israel, L. Silverberg, P.M., Sterner, C.J. and Wilson, G.M. Argon-Oxygen Nitrogen Three Component System, Experimental Vapor-Liquid Equilibrium Data Quarterly Progress Report IV, Contract No AF. 33(657)-8742, Air Products and Chemicals, Inc., Allentown, USA (July 1963) 3 Narinskii, G.B. Russ J Phys Chem (1969) 43 219 Levelt Sengers, J.M.H., Greer, W.L and Sengers, J.V. J Phys Chem RefData (1976) 5 1 Vasserman, A.A., Kazavchinskii, Ya. Z. and Rabinovieb, V.A. Thermophysical Properties of Air and Air Components (Moscow, 1966); English translation: Israel Program for Scientific Translations (Jerusalem, 1971) Vasserman, A.A. and Rabinovieb, V.A. Thermophysical Properties of Liquid Air and Its Components (Moscow, 1968); English translation: Israel Program for Scientific Translations (Jerusalem, 1970) Perry's Chemical Engineers' Handbook Sixth Edn, McGraw Hill, New York, USA (1984) 3 161

p = pc(0[1 + C 1 ( 0 ( - t)°355 + C2(~)t ]

(A2)

where plus refers to liquid and minus to vapour. Since ~ = 0 for pure fluid 2 (here nitrogen) and ~ = 1 for pure fluid I (here oxygen), in these limits Equation (AI) describes the vapour pressure curve and Equation (A2) describes the temperature~tensity saturation boundary for the pure fluid components. By definition, CI2) = Ci(O) and Cp I = C}1), i = 1 ... 6, are the coefficients of the fits of the pure fluid saturation properties to Equations (A l) and (A2). The coefficient C~ ) is proportional to the amplitude of the critical constant-volume specific heat divergence, which has been determined for nitrogen and oxygen by the correlations of Levelt Sengers et al. 25. With C~ ) thus constrained, the remaining coefficients are fit to pure fluid coexistence data by standard linear regression. Except for i = l, the ~ dependence is then given by linear interpolation, i.e.

Ci(~ ) =

C12) -1-

~I-CI 1)- CI 2)]

(A3)

On the critical line, it is assumed that x = l - ( . The critical line is represented by 1

1-x

x

R Yc(x ) - R Tel ~ R~2c2 -~- x(1 - x)

x IT, + (1 - 2x)T2 + (1 - 2x)2T3]

(A4)

1/T~I- I/T~(x) xx-

1/T¢I - I/T~2

P~(x)

/~T~(x) =

(A5)

(1 -- Xw)Pct q- xvPc2 - RTd R-T~2 + xT(1 - XT) X[Pl + (l -- 2xr)P2 + (1 -- 2xv)ZP3]

(A6)

p~(x) = (1 - xv)p~l + Xvpc2 + xv(l - xv) X [31 + (1 -- 2XT)32 + (1 -- 2XT)233]

(AT)

The parameters T~, Pi and Pi are determined from a standard linear regression fit to the experimental critical line, except that in the present application ~ =32 =

33~0. A(P, T) point is mapped into a ((, t) point with Equations (2) and (AI). Coexisting (molar) densities are then giyen by Equation (A2) and coexisting compositions

Cryogenics 1988 Vol 28 January

27

Vapour-liquid equilibrium: J.C. Rainwater and E. T. Jacobsen by (A8) where Q(~,t)= PTc d ( P c ) + Pc R T P c d( -~ RT~ [C~(-t) 1"9

-4-

C'4t

Pc d(1/T~) - (1 +t) R d( x [-- 1.9C3(--t) °'9 + C 4 + 2C5t + 3C6 t2]

+C'st2WC'6t3]-t

(A9) (AI0)

C; = CI1)- C~2)

dln Tc 1~(~, t) = C n t -

-

d~

(All)

and, finally, the g-dependence of C1 is given by

c?)+ ~Ec? ~ - c ? q C~(0 = 1 + Cx((1 -01(2(5 0)l/pc(0

(Al2)

Knowledge of the critical line and the pure fluid coexistence curves determines all parameters of the model except Cn and Cx. The latter are optimized, graphically and by visual criteria, to provide the best fit to the VLE data. Additional parameters have been introduced for mixtures with larger relative volatility4'5, but for nitrogen + oxygen the two parameters Ca and C x are sufficient.

Appendix B. Ancillary equations for dew and bubble line properties of air According to the present model, the coexistence locus in pressure, temperature and density for air is determined by numerical inversion of Equation (A8) for x = 0.7814, together with Equations (2), (A1), (A2), (A9) and (All). However, such a calculation is somewhat difficult and the results apply only over the extended critical region. It is thus convenient to make available ancillary equations which describe algebraically the pressure and density as functions of temperature along the complete coexistence locus of air. The dew and bubble line properties for air define the boundaries of the single phase and vapour surfaces of state, although these lines do not represent liquid vapour equilibrium in the same sense as the saturation line represents coexistence for a pure substance. (Liquid air on the bubble line is in equilibrium with mixtures of different compositions in the vapour phase, and air vapour on the dew line is in equilibrium with mixtures of different compositions in the liquid phase.) The location of the critical point for air also presents a different problem than that for a pure substance. Technically, the dew and bubble lines intersect at the mixture critical point. The critical point, maxcondentherm point, and maxcondenbar point do not coincide for air as they do for a pure substance. As a consequence, the phaseboundaries near the maxcondentherm appear as shown in Figure 14. The shape of the dew line near the critical point is uncertain and has not been defined by experiment. The development of the ancillary equations given here requires the use of a reference state for reducing the variables used in representing density and pressure as

28 Cryogenics 1988 Vol 28 January

functions of temperature. For this work, the maxcondentherm point at T = 132.6639 4- 0.0001 K, P = 3.7877 40.0003 MPa and p = 10.45 4- 0.02 kmol m - 3 was used as the reference point. The uncertainties quoted are those from the numerical search for the maxcondentherm according to the model, and are not meant to represent uncertainties in the true maxcondentherm point of air. Since the critical point density (Pc = 11.85 kmol m-3) and the critical point pressure (3.789 MPa) are both higher than the corresponding properties of the maxcondentherm point, the critical point temperature (132.56 K) is somewhat lower than the maxcondentherm temperature. Using the models given here, the prediction of vapour properties between the critical point and the maxcondentherm is best accomplished using the bubble line density and pressure equations, although this model interpretation is physically incorrect in that the interval between the critical and maxcondentherm is part of the dew curve. The ancillary equations for the dew and bubble line pressures and densities of air are given as Equations (B1) and (B2). Coefficients for these equations are given in Tables B1 and B2. The coefficients were determined by a least squares fit to selected data using a statistical selection algorithm to develop an optimum representation for the dependent variable. The functional form for bubble and dew line pressures as functions of temperature is

[ln(P/ej)] .(T/Tj)

25

= y, U,[1 - (T/Tj)] '/2

(B1)

i=l

where Pj and Tj are the pressure and temperature at the maxcondentherm point, respectively. The selected coefficients are given in Table B1. Coefficients not listed in the table are zero. The functional form for bubble and dew line densities as functions of temperature is 25

ln(p/pi) = ~ N i [ 1 - (T/Tj)] i/3

(B2)

i=l

where pj and Tj are the density and temperature, respectively, of the maxcondentherm point. The selected coefficients are given in Table B2. Coefficients not listed in the table are zero. We note that the reference temperature Tj in Equations (B1) and (B2) must be the maxcondentherm temperature rather than the temperature of the critical point or maxcondenbar point. In the latter two cases, temperatures for parts of the coexistence locus would be higher than the reference temperature, and Equations (B1) and (B2) would then call for the raising of a negative number to a fractional exponent. Table B1 Coefficients for equations for the dew and bubble line pressures of air Bubble line pressure

Dew line pressure

/ 1 2 3 19 24 25

N,. -0.1587873195 -0.5563463876 0.1590056463 -0.1128184577 0.3179621100 -0.3669957796

;

x 101 x 104 x 105 x 105

1 2 3 4 25

N, 0.2151013235 -0.6802016033 0.3030847776 -0.1824432102 0.1538723786

x x x x

101 101 101 102

V a p o u r - f i q u i d e q u i l i b r i u m . J.C. R a i n w a t e r a n d E, T. J a c o b s e n

Table B2 Coefficientsfor equationsfor the dew and bubble line densities of air Dew line density i 1 2 4 5 6 11 13 15 16

Ni -0.5249324675 -0.7155770519 0.9860549019 -0.3494367067 0.3606000885 --0.1373761128 0.3781087625 -- 0.5743242583 0.3203744098

Bubble line density

i

x 10 ~ × 102 × 10 3

x x × ×

1 03 104 104 1 04

× 10 4

Table B3 Pressure (MPa)

-0.3960870450 0.1116283855 -0.6954047683 0.2263076141 --0.1381840788 0.4281115030 --0.7834188581 0.8372907273 --0.4362490950 0.8573788641 --0.1540354710

Density (mol dm 3)

Temperature (K)

Source

Bubble line properties

Ni 1 2 3 5 6 7 8 9 10 12 14

Bubble and dew line properties for aim

x x x x x x x x x × ×

101 103 103 105 1 06 106 106 106 106 105 105

Coefficients for these equations were determined by weighted least squares fits to selected values for the dew and bubble line properties of air from Blanke 12 and from this work. The input for the ancillary equations is listed in Table B3. The results of this analysis are displayed graphically in Figures 6-13. A plot of the input to the ancillary equation, which shows the continuity between the modified Leung-Griffiths model and Blanke's data, is followed by a deviation plot of the input compared with the ancillary equation. Shown consecutively in the figures are the bubble curve in P - T space, the dew curve in P - T space, the bubble curve in p - T space, and the dew curve in p - T space. The estimated accuracies of the dew and bubble line pressures represented by Equation (B1) with coefficients from Table B1 are +0.4 and +0.2%, respectively. The estimated accuracies of calculated dew and bubble line densities are +_2 and _+3%, respectively. As shown in Figure 13, there is a discrepancy of up to 3 % between values of the dew line density of Blanke ~z and of this work at temperatures between 115 and 128K, which cannot be resolved. The model of this work is less accurate at temperatures removed from the critical state, and there are no known data from other sources in this range.

A p p e n d i x C. C o m m e n t s o n t h e d a t a o f M i c h e l s e t al. a n d s o m e r e l a t e d c o r r e l a t i o n s

For a comparison of our modified Leung Griffiths model and ancillary equations with the experiment of Michels et al. ~3, some care is needed in the selection of the appropriate data to examine. Both 'primary' and interpolated 'secondary' data are reported, and one of the primary data points has an atypically large uncertainty in density. Michels et al. present dew and bubble points for air in their Table VI. The first six entries were determined from the discontinuity in derivative of near-isochores on plots of P versus T and P V versus T, where V is the volume. The following three entries were determined from similar discontinuities of the slopes of isotherms on a plot of P versus P V (see their Figure 2). These nine points thus can be considered 'primary' data in the usual sense of the term for experiments of this

0.0071 0.06985 0.1403 0.2944 0,5821 0.9930 1.4664 1.641 8 1.7022 1.9642 2.2545 2.4335 2.5755 2.9306 3.1 225 3.2390 3.3254 3.541 5 3.7042 3.7547 3.7745 3.781 4

32.975 30.737 29,757 28.453 26.902 25.256 23.5348 23.136 22,8203 22.0410 21.1 737 20,708 20.1784 18.9733 18.2361 17.71 8 17.3349 16.0948 14.5883 1 3.6923 13.0443 1 2.6536

60.430 75.475 81.680 89.459 98.068 106032 11 2.7014 114.734 11 5.4207 11 8.1 533 120.9019 1 22.452 1 23.6709 1 26.4688 127.8844 128.667 129.31 84 1 30.7879 1 31.8801 132.2377 132.3961 1 32.4621

Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12

Blanke ~2

Blanke 12

Dew line properties 0.01183 0.02433 0.0251 5 0.04729 0.09898 0.22117 0.46198 0.94620 1.5344 1.6648 1.7741 2.0393 2.3315 2.6525 3.0043 3.0190 3.1 925 3.3890 3.5934 3.7250 3.7388 3.7917 3.7782 3.7898

0.021 562 0.041754 0.04291,2 0.077183 0.1 523 0.32099 0.64388 1.3040 2.2659 2.3875 2.6694 3.141 4 3.7041 4.3968 5.3008 5.2850 5.8880 6.6394 7.7301 8.9254 9.1 317 11.0220 9.9975 1 0.6345

67.204 71.374 71.689 75.739 81,526 88,977 97.340 107.262 11 5,2233 116.602 117,7474 120.3069 1 22,8834 125.4614 128.0230 1 28.075 1 29.2884 130.5329 1 31.7327 132.340 132.4892 1 32.641 0 132.6466 1 32.6611

Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12 Blanke 12

Blanke 12

Blanke 12

aproperties with no source listed are from this work

nature. The remaining 10 entries of their Table VI are based on graphically interpolated isotherms, and thus are not primary data and are omitted from a comparison with our correlations. The primary data points in general agree well with our model and ancillary equations, with one notable exception. The bubble point P = 1.963 MPa (19.376 atm), T = 118.15 K ( - 155°C) and p = 20.46 kmol m --3 (458.28 amagat) is low in density, compared with our model and ancillary equation, by more than 7 %. Upon examination of Table I and Figure 2 of Reference 13, it is seen that the slope of the lowest one-phase liquid isotherm is uncertain and, furthermore, a very small change in the slope could easily cause a 7 % shift in the

Cryogenics

1988 Vol 28 January

29

Vapour-fiquid equilibrium: J.C. Rainwater and E.T. Jacobsen 4

='

I

I

ILl

I

# o

1.0

oo

3

i

{~ LU

o o

1,11 <~

rr

e

0.. Z m

o° o

v

,9

o

,°

,

80

o

~

<

o o o~

l

~

o

~

n oo ~"~o~

U.I 0 I--- - 0 . 5

o

o

0

0

I-<

o

o o-

DO

o

z 0

o

{L

0.5

Z W

,

100

120

140

-Is17

0

T (K)

LU

-1.0~0

I 80

0.

Figure 6 Input points for Equation (B1), bubble curve; O, model points; ~ , data of Blanke12

I

I

100

120

140

T (K) Figure 9 Deviation plot for the pressure of the dew points compared to Equation (B1). Symbols as for Figure 7. Note that two Blanke points are off scale

LU n" 0.2

O) (0 UJ nn

I

I

I o

o

!

32

I

I

0.1 o

Z

28

o

z 0

o o

~"

-'0-~ -

0

0

0

0

~o

,0~

o% %

°oO

LU (:3

0

o

24

o%

E

o

o °o 0

2o

-0.1

I"Z LU

o

R" E

o

n" UJ

%

16

-0.2

I°

60

I

80

a

o

I

100

120

o o

140

8

12 60

T (K) Figure 7 D e v i a t i o n plot for t h e pressure of t h e b u b b l e p o i n t s c o m p a r e d to E q u a t i o n (B1) : O , m o d e l p o i n t s ; ~ , data of Blanke12; D , data of M i c h e l s e t a / , 13

{ 80

I 100

o

I 120

40

T (K) Figure 10

Input pointsfor Equation (B2), bubble curve. Symbols

as for Figure 6

>I

I

I

I'-

# o o

z uJ o

o

o

z z 0 I<~

o

D. o

1.0

I

I

I

0.5 O o

0

0

oo

LLI O

o

I'Z UJ 0

o o 0

6O

UJ - 1 . 0 100

120

140

T (K) Figure 8 Input points for Equation (B1), dew curve. Symbols as for Figure 6

30

o -0.5

Cryogenics 1988 Vol 28 January

n

0 o

o o

CC

C 80

oo

o o

-6.1 ~

60

] 80

I 100

1 120

140

T (K)

Figure 11 Deviation plot for the density of the bubble points compared to Equation (B2). Symbols as for Figure 7. Note that one Michels et al. point is off scale

Vapour-liquid equilibrium. J.C. Rainwater and E.T. Jacobsen 12

I

I

I

133

u

I

I

9

11

I

I

13

15

o o o

lO

8

To

o

E o

o E

6

Q.-

4

o @ o

~,,

o

v

o

oO°

2

132

I--

@ <>

0

6O

I~ 80

<>

I 100

I 120

140

T (K) Figure 12

Input points for Equation (B2), dew curve. Symbols as

for Figure 6

>I--

4.0

i

I

0 0

131

I 0

Z LU Q Z

7

D

p

(kmol/m

3)

O 2.0

Figure 14 Saturationcurve for air according to model: 17, maxcondentherm point; ~ maxcondenbar point; ©, critical point

[] <>

Z

o@

o o

o

°

for the saturation properties of air. The data of Michels

O

et al. are the only experimental input to their correlation

U.I

in the critical region. The table for saturated air in the Sixth Edition of 'Perry's Chemical Engineers' Handbook '2~ is, in turn, based on the correlation of Vasserman

Q(~LIJI-n¢ - 2 . 0 Z

UJ n

Pc

-4.C 60

I 80

I 100

I 120

et al. 140

T (K) 13 Deviation plot for the density of the dew points compared to Equation (B2). Symbols as for Figure 7 Figure

volume, and hence the density, of the bubble point. This problem is not present for the other data. Therefore, we have elected to omit this particular bubble point from our figures. The remaining six dew points and two bubble points are shown on our Figures 2 and 4 and the deviation plots. The bubble point P = 3.707 MPa, T = I 3 1 . 9 2 K and p = 13.68 kmoi m - 3 is low in density compared with the model and ancillary equation by ~ 6 %. However, unlike the omitted point, (~T/Op)x is very small and a small change in temperature causes a large change in density (see Figure 4), so this discrepancy is of much less concern. Otherwise, the data of Michels et al. agree with the model to within 0.15 K in temperature, 0.025 MPa in pressure and 0.17 kmol m - 3 in density. The probable error in the density of the omitted bubble point significantly affects the interpolated liquid data in Table VII and Figure 4 of Reference 13 for T < 125 K. In particular, the average of the liquid and vapour densities in that figure appears to decrease with decreasing temperature, whereas the expected result from our model as applied to this and other mixtures is that the average density should increase with decreasing temperature. An equivalent condition is that C S ) of Equation (A2) is negative. The appropriate increase in density would rectify this problem, which was earlier noted by Vasserman et al. (see p 218 of Reference 26). Vasserman et al. 26"27 have developed a correlation

Our model agrees with the coexistence points tabulated in Perry's Handbook to within 0.8% in pressure and density except at 120 K, about the limit of applicability of our model, where the model vapour density is higher by 2.7%. At this temperature, there is a comparable discrepancy between the model and the ancillary equation, cf. Figure 13. The critical pressure (3.769 MPa) and temperature (132.55 K) quoted in Perry's Handbook are also in good agreement with our model values (3.789 M Pa, 132.56 K). However, Perry's Handbook lists a critical density of 10.81 kmol m - 3, which is 8.8 % lower than our value of l l . 8 5 k m o l m -3. Vasserman et al. explain their choice of critical density on pp 220-221 of Reference 26. From this discussion, it is evident that the authors chose for Pc the density at the 'critical point of contact', i.e. maxcondentherm point, as determined by Michels et al. 13. This choice was made to optimize a theoretical saturation curve based on a reference equation for pure propylene. We therefore believe that our Pc is much closer to the actual critical density of air than the much lower value cited in Perry's Handbook. Finally, we mention a recent article by Pominov 29 on the bubble curves of the three binary mixtures formed by nitrogen, oxygen and argon. Isopleths are fit to the equation In P = A + B / T

(C1)

where A and B are composition-dependent coefficients. For many applications, Equation (C1) is a reasonable approximation for both the dew and bubble curves away from critical 3°. However, the equation does not describe even qualitatively the smooth transition from the dew to the bubble curve in the critical region and, therefore, is clearly inadequate near the critical locus.

Cryogenics 1988 Vol 28 J a n u a r y

31