(Vapour + liquid) phase equilibria of binary, ternary, and quaternary mixtures of CH4, C2H6, C3H8, C4H10, and CO2

M-2166 J. Chem.

1!%8,20.

Thermodynamics

413-428

(Vapour + liquid) phase equilibria of binary, ternary, and quaternary mixtures of CH,, C,H,, C,H,, C4H,,,, and CO, ALISDAIR

Q. CLARK

and KEITH

STEAD

Chemistry Department, University of Exeter. Exeter EX4 4QD, U.K. (Received 11 May 1987; injinal

form 3 August 1987)

(Vapour + liquid) phase equilibria have been studied for (ethane + propane), (ethane + n-butane), and (propane + n-butane) at 260.0, 270.0, and 280.0 K; (ethane + carbon dioxide), and (n-butane + carbon dioxide) at 260.0 K; (ethane + propane + n-butane) at 260.0, 270.0, and 280.0 K and at 0.50, 1.0, 1.5 MPa; (ethane + n-butane + carbon dioxide) at 260.0 K and at 0.50, 1.O, 1.5,2.0, and 2.5 MPa; (methane + ethane + n-butane) at 260.0 K and 1.0 MPa; (methane + n-butane + carbon dioxide) at 260.0 K and 1.0 MPa; and (methane + ethane + n-butane + carbon dioxide) at 260.0 K and 1.0 MPa. Calibration techniques using a g.c. have produced a decisive reduction in the calibrations needed for multicomponent mixtures. Nine points are adequate for a binary mixture, and it is shown that only 18 are needed for a ternary mixture and only 27 for a quaternary mixture. Calculations of the phase diagrams are given from a model using the Gibbons and Laughton equation of state, and the implications of this equation for the representation of pure fluids, and of binary and ternary mixtures are discussed.

1. Introduction The study of (liquid + vapour) equilibria of multicomponent mixtures is seldom attempted, the time and effort in calibration or analysis being often overwhelming. An attempt has been made to speed up this calibration and to use only binarymixture calibrations in the study of (liquid + vapour) equilibria of ternary and quaternary mixtures. It is shown that nine calibration points are adequate for the analysis of binary mixtures. (l) For the characterization of a ternary mixture, a method which requires only 18 standard binary mixtures, i.e. the nine calibrations for each of two of the binary mixtures, has been developed. Extending this to a quaternary mixture requires only the 27 calibration points necessary to analyse three of the constituent binary mixtures and no additional calibration. Measurements have been made on (ethane + propane), (ethane + n-butane), and (propane + n-butane) at 260.0, 270.0, and 280.0 K and on (ethane + carbon dioxide) and (n-butane + carbon dioxide) at 260.0 K; on (ethane + propane + n-butane) at 260.0, 270.0, and 280.0 K at pressures of 0.50, 1.0, and 1.5 MPa; on (ethane + n-butane + carbon dioxide) at 260.0 K and at 0.50, 1.0, 1.5, 2.0, and 0021-9614/88/040413+

16 %02.00/O

~3 1988 Academic Press Limited

414

A. Q. CLARK

AND

K. STEAD

2.5 MPa; on (methane + ethane + n-butane) and (methane + n-butane + carbon dioxide) at 260.0 K and 1.0 MPa; and on (methane + ethane + n-butane + carbon dioxide) at 260.0 K and 1.0 MPa. Use of, and comparison with, literature values is made where possible; excellent agreement is obtained. Some theoretical studies have also been made using the recent Gibbons and Laughton equation of state. Studies on the non-uniqueness of the values of the parameters X and Y in their equation, the fit of the equation to some binary mixtures containing CO,, and the extension to (vapour + liquid) equilibria of ternary mixtures are presented. Whilst there are some remarkable correlations possible, it must be concluded that this equation of state has some disquieting features.

2. Apparatus The apparatus is as described previously”) with some slight modifications. It is of the dynamic type in which vapour is recirculated through the liquid to accelerate equilibrium. All problems with leaks in the compression-type couplings seem to have been eliminated by the judicious use of small amounts of indium wire between the ferrule and its seat. The multi-turn cooling coil in the cryostat was replaced by an annular jacket with a greater capacity and this enabled low temperatures to be reached in a shorter time with the liquid-nitrogen coolant. A pressure transducer (R.S. Components) was used to measure the pressure of the sample entering the g.c. When the bridge circuit of the transducer was held at a constant potential difference, and a value of 8.501 V was chosen, the transducer gave remarkable reliability and reproducibility. The g.c. output was fed into a B.B.C. Model B computer so that integration of peak areas and recording of peak heights was automatic. The resolution of the analogue port of the computer was improved with an EPROM 2764 silicon chip containing a machine-code program. The program averaged 16 points every 0.16 s and stored the averaged point. Peaks, their heights, and their areas, were output to a Sanyo Visual-Display monitor and calculated results were printed on an Ap40 printer (Able Systems Limited). The changes were made to obtain a faster operation of the equipment and also to improve both its reliability and reproducibility.

3. Experimental Standard mixtures were always made by the barometric method previously described.“’ The method was easily adapted to ternary and quaternary mixtures but the number of calibration mixtures required for a completely satisfactory calibration was large and daunting. It was realised that a method could be devised for ternary and quaternary mixtures which would use only binary-mixture calibrations so long as the whole process was reproducible and a little faster than normal. The reproducibility was improved as explained in the previous section and the speed of operation was increased by incorporating a calibration-mixture manifold.

(VAPOUR + LIQUID)

415

PHASE EQUILIBRIA

This allowed nine calibration mixtures to be made in a day and sampled on the next day, producing very satisfactory calibration curves for binary mixtures. Many checks were carried out on both the mixing and sampling of this most crucial operation. The method is illustrated for a mixture (A + B + C) where calibration curves for (A+B) and (A+C) were used to calculate mole fractions of A, B, and C in the ternary mixture. When the ternary mixture was passed through the g.c. three peaks were obtained and using pairs of peaks for (A, B) and (A, C) pseudo-binary mole fractions x,(AB) and x*(AC) of A in the corresponding binary mixture could be calculated. Relating these mole fractions to the true mole fraction x,(ABC) of A in the ternary mixture produces the equation: l/x,(ABC) The mole fraction x,(ABC)

= l/x,(AB)

+ l/x,(AC)

of B in (A +B+C)

- 1.

is then obtained as

x,(ABC) = x,(ABC)( l/x,(AB)

- 1),

and xc(ABC) is obtained by making the sum of the mole fractions equal to unity. The former equation may be derived as follows. If the definition: x,(ABC) = n,/(n,+ n,+n,), is inverted then it is easily separated as l/x,(ABC) = {(nA + n,)/n,} + {(a, + nc)/~,> - (n&J, which is the former equation above with x,(AB) = rtA/(nA+ng) and x*(AC) = n&n,+ nc). The same method was used for mixtures of (A+B+C+D), using values for (A+B), (A+C), and (A+D). Substance A was chosen as that giving the tallest peak on the g.c. since this was found to give the best results for both ternary and quaternary mixtures. Some of the results of testing this method are shown in table 1. The sample of methane was supplied by Argo Ltd and quoted as 99.995 mass per cent purity. Analysis by g.c. revealed no impurities. Ethane was supplied by the British Oxygen Company and was their C.P. grade of 99.0 mass per cent purity. Two tiny impurity peaks were revealed after the main ethane peak, and thought to be a C, and a C, hydrocarbon. Both propane and n-butane were supplied by Cambrian Gases (Matheson) and quoted as of 99.0 mass per cent purity. Whilst the TABLE 1. Some mixtures used to test the binary-to-ternary and binary-to-quaternary analytical technique, where x is the true mole fraction, x,, is that derived from the height-fraction calibration, and x, is that derived from the area-fraction calibration x 0.324 0.813 0.505 0.602 0.598 0.294 0.298

XII

Cd%

0.297 0.824 0.506 CH, 0.610 0.589 0.306 0.295

X,

X

0.313 0.812 0.508

0.331 0.094

0.628 0.623 0.315 0.319

XII

C,Hs 0.351 0.089

X,

X

0.345 0.093

0.345 0.093 0.201

CA 0.199 0.305 0.099

0.200 0.323 0.097

0.182 0.325 0.087

0.099 0.203 0.098 0.206

-Yh

GH,,

0.372 0.087 0.227

(31,

0.096 0.211 0.088 0.209

-?a

0.342 0.095 0.218 0.078 0.195 0.080 0.179

X

XII

X,

co2 0.294 0.299

0.267 co* 0.294

0.274 0.294

0.303 0.397

0.283 0.399

0.280 0.415

416

A. Q. CLARK

AND

K. STEAD

propane showed no impurity peaks the n-butane showed a very small peak preceding the n-butane and merging into it; this was thought to be a butene. The carbon dioxide came from The Distillers Company Ltd, of stated purity 99.8 mass per cent; no impurities were detected. A freeze-thaw procedure of degassing was used on all samples. Modifications to the procedure previously adopted,“) necessary to speed up the calibration, caused problems for mixtures containing methane. In the transfer of material, it was not possible to cool the methane to a temperature at which the vapour pressure could be declared negligible. At 77 K, the vapour pressure of methane is about 1 kPa. Corrections were made for this when making standard mixtures in the manifold; they were in the range 0.001 to 0.003 in the mole fractions. The uncertainties in the measurements based on calibration checks and many other considerations are given as + 1, +6, and f20 kPa for pressures in the intervals 0 to 1100, 1100 to 4100, and 4100 to 15000 kPa. respectively, and kO.008 and fO.O1O K for temperatures below and above 273 K, respectively. For mole fractions, a very detailed analysis was carried out on calibration curves for binary mixtures determined on both peak-height and peak-area fractions. At best the uncertainty was approximately 0.005 and at worst 0.02, dependent upon peak shape and upon curvature of the calibration curve. From the method of analysing the ternary and quaternary mixtures the average uncertainty was found to be at worst fO.O1 and kO.02, respectively. This may be seen from the results given in table 1 above.

4. Results As a ternary mixture (ethane + propane + n-butane) was chosen because the three binary mixtures show only small deviations from ideality and such a simple mixture would test the analytical procedures. (Liquid + vapour) equilibria were studied at temperatures of 260.0, 270.0, and 280.0 K and at pressures of 0.50, 1.0, and 1.5 MPa. The binaries were studied at the three temperatures. Results for the binaries are given in table 2 and for the ternary in table 3. The binary peak-areafraction calibration curves were used to analyse the ternaries. A survey of the literature produced no previous results for the ternary and only sparce references for the binaries. For (ethane + propane), five references@-@ were found at about the temperature range studied and whilst no results exactly matched the chosen temperatures those of Hirata er ~1.‘~’ at 273.15 K showed excellent agreement. For (ethane + n-butane), there were only two”,*’ previous sets of results for temperatures around 273 K, and the second was published just after these studies were completed. Again comparison of these results with those of Kaminshi et u/.@) at 273.15 K showed excellent agreement. For (propane + n-butane), three relevant reference@ 5**) were found and comparison of the results of Skripka et ~1.‘~) at 273.15 K again showed excellent agreement. For the ternary there are no previous results for temperatures below 300 K. Figure l(a) shows the (vapour + liquid) equilibrium at the three pressures and all at 280 K. The individual tie-lines are not shown but some experimental points are; the smoothness of the curves is very evident.

(VAPOUR + LIQUID)

PHASE EQUILIBRIA

417

TABLE 2. (Liquid + vapour) equilibria for binary mixtures of ethane, propane, and n-butane at various temperatures T and pressures p

T/K

PIMPa

260.0

0.3145 0.3620 0.5017 0.4344 0.4927 0.6533 0.5855 0.6426 0.8327 1.026

x

Y

p/MPa

x

{xC,H,+(l -x)C,H,}(U = {~Cd-b+(l

270.0 280.0

260.0

0 0.170 0.452 0 0.160 0.417 0 0.145 0.382 0.539

0.1969 0.5000 0.6780 0.1339 0.2367 0.5883 0.7938

{xC,H,+(I 0 0.078 0.265 0.392 0 0.067 0.260 0.367 0 0.063 0.250 0.330

-x)C,H,,}(l) = (C,H,+(l-y)C,H,,}(g) 0 0.7975 0.558 0.951 0.648 1.000 0.657 0.968 0.880 1.207 0.766 0.980 0.924 1.357 0.857 0.975 0 1.000 0.549 0.944 0.571 1.202 0.638 0.956 0.852 1.509 0.750 0.972 1.703 0.851 0.981 0.900 1.110 0.505 0.915 0 0.470 1.409 0.607 0.942 0.805 1.832 0.738 0.963 0.976 0.871 2.072 0.823

0.0611 0.0759 0.1086 0.1464 0.0932 0.1120 0.1540 0.2046 0.1339 0.1598 0.2099 0.2779

IxC,H,+(l-x)C,H,,}(I) 0 0 0.070 0.229 0.201 0.542 0.363 0.734 0 0 0.066 0.224 0.195 0.520 0.362 0.702 0 0 0.065 0.200 0.187 0.470 0.349 0.669

270.0

280.0

260.0

270.0

280.0

0.1671 0.2134 0.2368 0.2815 0.2320 0.2962 0.3381 0.3899 0.3230 0.3994 0.4557 0.5253

0.283 0.501 0.672 0.270 0.483 0.668 0.472 0.665 0.825 0.913

= {~C,b+(l 0.466 0.649 0.740 0.904 0.452 0.624 0.774 0.902 0.465 0.631 0.770 0.899

x

Y

0.855 0.929 1 0.852 0.920 1 0.954 1

0.968 0.980 1 0.951 0.976 1 0.980 1

1.576 1.699

0.952 1

0.994 1

2.010 2.195

0.948 1

0.991 1

2.501 2.791

0.938 1

0.991 1

0.937 1

0.983 1

0.934 1

0.980 1

0:941 1

0.981 1

PIMPa

-~)C,Hsl(g) 0.612 1.410 0.784 1.537 0.899 1.699 0.579 1.798 0.780 1.970 0.887 2.195 0.751 2.615 0.863 2.791 0.940 0.964

0 0.060 0.168 0 0.057 0.162 0 0.057 0.151 0.266

0.0611 0.1613 0.4029 0.5694

0.6336 0.9366 1.156 0.8182 1.163 1.478 1.444 1.871 2.239 2.378

Y

-yGH,oNs) 0.804 0.2929 0.894 0.3145 0.921 0.972 0.770 0.4039 0.883 0.4345 0.926 0.967 0.770 0.5520 0.869 0.5855 0.919 0.965

The mixture (ethane + n-butane + carbon dioxide) contains one binary (carbon dioxide + ethane) which shows positive azeotropy. The study was carried out at 280 K and at pressures between 0.5 and 2.5 MPa for the ternary. Results for (ethane + n-butane) are given in table 2. Results for (ethane + carbon dioxide) and (n-butane + carbon dioxide) are given in table 4 and for the ternary in table 5. Again peak-area fractions of the binary calibration curves were used for the ternary analysis.

418

A. Q. CLARK

AND

K. STEAD

TABLE 3. (Liquid + vapour) equilibria for ~~,C,H,+X~C~H~+(~-XI-XZ)C~H~O}(~)={Y,C~H,+Y,C,HS+(~-~‘,-Y,)C,H,,}(~) at various temperatures T and pressures p

T/K

PIMPa

xl

260.0

0.500

0.168 0.196 0.203

260.0

l.ooo

260.0

Yl

)I2

0.832 0.665 0.631

0.464 0.534 0.564

0.536 0.448 0.415

0.208 0.269 0.353

0.608 0.289 0

0.574 0.777 0.914

0.399 0.159 0

0.554 0.564

0.446 0.384

0.855 0.872

0.145 0.125

0.579 0.656

0.265 0

0.901 0.970

0.086 0

1.500

0.900 0.906

0.100 0.057

0.971 0.980

0.029 0.017

0.919

0

0.982

0

270.0

0.500

0.063 0.092 0.111

0.937 0.743 0.683

0.173 0.293 0.358

0.827 0.668 0.592

0.193 0.207 0.260

0.286 0.253 0

0.657 0.692 0.852

0.227 0.186 0

270.0

1.000

0.388 0.382 0.384

0.612 0.595 0.393

0.704 0.705 0.814

0.296 0.292 0.160

0.452 0.549

0.234 0

0.876 0.944

0.083 0

270.0

1.500

0.673 0.676 0.710

0.327 0.301 0.210

0.888 0.891 0.931

0.112 0.107 0.058

0.678 0.731

0.118 0

0.944 0.970

0.040 0

280.0

0.500

0 0.024

0.853 0.621

0 0.075

0.948 0.830

0.115 0.178

0.237 0

0.505 0.760

0.293 0

280.0

1.000

0.241 0.317

0.759 0.436

0.510 0.686

0.490 0.268

0.398 0.451

0.146 0

0.854 0.911

0.064 0

280.0

1.500

0.492 0.497 0.533

0.508 0.463 0.304

0.763 0.799 0.855

0.237 0.195 0.123

0.574 0.635

0.085 0

0.922 0.949

0.032 0

+ carbon

dioxide)

TABLE

4. (Liquid

pIMPa

x

+ vapour)

Y

PIMPa

equilibria for (carbon dioxide + n-butane) at 260.0 K and various pressures p x

Y

PiMPa

(.~CO,+(l-x)C,H,,}(l) 0.061 0.233 0.500

0

0

0.760

0.025 0.077

0.768 0.886

1.000

2.418 2.494 2.648

0

0

0.030 0.110

0.054 0.165

2.781 2.795 2.784

0.187 0.250

0.923 0.938

0.287 0.418 0.454

Y

and (ethane

PIMPa

x

Y

PIMPa

x

Y

0.586 0.737

0.971 0.984

2.221 2.418

0.903 1

0.992 1

0.721 0.799 0.880

1.699

1

1

= WO,+(l-yGH,,)(g)

1.175 1.500

0.368 0.465

2.757 2.647 2.493

0.545 0.636 0.742

{xC,H,+(l-xW,J(1)

0.254 0.460 0.487

x

Y2

Yl

0.961 0.967

= {yC,H,+U

0.485 0.565 0.649

1.755 2.000

-y)CO,)W

2.308 2.103 1.940

0.821 0.893 0.943

There are three relevant previous studies (9-11) for (n-butane + carbon dioxide), and a comparison of these results with those of Weber(“) is shown in figure 2. For (ethane + carbon dioxide), there is an extensive literature but little agreement.099V ’ 2-23) Th e results of this work at 260.0 K are shown in figure 3 and compared with the work of Fredenslund and Mollerupo3’ at 263.15 K. No previous results exist for the ternary mixtures. The results are shown in figure l(b). Some

(VAPOUR + LIQUID)

(a) ---=-------=r ---7---I

PHASE EQUILIBRIA

419

1 I I

;-===&-===7-j

FIGURE 1. The phase diagrams of (a), (ethane + propane + n-butane) at 280.0 K and (b). (ethane + n-butane + carbon dioxide) at 260.0 K.

experimental points, but no individual tie-lines, are shown in the figure; the smoothness of the curves is obvious. For the study of the quaternary mixture (methane + ethane + n-butane + carbon dioxide) a constant temperature of 260.0 K and a constant pressure of 1.0 MPa were chosen. Of the four ternaries, one has just been given, two were then studied at this temperature and pressure, and the results are given in table 6, while the fourth ternary (methane + ethane + carbon dioxide) was wholly gaseous under these conditions. Results for the quaternary study are given in table 7. In these analyses it was found that peak-height fraction gave the more reliable results.

420

A. Q. CLARK TABLE

AND

K. STEAD

+ vapour) equilibria for = {y,C,H,+yzC,H,,+(l-y,-y,)CO,}(g) at 260.0 K and various pressures p

{xtC,H,+x,C,H,,+(l-x,-x,)CO,}(l)

5. (Liquid

Xl

x2

Yl

Y2

X1

x2

Yl

Y2

0.500

0 0.056

0.923 0.881

0 0.111

0.114 0.082

0.160 0.201

0.794 0.778

0.326 0.614

0.084 0.090

0.304

0.696

0.914

0.086

1.000

0 0.067

0.750 0.678

0 0.087

0.062 0.035

0.215 0.374

0.669 0.565

0.260 0.523

0.034 0.037

0.460 0.656

0.512 0.344

0.735 0.970

0.037 0.030

1.500

0 0.294

0.535 0.431

0 0.256

0.033 0.015

0.628 0.703

0.278 0.237

0.705 0.764

0.014 0.013

0.919

0.081

0.982

0.018

2.000

0 0.174

0.263 0.252

0 0.163

0.016 0.012

0.374 0.530

0.221 0.206

0.286 0.438

0.011 0.009

0.604 0.936

0.156 0

0.523 0.831

0.010 0

2.500

0.030 0.109 0.239

0 0.041 0.064

0.054 0.130 0.214

0 0.002 0.005

0.319 0.443 0.567

0.064 0.062 0.042

0.256 0.365 0.439

0.004 0.005 0.005

0.742

0

0.649

0

i

2

1

B -5 1

0

I

I

I

0.2

I

I

0.4

I

I

0.6

I

,

0.8

X FIGURE 2. Experimental 0, This work; 0, reference

(liquid 11.

+

vapour)

equilibrium

for

{(1 -x)C,H,,+xCO,}

at 260.0 K.

(VAPOUR

+ LIQUID)

I

PHASE

,

I

0.2

421

EQUILIBRIA

I

I

0.4

I

0.6

I

I

0.8

X

FIGURE 3. Experimental (liquid + vapour) 260.0 K; 0, reference 13 at 263.15 K. TABLE Xl

0 0.007

6. (Liquid

X2

Yt

fvapour) Y2

{.~,CH,+x,C,H,,+(l-x,-x,)CO,}(1) 0.750 0 0.062 0.801 0.100 0.058

equilibrium

equilibria

for two ternary

XI

x2

Yl

0.042 0.086

0.813 0.899

0.053 0.080

0.601 0.892

(x,CH,+x,C,H,,+(l-x,-x,)C,H,}(I)

0 0.024

0.344 0.472

0 0.314 TABLE

0.030 0.034 7. (Liquid

for ((1 -x)CO,

+ vapour)

mixtures

+xC,H,}.

0, This work

at

at 260.0 K and 1 .OO MPa x2

Yl

Y2

= {Y,CH,+Y,C,H,,+(~-Y~-Y,)CO,}(~) 0.392 0.067 0.104 0.746 0.072

0.896

0.919

0.081

= {y,CH,+y,C,H,,+(l-y,-y,)C,H,J(g) 0.624 0.062 0.104 0.862 0.080

0.896

0.919

0.081

equilibria

YZ

XI

for the quaternary

mixture

{x,CH,+X,C,H,+X~C,H,,+(~-x,-x,-~,)Co,}(l) = {Y,CH,+Y,C,H,+Y,C,H,,+(~ at 260.0 K and 1.00 MPa -x I

X2

X3

Yl

YZ

Y3

Xl

0.036 0.011 0.012 0.007

0.341 0.434 0.339 0.466

0.604 0.545 0.584 0.502

0.455 0.165 0.175 0.101

0.304 0.709 0.320 0.758

0.048 0.024 0.036 0.025

0.006 0.020 0.007

-Y,

X2

X3

0.490 0.368 0.278

0.490 0.599 0.564

-y,--y,)CO,Hg)

Yl

Y2

Y3

0.108 0.360 0.105

0.795 0.332 0.263

0.022 0.049 0.031

422

A. Q. CLARK

AND

K. STEAD

There were no literature results to compare with our results, which are plotted in figure 4, showing the tie-lines measured. The smoothness of the surfaces and evenness of the tie-lines are apparent.

5. Theoretical analysis The extension of the binary-mixture

Raoult’s-law

equation:

where p is the vapour pressure of the liquid mixture with mole fractions xA of A and xg of B and pX and pi are the vapour pressures of pure A and B to ternary mixtures as P = X*PX + X,P,* + X,P2t 1

is simply to predict that liquidus and vapourus lines in the constant T and p slices of a ternary diagram are straight lines. The equation of a straight line in the usual triangular diagram has the same form as the second equation above. Whilst it is obvious from figure l(a) that some simple binary mixtures studied here closely follow Raoult’s law, the ternary mixture shows linearity only in the vapourus line. The success of the recent equation of state of Gibbons and LaughtorP4) in predicting phase equilibria suggested that it would be useful for these ternary mixtures. However, in the work necessary before using their equation for ternary

FIGURE 1.00 MPa.

4. The phase diagram

of (methane

+ ethane

+ n-butane

+ carbon

dioxide)

at 260.0 K and

(VAPOUR

mixtures some interesting mixtures.

f LIQUID)

PHASE

EQUILIBRIA

423

results arose on both pure substances and binary

PURE SUBSTANCES

The equation of state is where

P = RT/(I/,-b)-a~/~~/,(~~++), a = 1 +x(T,-

l)+ Y(&i’2-

l),

and TR is T/T”, the reduced temperature. In reference 24 the authors state that X and Y are “chosen by minimizing the root-mean-square error in the calculated vapour pressures for the whole range of the liquid vapour-pressure curve”. That is not possible since there seems to be no simple single minimum in the error surface. Detailed calculations on several pure substances always produced several values of X and Y. An exact value of a for each p(T) point was first calculated using methods mentioned elsewhere.“‘* 26) A r earrangement of the equation of state produced a cubic equation in V, which will have three real roots at the experimental values of T and p, if it is satisfactory equation. This limits the range of values of a and guides the calculation to an initial value of a. With this initial a, and at a particular temperature, a pressure was found such that the chemical potentials of the liquid and gas phases were equal. The value of a was then changed to produce a pressure closer to the experimental vapour pressure at the chosen temperature. Using the Newton-Raphson iterative method both for the equalizing of the chemical potentials and for the equalizing of the calculated and experimental vapour pressure produced a satisfactory value of a. The values of a reproduced the experimental pressure exactly; the problem arose in trying to fit the exact values of a to the equation in X and Y. When four different fitting techniques were used four different values of X and Y were obtained each giving the same root-mean-square error as suggested by Gibbons and Laughton. However, when these values of X and Y were plotted they were found to fit a straight line. For water the equation: Y = -1.789X-2.169, over the range X = 0.13 to 0.17 gives an error of 0.54 per cent comparable with X = 0.165, Y = -2.465, and an error of 0.53 per cent given in reference 24. For carbon dioxide the equation: Y = - 1.863X- 1.737, over the range X = 0.1 to 0.3 gives an error of 0.30 per cent comparable with X = 0.196, Y = -2.102, and an error of 0.22 per cent given in reference 24. As a preliminary to finding accurate values of X and Y for CJH, and C,Hlo, a plot of X (or Y) against the number of carbon atoms in the hydrocarbon was found to give a straight line for C,H,, C5H12, and C8H1s, using X and Y from reference 24. Interpolating for C,H, gave X and Y as 0.361 and -2.183, and for C,Hlo the values were 0.461 and -2.548. After much searching on the error-

A. Q. CLARK AND K. STEAD

424

TABLE 8. The critical pressure p’, critical temperature T’, and values of X and Y used in the calculations

p’lA4Pa ’

Tc/K a

X

Y

4.620 4.880 4.2250 3.800 7.380

190.60 305.40 369.80 425.20 304.20

- 0.070 B 0.262 b 0.346’ 0.453 * 0.196b

-0.907 b -1.810” -2.158’ -2.516’ -2.1026

Methane Ethane Propane n-Butane Carbon dioxide ’ Taken b Taken ’ Based * Based

from reference 27. from reference 24. on vapour pressures from references 28 to 3 1. on vapour preSsures from references 31 to 33.

surface “best values” for these substances were obtained, very close to these interpolated values. Table 8 contains the critical properties and values of X and Y for the substances used in these calculations. BINARY

MIXTURES

Calculations on the phase equilibria of (ethane + carbon dioxide) were carried out with the Gibbons and Laughton equation of state but with the method and mixing rules given before. (25,26) The change of mixing rules entirely in the b term was shown to have no significant effect on the results. Since the equation of state had been shown to be so good for other binary mixtures it was hoped that its predictive power for this mixture would be useful. However, when the phase diagram was calculated for a temperature of 260.0 K the curves were disappointing. In the combining rules there is the fitting paramerer 5 for the deviation from the geometric mean of the coefficient cAB {it is the same as (1 - k,) of equation (13) of reference 24) and with 5 = 1 the predicted phase diagram was almost according to Raoult’s law; the experimental curve shows a positive azeotrope. Variation of 5 produces some sort of fit with experiment but not a very good one; see figure 5. Similar attempts to fit (n-butane + carbon dioxide) at 260.0 K needed 5 = 0.80 to achieve any reasonable agreement between experimental (table 4) and calculated results. The large deviations from unity of < for these mixtures contrasts with the zero deviation required to fit some (carbon dioxide + water) mixtures.(24’ The azeotropic point was calculated for various values of 5 and is compared with experiment in table 9. TERNARY

MIXTURES

In extending the calculations to ternary mixtures substance in the mixture was given by Pi

=

‘%

+

t1

-Xi)(aAm/aXi)T,y,,xj/x~

the chemical potential +PKlT

of each

(VAPOUR + LIQUID)

425

PHASE EQUILIBRIA

FIGURE 5. The calculated and experimental (liquid + vapour) equilibrium for {xC,H, +(l -x)CO,J at 260.0 K. 0, Experimental points from table 4; , the calculated lines for various values of c,

where i, j, and k refer to the substances in the mixture and the constraint xJ/xL indicates that this ratio is constant. With this constraint and the use of combining and mixing rules for only pair interactions, calculations of the chemical potential of each substance in the phases could be made in the usual way. The use of the constant ratio (Xj/X,) was a mathematical device to avoid problems of differentiating with respect to x in the ternary mixture. Its use was suggested by Van Ness.(34) The equations for chemical potential are then very similar to equations for the binary case, except that there are three different derivatives of the terms a(x) and b(x), (25,26) with respect to the mole fraction of each component. The actual manual search for the simultaneous equality of the chemical potential for all three substances was very tedious. An area where one phase was expected was compared with points along a straight line of constant (xj/xL) where the second phase was expected. The size of the area and length of the line were decreased as the TABLE 9. Calculated and experimental azeotropes for {xC,H, +(l -x)CO,} -.-~-~.

Calculated Experiment”

t

0.95 0.90 0.85 -

PIMPa

2.4263 2.5855 2.8159 2.825

’ From interpolation of results in references 13, 9, and 22. and this work.

x

0.923 0.750 0.687 0.678

at 260.0 K

A. Q, CLARK

426

AND

K. STEAD

(4

0

1

0.8

0.6

0.4 xC,H

0.2

1

0.8

I 01

0.6

0.4

0.2

~(C4H10)

FIGURE 6. The calculated and experimental (liquid + vapour) equilibrium for (a), (ethane + propane + n-butane) at 270.0 K and 0.50 MPa and (b), (ethane + n-butane + carbon dioxide) at 260.0 K and 2.00 MPa. 0, This work; -, calculated.

step size in mole fraction was also decreased until a satisfactory answer to 0.001 in mole fraction was reached. Results for (ethane + propane + n-butane) at 270.0 K and 0.5 MPa are given in figure 6(a). All the values of r are unity. Results for (ethane + n-butane + carbon dioxide) at 260.0 K and 2.0 MPa are given in figure 6(b). The value of 5 for (ethane + carbon dioxide) was 0.85 and for (n-butane + carbon dioxide) it was 0.80. The agreement is not perfect. The authors gratefully acknowledge the help of the Royal Society and British Gas plc in purchasing equipment and of the S.E.R.C. in the award of a studentship to one of us (A.Q.C.). We wish to thank Mr T. Booty and Mr R. Batten for their help with the work, and Dr R. J. Williams and Dr P. W. Fowler for many helpful contributions. REFERENCES 1. Stead, K.; Williams, J. M. J. Chem. Thermodynamics 1980, 12, 265. 2. Hirata, M.; Suda, S.; Hakuta, T.; Naghama, K. Mem. Pac. Technol.

Tokyo

Mefrop.

Univ.

1969,

19,

103. 3. Price, A. R. Ph.D. Thesis, Rice University, Houston, TX. 1957. 4. Miksovsky, J.; Wichterle, J. CoIleel. Czech. Chem. Commun. 1975,40, 365. 5. Skripka, V. G.; Nikitina, I. E.; Zhdanovich, L. A.; Sirotin, A. G.; Benyaminovick, 0. A. Gazov. Prom.

1970, 15, 35.

6. Budenhobser, R. A.; Djordjevich, L. J. Chem. Eng. Dara 1970, 15, 10. 7. Kay, W. B. Ind. Eng. Chem. 1940, 32, 353. 8. Kaminishi, G.; Yokayama, C.; Takahashi, S. Sekiyu Gukkuishi 1986, 29, 32. 9. Nagahama, K.; Konishi, H.; Hoshino, D.; Hirata, M. J. C/rem. Eng. Jpn 1974, 7, 323. 10. Kalra, H.; Krishnan, T. R.; Robinson, D. B. J. Chem. Eng. Data 1976, 21, 222. Il. Weber, L. A. Cryogenics 1985, 25. 338.

(VAPOUR

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

+ LIQUID)

PHASE

EQUILIBRIA

421

Khazanova, N. E.; Lesneskaya, L. S. Russ. J. Phys. Chem. 1%7,41, 1276. Fredenshmd, A.; Mollerup, .I. .I. Chem. Sot. Furaday Trans. I1974, 70, 1653. Gugnoni, R. J.; Eldridge, J. W.; Okay, V. C.; Lee, T. J. AIChE. J. 1974, 20, 357. Hamam, S. E. M.; Lu, B. C. Y. Can. J. Chem. Eng. 1974, 5. 283. Mollerup, J. J. Chem. Sot. Faraday Trans. I 1975, 71, 235 1. Davalos, J.; Anderson, W. R.; Phelps, R. E.: Kidnay, A. J. J. Chem. Eng. Data 1976, 21, 81 Ohgaki, K.; Katayama, T. Fluid Phase Equilibria 1977, I, 27. Kuenen, J. P.; Robson, W. G. Phil. Mug. 1902, 6, 116. Clark, A. M.; Din, F. Disc. Faraday Sot. 1953, 202. Jensen, R. H.; Kurata, F. AIChE. J. 1971, 17, 357. Hakuta, T.; Nagahama, K.; Suda, S. Kaguku Kogaku 1%9,33. 904. Robinson, D. B.; Kalra, H. Proc. Ann. Conv., Gus Process. Assoc. Tech. Pap. 1974, 53, 14. Gibbons, R. M.; Laughton, A. P. J. Chem. Sot. Faraday Trans. II 1%4,80, 1019. Stead, K.; Williams, J. Chem. Sac. Faraday Trans. II 1980, 76, 1045. McGlashan, M. L.; Stead, K.; Warr, C. J. Chem. Sot. Faraday Trans. II 1977, 73, 1889. Reid, R. C.; Prausnitz, J. M.; Sherwood, 7. K. The Properties of Gases and Liquids. 3rd edition. McGraw-Hill: New York. 1977. Kemp, J. D.; Egan, C. J. .I. Am. Chem. Sot. 1938,60, 1521. Kuloor. N. R.; Newitt, D. M.; Bateman, J. S. Thermodynamic Functions of Gases. Vol. 2. Din, F.: editor. Butterworth: London. 1962, p. 115. Thomas, R. H. P.; Harrison, R. H. J. Chem. Eng. Dafa 1982. 27. I. Tickner, A. W.; Lossing, F. P. J. Phys. Chem. 1951, 55, 733. Das, T. R.; Reed, C. 0.; Eubank, P. T. J. Chem. Eng. Data 1973, 18, 244. Aston, J. G.; Messerly, G. H. J. Am. Chem. Sot. 1940, 62, 1917. Van Ness, H. C. Classical Thermodynamics qf Non-electrol.yre Solutions. Pergamon: Oxford. 1964.