Vapour-liquid equilibria calculations for normal fluid systems using a new cubic equation of state

Fluid Phase Equilibria, I5 (1983) 33-66 Elsevier Science Publishers B.V.. Amsterdam

33 -

Printed

in The Netherlands

VAPOUR-LIQUID EQUILIBRIA CALCULATIONS FOR NORMAL FLUID SYSTEMS USING A NEW CUBIC EQUATION OF STATE RICHARD

FREZE

and JEAN-LOUIS

Laboratoire de GEnie Chimique, Marseille I4 (France) ANDRE

PENELOUX

FacuM

and EVELYNE

CHEVALIER des Sciences

et Techniques

February

15, 1983; accepted

13397

RAUZY

Laboratoire de Chimie Physique, Facultk des Sciences de Luminy, (Received

de St. Jerome,

1328% Marseille 9 (France)

in final form June 15, 1983)

ABSTRACT Freze, R., Chevalier, J.-L., Penelow, A. and Rauzy, E., 1983. Vapour-liquid equilibria calculations for normal fluid systems using a new cubic equation of state. Fluid Phase Equilibria, 15: 33-66.

A new cubic equation of state is presented with a pseudocritical compressibility factor taken as substance-dependent. This equation leads to good phase-behaviour prediction for normal fluid mixtures up to the critical state, even in the case of binary systems involving a light and a heavy alkane, for which the equations of Redlich-Kwong-Soave and Peng-Robinson give poor results. With the volume correction proposed by Peneloux et al., this method also gives good estimates of the volumetric properties of pure compounds and mixtures, except in the neighbourhood of the pure-component critical points.

INTRODUCTION

In the last ten years, numerous methods have been published in the literature for predicting the thermodynamic properties of pure compounds and mixtures from a cubic equation of state. Among these, the Redlich-Kwong-Soave and the Peng-Robinson methods are certainly those used most frequently in the petroleum industry and related fields for the prediction of vapour-liquid equilibria. In most cases these methods give fairly good results, except for mixtures of methane with a heavy hydrocarbon. We here propose a method founded on a new cubic equation of state, which improves the prediction of the phase behaviour and volumetric properties of such mixtures. 0378-3812/83/$03.00

0 1983 Elsevier Science Publishers

B.V.

34 THE EQUATION OF STATE

The general cubic equation given by

of state (see Appendix

I for more details) is

P=RT/(U-b)-a/(o-r,)(u-r,)

(1)

where r, and r, are the roots of the quadratic v2+/3o+6=0

equation

in volume (2)

Some general features of a cubic equation as represented by eqn. (1) have been discussed by Abbott (1973). Using the classical conditions at the critical point, for a given compound, a =f&

T,, %)

(3)

b =f2(pCt

T,, $9 Z:)

(4)

ri

=fdPc, T,, Q,, Z,*>

(5)

r, =f,(Pc, T,, a,, z:)

(6)

Therefore the cubic equation of state is defined by only two parameters: s1,, which defines the family of equations, and Zz, a pseudocritical compressibility factor which characterizes a particular member of the family. Selection of parameter Q, In order to use an equation of state in the calculation of vapour-liquid equilibria it is not necessary that the calculated volumes be well represented, but it is required that the vapour pressure of the pure compound be obtained with good accuracy. To fulfill this condition with a cubic equation of state it is absolutely necessary that the parameter a be taken as dependent on temperature. In previous work on cubic equations of state (Peneloux et al., 1982), it has been shown that when the parameter a is correlated with reduced temperature using the equation proposed by Soave (1972), i.e., ( a/a,)“2

= 1 + m( 1 - T,‘/‘)

(7)

m=m,+m,w+m,w2 a,=a(T,= Q2,= a:/’

1)

(8) (see Appendix

I) (9)

where w is Pitzer’s acentric factor, the deviations between the experimental and calculated vapour pressures present a minimum for a certain value of the parameter P,.

35

For the first ten compounds in the n-alkane series, using vapour pressures and critical constants tabulated by the American Petroleum Institute (1974), the best results are obtained using the equation of state with the value of 9, equal to 0.77 (mean relative error 0.6%; relative error < 1% for all compounds). On the basis of this result we selected the parameter s1, = 0.77 for all subsequent work, and name the corresponding equation the C- 1 equation. It is interesting to note that this equation is very close to the family of the Peng-Robinson equation (for which S2, = 0.77039; see Table I and Appendix I). Selection of parameter Z: In order to keep the equation of state simple, we take the parameter Zz as constant for a given compound. As did Usdin and McAuliffe (1976), we first attempted to determine the optimal value of Z,* for each pure compound using the liquid volumes. Nevertheless, using the equation of state so defined it was not possible to predict the vapour-liquid equilibria of hydrocarbon mixtures with the expected precision, at least using the customary combination rules. Therefore a different approach was chosen. We selected the following combination rules for the application of the equation to mixtures: a = C C xixja,i 1 j aij= (a,aj)“‘(l

(10) - ki,)

b = &,b,

r,a/b

(11) (12)

=c Cxixj(r,a/bIlj i

j

r2a/b=CCx,x,(r2a/b)ii i

j

(rza/b)ij= [(r2,aj/b,)(r2jaj/b,)11’*

(16)

where kjj is a binary interaction parameter. In this work we take kij = 0 for all alkane-alkane binary systems. Using vapour-liquid equilibrium data for methane-n-alkane systems, we determined the optimal value of Z,* for each n-alkane (C, to C,,) by minimizing the error between the experimental and calculated pressures. For methane we fixed Z,* = 0.319, which is the value giving the best

0.76ooo 0.77000 0.78009 0.79000 0.80000 0.81000

Other examples

(1977).

0.75000 0.75000 0.75331 0.77039 0.78454

Van der Waals Clausius Redlich-Kwong Peng-Robinson Harmens ’

aHamtens

Q,

Equation

0.37500 z, l/3 0.30740 0.28619

0.12500 eqn. (I-16) 0.08664 0.07779 0.07073

6

Characteristic parameters of some cubic equations of state

TABLE 1

0 eqn. (I-17) 0 0.03222 0.03973

0 eqn. (I-18) - 0.08664 - 0.18782 -0.25189

‘2

0.30400 0.27611 0.25490 0.23700 0.22111 0.20659

0.37500 0.37500 l/3 0.27517 0.24647

z:

0.06400 0.04611 0.03490 0.02700 O.G2111 0.01659

0.12500 0.12500 0.08664 0.04557 0.03101

6

Basic term (eqn. (I-22))

-0.15200 -0.21778 - 0.27020 -0.31600 - 0.35777 - 0.39682

0 0 - 0.08664 - 0.22004 - 0.29159

4

37 TABLE

2

Optimal

values of ZT and physical

properties

of compounds

used in this work B

Compound

Z,*

r, (K)

P, (atm.)

w

Z,

Methane Ethane Propane n-Butane i-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane Benzene Toluene Nitrogen Carbon dioxide Hydrogen sulphide Ammonia Water

0.319 0.315 0.307 0.304 0.304 0.300 0.297 0.296 0.294 0.293 0.292 b b b

190.55 305.43 369.82 425.16 408.14 469.65 507.30 540.10 568.76 593.60 617.50 561.99 591.72 126.20 304.19 373.20 405.50 647.27

45.41 48.11 41.92 37.46 36.07 33.25 29.71 29.98 24.54 22.60 21.30 48.19 40.52 33.56 72.83 89.65 112.05 218.28

0.0112 0.0977 0.1537 0.2009 0.1846 0.2519 0.3005 0.3499 0.3983 0.4535 0.5020 0.2098 0.2642 0.0373 0.2229 0.1126 0.2521 0.3445

0.2968 0.2843 0.2807 0.2738 0.2832 0.268 1 0.2640 0.2629 0.2589 0.2600 0.2470 0.268 1 0.2646 0.2986 0.2744 0.2840 0.2429 0.2330

a References (1978). b Compound

properties

for physical

b b b b

given in Peneloux

not used in determination

et al. (1982) and Freze and Chevalier

of Zz.

representation of the liquid volume in the range 0.55 < T, -c 0.95, P < 700 atm. The results are presented in Table 2. It can be seen that the values of Zz differ from the experimental critical compressibility factors. For compounds having an acentric factor greater than 0.090, the parameter Z,*, may be correlated with fairly good accuracy with w by the relation Z; = 0.2910 + 0.04421 exp( -6.46361

U)

(17)

For compounds having o < 0.090, we fixed Zz = 0.319, because this value gives better results in vapour-liquid equilibria calculations than that obtained from eqn. (17). Adopting this correlation, the C-l equation of state is generalized and can be used in phase-behaviour prediction for hydrocarbon mixtures. APPLICATIONS

We now discuss further applications of the C-l equation of state presented above. All the formulas used in this method are given in Appendix II.

38

The deviations used to compare the calculated and experimental values of pressure, composition or volume for the cases considered here are defined in Appendix III. Vapour pressures of pure compounds

The results obtained using the C-l equation in vapour-pressure computations are presented in Table 3 for a series of seventeen compounds including water and ammonia. The comparisons are made with the equations of Soave (1972) Peng and Robinson (1976) and Graboski and Daubert (1978). It can be seen that the C-l equation is a little better than the Peng and Robinson equation, and significantly better than the other two. Vapour-liquid

equilibrium

In order to test the various methods for vapour-liquid equilibria prediction, we collected a set of vapour-liquid equilibrium data for systems related to the petroleum industry. For better comparison and to display the weakness of each method for particular types of systems, this set was divided into eight subsets: (1) subset SO, which includes binary systems containing alkanes with molecules of not too different sizes: methane-ethane, methane-propane, and binary systems containing ethane and propane with alkanes from C, to Cl& (2) subset S14, an intermediate subset which includes the binary systems methane-butane, methane-pentane and methane-hexane; (3) subset S17, including systems containing alkanes having very different molecular sizes, starting with the system methane-heptane; (4) subset SK, comprising the binary systems containing the alkanes with aromatic compounds, carbon dioxide and hydrogen sulphide, and also the systems methane-nitrogen, benzene-carbon dioxide and hydrogen sulphide-carbon dioxide; (5) subset SN, comprising all the binary systems containing nitrogen except methane-nitrogen; (6) subset ST, comprising systems having more than two constituents and containing alkanes only; (7) subset STK, comprising systems having more than two constituents and containing alkanes, aromatic compounds, carbon dioxide and hydrogen sulphide; and (8) subset SY, including the Yarborough mixtures, which contain alkanes, nitrogen, carbon dioxide and hydrogen sulpbide. For all these sets only isothermal data were selected. The temperatures

39 TABLE

3

Mean relative deviations d ,,,,,(I’) (%) between experimental and predicted (for each compound 100 points between T, = 0.55 and Tr = 0.98)

vapour

pressures

Compound

Experimental values

Method ’ C-l

RKS

PR

GB

Methane

American Petroleum Institute (1974)

0.46

1.59

0.68

2.05

Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane Benzene Toluene Nitrogen Carbon dioxide b Hydrogen sulphide Ammonia Water

-

0.62 0.43 0.44 0.46 0.96 0.74 0.72 0.3 1 0.60 1.29 0.72 0.45 0.44 1.23 0.53 2.58

0.99 1.37 1.28 1.38 1.77 0.98 1.49 1.87 2.46 0.66 1.91 1.03 0.86 1.94 1.79 4.02

0.83 0.48 0.41 0.50 1.03 0.89 I .08 0.86 1.07 1.40 0.63 0.92 0.40 1.01 0.58 2.34

1.23 1.53 1.42 1.43 1.78 0.95 1.45 1.79 2.36 0.65 1.96 1.39 0.81 2.28 1.82 4.00

0.76

1.61

0.89

1.70

Ambrose

et al. ( 1970)

Wagner (1973) Kennedy and Thodos ( 1960) West (1948) Baehr et al. (1976) Ambrose and Lawrenson (1972)

Mean relative deviation * RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; text. b 100 points between T, = 0.713 and T, = 0.98.

GB, Graboski-Daubert;

for C-l,

see

TABLE 4 Description

of systems forming

subset SO

Code

System

Reference

so- 1

C, +C,

Price and Kobayashi

so-2

C, +C,

Wichterle

SO-3 so-4

C, +c* C, +c,

Davalos et al. (1976) Akers et al. (1954a)

T(K)’ (1959)

and Kobayashi

(1972a)

199.8 255.4 130.4 158.2 186.1 190.9 193.9 199.9 250.0 157.6 194.8 226.5 256.5

[Cl; 227.8 [Cl; [Cl; 283.1 [C] [4]; 144.3 [7]; [IO]; 172.0 [9]; [II]; 189.7 [13]; [15]; 192.4 [Cl; [Cl; 195.4 [Cl; [13, C] [7] [3]; 174.3 [3]; [6]; 213.2 [9, C]; [9, C]; 241.5 [12, C]; [12, C]; 273.2 (13, C]

40 TABLE 4 (continued) Code

System

Reference

T(K)*

so-5

Roof and Baron ( 1967)

SO-6

c,+c3 c,+c3

so-7

c,+c,

Miksovsky

SO-8

c, + n-c,

Mehra and Thodos

so-9

c, + n-c,

Reamer et al. (1960)

so-10 so-11

c, + n-C,

c, + n-c,

Ohgaki et al. (1976) Mehra and Thodos ( 1%Sb)

so-12

c, + n-c,

Rodrigues

so-13

c, + n-c,,

Reamer

so-14

c, + n-c,

Kreglevski

so-15

c, + n-c,

Vejrosta

and Wichterle

SO-16

c, + n-c,,

Reamer

and Sage (1966)

305.0 328.0 144.3 172.0 191.0 217.3 303.2 343.2 338.7 394.3 277.6 344.3 410.9 298.2 338.7 394.3 449.8 313.2 348.2 277.6 344.3 410.9 477.6 387.5 419.7 336.6 360.9 369.7 383.2 277.6 344.3 410.9 477.6

Wichterle

and Kobayashi

and Wichterle

of determinations

(1975)

(1965a)

et al. (1968) and Sage (1962)

and Kay (1969)

Total 633 points (966 values of y between 50 critical data ’ Number

(1972b)

(1974)

[Cl; 316.6 [Cl; [Cl; 344.8 [C] 161; 158.2 [8]; [S]; 187.5 [9]; [lo]; 195.2 [13]; 111, C] [18, C]; 323.2 [15, C]; (41 [6, C]; 366.5 [8, C]; [3, C] (71; 310.9 [12, C]; [13, C]; 377.6 [13, C]; [11, C]; 444.3 [C] [6] (7, C]; 366.5 [7, C]; [6, C]; 422.0 [6, C]; (6, C] [13]; 323.2 [12]; [9]; 373.2 [9] [7]; 310.9 [7, C]; [I 1, C]; 377.6 [15, C]; [16, C]; 444.3 [4, C]; [15, C]; 510.9 114, C] [Cl; 399.3 [Cl; [Cl; 440.0 [C] [15]; 344.3 1141; [14]; 365.7 [13]; [I 11; 373.2 181; [5] [3]; 3 10.9 [3]; [5]; 377.6 17, C]; [8, C]; 444.3 (8, C]; [8, C]; 510.9 [7, C]

0.01 and 0.99, 300 values of y outside

this interval)

(C, critical data) given in brackets.

and ‘numbers of determinations for each isotherm are given for all the systems in Tables 4- 11. We have divided the vapour molar fractions into two groups, those between 0.01 and 0.99, and those lower than 0.01 or greater than 0.99, because in the second group the differences between the

41 TABLE

5

Description

of systems forming

subset S14

Code

System

Reference

T(K)=

s14-1

C, + n-C,

Sage et al. (1940)

S14-2 s14-3

c, + n-c, C, + n-C,

Roberts et al. (1962) Wiese et al. (197Oa)

s14-4

c, + n-c,

Kahre (1974)

s14-5

c, + n-c,

Elliot et al. (1974)

Sl4-6

c, + n-c,

Sage et al. (1942)

s14-7

c, + n-q

Chu et al. (1976)

Sl4-8

C,+n-C,

Poston

294.3 [Cl; 310.9 [Cl; 327.6 [Cl; 344.3 [Cl; 360.9 [Cl; 377.6 [Cl; 394.2 [C] 210.9 [Cl; 244.3 [Cl; 277.6 [C] 277.6 [6]; 310.9 [6]; 344.3 161; 377.6 [3] 166.5 [8]; 177.6 [8]; 186.0 191; 194.1 [6]; 211.0 [9]; 227.6 171; 255.4 [7]; 283.2 [9] 144.3 [3]; 166.5 [5]; 177.6 191; 190.6 [ 121; 199.9 [Cl; 210.9 (4, C]; 222.1 [Cl; 233.2 [7, C]; 244.3 [Cl; 255.4 [11, C]; 227.6 112, C] 310.9 [17, C]; 344.3 [14, C]; 377.6 [lo, C]; 410.9 [7, C]; 444.3 [C] 176.2 141; 194.2 [Z]; 199.9 [4]; 223.9 17); 248.3 [lo]; 273.2 [7, C] 310.9 [IO]; 344.3 [6]; 377.6 [S]; 411.0 141; 444.3 13)

and McKetta

(1966)

Total 253 points (326 values of y between 23 critical data a Number

of determinations

0.01 and 0.99, 180 values of y outside

(C, critical

this interval)

data) given in brackets.

experimental and calculated values are always smaller and the overall mean error is thus lower. For some systems the number of points used is lower than the number of determinations published by the authors. In these cases we excluded data at high pressures in the vicinity of the critical region, for which our computer program for bubble-point determination failed to converge. This is not too important, because we made specific tests for critical data. The whole data bank contains 3588 vapour-liquid equilibrium determinations and 143 critical data sets. For comparison we have chosen the Redlich-Kwong-Soave and Peng-Robinson methods, which are those used most frequently in high-pressure vapour-liquid equilibrium calculations. The results are presented in Tables 12-19 and in Table 20 for the critical data. For the reason explained

42 TABLE 6 Description

of systems forming

subset S17

Code

System

Reference

s17-1

C,+n-C,

Reamer

Sl7-2

c, + n-c,

Kohn and Bradish (1964)

s17-3

c, + n-c,

Shipman

s17-4

c, + n-c,,

Reamer

s17-5

c, + n-c,,

Beaudoin

Sl7-6

C, + n-C,,

Wiese et al. (1970b)

T(K)’ et al. (1965)

and Kohn (1966)

et al. (1943)

and Kohn ( 1967)

277.6 344.3 410.9 477.6 298.2 348.2 423.2 223.2 273.2 323.2 373.2 310.9 377.6 444.3 5 10.9 348.2 423.2 277.6 477.6

(13, C]; 310.9 [14, C]; [14, C]; 377.6 113, C]; (11, C]; 444.3 [IO, C]; [8, C]; 510.9 [C] [7]; 323.2 171; [6]; 373.2 [7]; [6] [7]; 248.2 [lo]; [19, C]; 298.2 (19, C]; [19, C]; 348.2 [19, C]; [IO]; 423.2 [lo] [Cl; 344.3 [Cl; [C]; 410.9 [Cl; [C]; 477.6 [Cl; [C] [lo]; 373.2 [lo]; [7] [S]; 310.9 [S]; 410.9 [5]; [4]; 510.9 [3]

Total 323 points (272 values of y between 0.01 and 0.99, 374 values of y outside this interval) 19 critical data a Number

of determinations

(C, critical data) given in brackets.

above, the deviations between the experimental and predicted values are given only for vapour molar fractions between 0.01 and 0.99. For all of the alkane-alkane binary systems the interaction parameters ki, were taken equal to zero. For the other binary systems involving an alkane and a nonalkane or two nonalkanes, the values of k,, were determined from experimental data, and considered to be independent of system temperature, pressure and composition. For this purpose we used a flash calculation program which minimizes the sum of the squared deviations between the experimental and predicted vapour and liquid mole fractions. The kij values are listed in Tables 21-23. Because of lack of experimental data, or because of their temperature dependence, some values have been estimated and are given in brackets. We now consider the comparisons for the various subsets of data.

TABLE

7

Description Code SK-

1

of systems forming System

Reference

T(K)a

c, +

Elbishlawi and Spencer (1951) Ohgaki et al. (1976) Glanville et al. (1950) Butcher et al. (1972)

338.7 1151

benzene SK-2

c2+

benzene SK-3 SK-4

subset SK

C3 + benzene n-c, + benzene

SK-5

c, +co,

SK-6

c, +co,

SK-7

c, + co,

SK-8

c, + co,

SK-9

c, + co,

SK-10

c, + co2

SK-11

c, + co,

SK-12

c, + co,

SK-13

n-c, + co,

SK-14

n-c,

SK-15

i-c, + cos

SK-16

n-c, + co,

SK-17

n-c, + co,

SK-18

n-c, + co,

SK-19

n-c,

SK-20

n-c ,a + co,

+ co,

+ co,

Davalos et al. (1976) Somait and Kidnay (1978) Hamam and Lu (1974) Fredenslund and Mollerup (1974) Davalos et al. (1976) Roof and Baron (1967) Nagahama et al. (1974) Hamam and Lu (1976a) Olds et al. (1949) Nagahama et al. (1974) Nagahama et al. (1974) Poettman and Katz (1945) Besserer and Robinson (1973) Oh&i and Katayama (1976) Kalra et al. (1978) Reamer and Sage (1963)

298.2 [7] 310.9 410.9 383.2 428.2 473.2 230.0

161; 344.3 [lo]; 377.6 114, C]; 115, C]; 444.3 [I& C]; 477.6 [13, 61 [9]; 398.2 [8]; 413.2 [9]; 191; 443.2 [lo]; 458.2 [lo]; 19); 488.2 [7] [12]; 250.0 181; 270.0 [7]

270.0 [7] 222.0 288.7 223.2 283.2

1121; 244.3 [9]; 266.5 [12]; (21 191; 243.2 [8]; 263.2 [8]; [8]

250.0 [13] 305.0 [Cl; 305.9 [Cl; 208.2 [Cl; 327.8 [Cl; 345.2 [C] 253.0 [lo]; 273.2 [IO] 244.3 [IO]; 266.5 [I l] 325.9 [Cl; 351.7 [Cl; 377.2 [Cl; 398.8 [C] 273.2 [13] 273.2 [18] 314.9 404.9 277.7 377.6 298.2

[Cl; 325.7 [Cl; 368.5 [Cl; [Cl (lo]; 311.0 [14]; 344.2 [15]; [9] [lo]; 313.2 [lo]

310.7 477.2 277.6 377.6 477.6

[22]; 352.6 [17]; 394.3 1161; [S] [9]; 310.9 [IO]; 344.3 [7, C]; [9, C]; 410.9 [ 11, C]; 444.3 [ 11, C]; [I I, C]; 510.9 19, C]

44 TABLE 7 (continued) Code

System

Reference

T(K)’

SK-2 1

C, +H,S

277.6 [23, C]; 310.9 [20, C]; 344.3 [IO, C]

SK-22 SK-23

C, +H,S C,+H,S

Reamer et al. (1951) Kalra et aI. (1977a) Brewer et al. (1961)

SK-24

i-C4 +H,S

SK-25

n-C, + H,S

SK-26

n-C,, + H,S

SK-27

C, +N,

SK-28

C, +N,

Kidnay (1975)

SK-29 SK-30

co, + benzene CO, +H,S

Ohgaki and Katayama (1976) Bierlein and Kay (1953)

SK-3 1

CO, +H,S

Sobocinski and Kurats (1959)

Besserer and Robinson (1975a) Reamer et al. (1953a) Reamer et al. (1953b) Stryjek et al. (1974a) et al.

227.9 260.2 299.8 322.0 277.6 377.6 277.6 377.6 277.6 377.6 113.7 138.4 172.0 112.0 140.0 170.0 298.2

[7]; 255.3 [S]; 283.2 [ 1I] [2]; 272.0 [I]; 283.2 [2]; [I]; 310.9 [I]; 316.5 [l]; [I]; 327.6 [l] [a]; 310.9 [9]; 344.4 (81; [3] [7]; 310.9 (71; 344.3 [7]; [Ill; 410.9 [9] [7]; 310.9 [7]; 344.3 [7]; [6, C]; 410.9 (8, C]; 444.3 [9, C] [a]; 122.0 [ll]; 127.8 [15, C]; [9]; 149.8 [8]; 160.9 191; [7, C]; 177.6 [C] [3]; 120.0 [2]; 130.0 [IS]; [13]; 150.0 [ll]; 160.0 [14]; [IO]; 180.0 [4] [8]; 313.2 [9]

293.2 316.9 333.2 347.6 266.5 299.8 332.2

[4]; 303.2 [Cl; 323.2 [3]; 337.9 [C] 131; 277.6 [4]; 310.9 [2]

[4]; 313.2 [5]; [4]; 330.1 [Cl; [Cl; 343.2 [2]; (41; 288.7 [4]; 141; 322.0 [3];

Total 951 points (1722 values of y between 0.01 and 0.99, 180 values of y outside this interval) 36 critical data ’ Number

of determinations

(C, critical data) given in brackets.

Subset SO (Tables 12 and 20) The bubble points and vapour mole fractions are predicted with good accuracy by the three methods, which all give similar results. The critical properties are also estimated correctly. Subset 514 (Tables 13 and 20) The predictions are less good than for subset SO, the C-l method giving the best results.

45 TABLE 8 Description

of systems forming

subset SN

Code

System

Reference

T(KIa

SN- 1 SN-2 SN-3 SN-4 SN-5 SN-6

C, +N, C, +N, G+Nz C, +N, n-C, + N, n-C.,+ N,

149.2 200.0 311.6 230.0 310.9 310.9

SN-7

n-C, + N,

Stryjek et al. (1974b) Grauso et al. (1977) Roof and Baron (1967) Grauso et al. (1977) Akers et al. (1954b) Lehigh and McKetta (1966) Kalra et al. (1977b)

SN-8

n-C, + N,

SN-9

n-C, +N,

SN-IO

n-C,, + N,

SN-11

CO, + N,

SN-12

CO, + N,

SN-13

CO2+ N,

SN-14

H,S+N,

Poston and McKetta (1966) Peters and Eicke (1970) Azarnoosh and McKetta (1963) Zenner and Dana (1963) Muirbrook and Prausnitz (1965) Somait and Kidnay (1978) Besserer and Robinson (1975b)

[Cl; [9]; [Cl; [9]; [Cl; [C]

172.0 230.0 327.1 260.0 366.5

[Cl; 194.3 [C] 191; 260.0 [7] (C] [lo]; 290.0 [ 131 [Cl; 399.8 [C]

277.4 [7]; 310.7 1141; 344.3 (II]; 377.6 [lo] 310.9 [ 111; 344.3 [1 11; 377.6 (1 I]; 410.9 [ll, C]; 444.3 [6, C] 376.4 16, C]; 413.2 18, C]; 453.2 [7, C 310.9 (231; 344.3 [23]; 377.6 1231; 410.9 123) 218.2 191; 232.9 [l I]; 273.2 [7] 273.2 [C] 270.0 [31] 321.9 [IO]; 300.0 [ 1 I]; 277.7 [12]; 256.4 [12]

Total 365 points (564 values of y between 0.01 and 0.99, 166 values of y outside 15 critical data a Number

of determinations

this interval)

(C, critical data) given in brackets.

Subset SI 7 (Tables 14 and 20) Here a great difference between the three methods can be seen. For the Redlich-Kwong-Soave and Peng-Robinson methods the predictions of bubble-point pressures are very poor, but the C-l method gives good estimates (mean error c 3%). The critical properties are estimated correctly by all three methods, which give similar results. For the system methane-n-nonane (S17-3) the results for eight isotherms (223.2-423.2 K) are presented in Table 24. The C-l method is definitely superior at all temperatures. In fact, this is the main advantage of the C-l method: it can predict with good accuracy vapour-liquid equilibria for binary systems containing a light and a heavy alkane up to the critical point.

46 TABLE 9 Description

of systems forming

subset ST

Code

System

Reference

T(K)s

ST-l

c,+c,+c,

158.2 [2]; 172.0 171; 185.9 [9]

ST-2

C, +C, + n-C,

ST-3

c, + c, + n-c,

ST-4 ST-5

C, +C, + n-C,, c, + n-c, + n-c,,

ST-6

C, + n-C, + n-C,

ST-7

c, + n-c, + n-c,

ST-8

C, + n-C, + n-C,

Wichterle and Kobayashi (1972~) Van Horn and Kobayashi ( 1967) Van Horn and Kobayashi ( 1967) Wiese et al. (1970b) Reamer et al. (1951b) Mehra and Thodos (1966) Mehra and Thodos (1968) Dastur and Thodos (1964)

222.0 [9]; 233.2 [l I]; 244.3 [I21 199.8 [4]; 210.9 [S]; 222.0 [13] 277.6 [16]; 310.9 [16]; 410.9 [I I] 277.6 [S]; 344.3 [12]; 410.9 1121 394.3 [21]; 366.4 [20]; 338.7 [14] 422.0 [28]; 449.8 [22] 338.7 [30]; 366.5 [33]; 394.3 [37]

Total 492 points (1276 values of y between 0.01 and 0.99, 200 values of y outside a Number TABLE

of determinations

STK-

1

given in brackets.

10

Description Code

this interval)

of systems forming

subset STK

System

Reference

T(K)’

c, +c, +co,

Im and Kurata (1971) Im and Kurata (1971) Wang and McKetta (1964) Im and Kurata (1971) Im and Kurata (1971) Hamam and Lu (1976b) Im and Kurata

166.5 (21; 183.2 [2]; 199.8 [l]

STK-2

c, +c,+co,

STK-3

C, + n-C, + CO,

STK-4

C, + n-C4 +CO,

STK-5

c,+c,+co,

STK-6

c,+c,+co,

STK-7

c,+c,+c,+co,

150.2 [2]; 165.2 [2]; 185.2 [2] 177.6 131; 210.9 [l I]; 244.3 [20] 185.2; [2]; 200.2 [2]; 205.2 [3] 190.2 [2]; 200.2 [2]; 205.2 [2] 244.3 [18]; 266.5 (301 190.2 [4]; 200.2 [6]; 205.2 (61

(1971) Total 188 points (532 values of y between 0.01 and 0.99, 52 values of y outside this interval) ’ Number

of determinations

given in brackets.

47

TABLE

11

Description

of systems forming

subset SY (Yarborough

mixtures)

(Yarborough,

1972)

Code

System

Mixture number

T(K)’

SY-1

c, +c, +c, + n-c,+ n-c,+ n-c,,

3

366.5 [IO]

4 5 6 I 8 8C

394.3 366.5 338.7 366.5 366.5 366.5

[6] [9] [7]; 366.5 [8] [12] (141 [IO]

8D 8D2 8E 9

366.5 366.5 310.9 338.7

(81 [2] Ill]; 338.7 [l I] [8]; 366.5 [12]

10 II

366.5 [9]; 394.3 [S] 366.5 [7]; 394.3 [8]

14 148 16 168 20

310.9 366.5 283.2 283.2 230.4 310.9

(71; 338.7 [7]; [lo] (51 161; 225.4 [4] [l I] 1131

20A 23 23A IIA

310.9 394.3 394.3 366.4

[lo] 191 [SJ [ 141

12 13 15 15A ISB 15c 15D 18 19 208 23B

338.7 310.9 310.9 366.5 255.4 255.4 255.4 227.6 283.2 310.9 394.3

[8]; 366.5 [14] [4]; 366.5 1131 [7]; 338.7 [12] [ 121 [5] [4] [6] [7] [8] [ll] [8]

+ toluene SY-2 SY-3 SY-4 SY-5 SY-6 SY-7

c, +c, +c, + n-c,+ nc, + n-c,, c,+c,+c,+n-c,+n-c,+n-c,,+ N, + CO,

SY-8 SY-9 SY-10 SY-11

c,+c,+c,+n-c,+n-c,+n-c,()+

W SY-12 SY-13 SY-14 SY-15 SY-16 SY-17 SY-18

C, + C2 + C, + n-C, + n-C, + n-C,, + toluene + H z S c,+c,+c,+n-c,+n-c,+n-c,,+ N, +CO, + H,S

c, +c, +c, + n-c, + n-c,+ n-c,,+ toluene+N2 + CO,

SY-19 SY-20 SY-21 SY-22

c, + cz + c, + n-c,+ n-c,+ n-c,,+ toluene+

N2 + CO, + H,S

SY-23 SY-24 SY-25 SY-26 SY-27 SY-28 SY-29 SY-30 SY-31 SY-32 SY-33 Total

383 points (2326 values of y between 0.01 and 0.99, 978 values of y outside this interval) * Number

of determinations

given in brackets.

48

TABLE

12

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(v) (W) for systems of subset SO System a

d,(P)

b

pressures

d,(P)

(fg) and

d(v) =

C-l

RKS

PR

C-l

RKS

PR

so-2 so-3 so-4 SO-6 so-7 SO-8 so-9 so-10 so-11 so-12 so-13 so-15 SO-16

1.4 1.3 3.1 1.5 1.4 4.7 1.9 3.5 3.1 2.4 2.8 2.6 3.3

1.7 1.4 3.8 3.3 1.2 4.9 2.1 2.5 2.7 5.8 4.6 2.0 2.6

1.4 1.8 4.7 4.8 1.4 4.8 2.5 2.6 2.5 6.4 4.5 2.8 2.5

0.38 0.48 0.72 0.08 0.46 0.93 0.73 0.38 1.17 1.04 1.34 0.44 0.40

0.59 0.47 0.69 0.16 0.43 0.90 0.77 0.42 0.54 1.05 1.07 0.44 0.33

0.30 0.49 0.70 0.28 0.44 0.87 0.75 0.34 0.87 I.00 1.27 0.43 0.32

Mean for subset

2.4

3.0

3.4

0.69

0.67

0.68

s

a See Table 4 for description of systems. ’ RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

TABLE

for C-l, see text.

13

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (%) for systems of subset S14 System a

d,(P)

pressures

d,(P)

(%) and

d(y)C

b

C-l

RKS

PR

C-l

RKS

PR

s14-3 s14-4 s14-5 Sl4-6 s14-7 Sl4-8

3.6 3.5 6.2 1.8 7.3 4.5

6.6 5.4 3.9 4.9 9.3 9.1

7.2 8.1 5.2 5.4 12.0 9.7

1.03 0.32 0.47 0.68 0.06 2.56

1.40 0.36 0.65 1.26 0.36 1.85

1.15 0.60 0.42 1.06 0.29 1.89

Mean for subset

4.4

6.0

7.6

0.97

1.10

1.07

a See Table 5 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

for C- 1, see text.

49 TABLE

14

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (9) for systems of subset S17 System ’

d,(P)

b

pressures

d,(P)

(%) and

d(u)’

C-l

RKS

PR

C-l

RKS

PR

s17-1 Sl7-2 s17-3 s17-5 Sl7-6

2.9 2.0 2.8 3.6 3.8

8.2 12.7 15.0 11.5 13.1

9.1 13.6 17.4 12.2 14.2

0.63 0.80 0.54 0.33 0.97

1.12 0.82 0.65 0.22 1.11

0.82 0.95 0.54 0.31 0.96

Mean for subset

2.9

13.2

15.0

0.65

0.93

0.80

a See Table 6 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

for C-l, see text.

Subset SK (Tables 15 and 20) For these systems an adjusted binary interaction parameter k,, has been used. The results are good for all three methods, with a mean relative error inthe bubble-point pressure close to 2.5%. Subset SN (Tables 16 and 20) For these systems, and particularly for the nitrogen-n-hexane and nitrogen-n-heptane binary systems, it is not possible to predict vapour-liquid equilibrium up to the critical point with a constant value of the binary interaction parameter ki,. Thus the overall results are poor for all three methods. Nevertheless, the C-l method gives a slightly better prediction than the other two. Subsets ST and STK (Tables 17 and 18) For these multicomponent systems the C-l method gives better results, with a mean relative error in the bubble-point pressure close to 4%. Subset SY (Table 19) Similar medium-quality

results are obtained

for all three methods.

Considering the whole data bank, it can be seen that the new method gives better results in vapour-liquid equilibrium prediction.

50 TABLE

I5

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (W) for systems of subset SK System ’

d,(P)

b

pressures

d,(P)

d(y)’

C-l

RKS

PR

C-l

RKS

PR

0.7 3.0 2.7 2.1 2.7 0.3 1.5 1.3 1.2 4.6 2.5 2.5 7.5 4.2 6.4 2.4 3.6 3.8 2.2 1.6 1.9 1.8 4.1 0.7 0.7 3.4 1.3 1.7

2.6 5.4 2.9 1.7 1.4 2.6 1.0 0.7 0.4 4.5 2.1 2.5 7.1 3.7 5.5 2.1 3.5 3.6 1.3 2.5 1.7 2.6 4.3 0.8 0.9 2.1 1.1 1.2

3.2 4.8 3.0 1.3 1.9 2.8 1.1 0.6 0.5 4.5 2.0 2.4 7.0 3.3 5.6 2.4 3.6 3.7 1.7 2.8 1.4 2.3 4.4 0.8 0.7 2.1 1.2 1.5

1.12 0.11 1.00 0.63 1.77 0.68 1.85 0.87 0.71 1.97 0.59 0.86 2.58 1.42 0.42 1.26 0.81 1.75 1.91 1.25 2.36 1.60 0.61 0.44 0.34 0.22 1.02 1.44

0.81 0.08 1.02 0.56 1.21 0.80 1.52 0.58 0.47 1.89 0.56 0.87 2.60 1.39 0.44 0.72 1.84 1.53 1.42 2.56 1.46 0.45 0.52 0.43 0.29 1.11 1.48

0.88 0.15 1.04 0.59 1.29 0.67 1.67 0.61 0.55 1.77 0.5 1 0.83 2.52 1.45 0.42 1.29 0.86 2.45 1.75 1.24 2.47 1.60 0.75 0.43 0.34 0.22 0.95 1.38

Mean for subset 2.7

2.4

2.4

1.10

1.02

1.05

SK-1 SK-2 SK-3 SK-4 SK-5 SK-6 SK-7 SK-8 SK-9 SK-11 SK-12 SK-14 SK-15 SK-17 SK-18 SK-19 SK-20 SK-21 SK-22 SK-23 SK-24 SK-25 SK-26 SK-27 SK-28 SK-29 SK-30 SK-31

(Sg) and

a See Table 7 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

1.02

for C-l, see text.

Volumes Pure jluids The C-l method gives adequate results for methane (see above) and for compounds which have a small acentric factor w. However, as w increases, the calculated volumes become smaller than the experimental values and it is necessary to correct them.

51 TABLE

16

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (%) for systems of subset SN System a

d,(P)’

pressures

d,(P)

(W) and

d(y)’

C-l

RKS

PR

C-l

RKS

PR

SN-2 SN-3 SN-7 SN-8 SN-9 SN-10 SN-11 SN-13 SN-14

5.4 6.0 4.5 1.5 9.2 3.6 3.2 1.1 4.9

5.1 6.4 4.8 11.1 12.8 4.5 4.2 2.3 5.4

5.8 6.5 4.8 9.8 10.0 3.8 4.6 2.4 5.4

1.55 1.09 1.00 1.44 3.10 I .67 2.92 0.84 1.12

2.04 1.87 1.12 2.59 2.82 1.60 1.74 1.99 1.64

1.72 0.90 2.02 2.72 I .59 1.86 1.63 1.37

Mean for subset

4.8

6.4

5.9

1.50

1.82

1.55

* See Table 8 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

1.47

for C-l, see text.

In a recent publication it was shown by Peneloux et al. (1982) that it is possible to improve the volumes calculated using an equation of state by a translation of the form d=v+c TABLE

(18) 17

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (W) for systems of subset ST System ’

d,(P)

pressures

d,(P)

(‘~6)and

d(y)’

b

C-l

RKS

PR

C-l

RKS

PR

ST-I ST-2 ST-3 ST-4 ST-5 ST-6 ST-7 ST-8

2.2 3.0 2.3 2.3 3.5 3.6 4.9 5.0

2.9 12.8 14.5 7.7 6.2 3.8 5.0 5.3

3.9 16.1 18.2 9.3 8.3 3.8 4.9 5.1

0.57 0.49 0.12 0.58 0.59 0.60 2.34 1.21

0.65 0.61 0.22 0.93 0.8 1 0.39 1.83 1.00

0.50 0.93 0.33 0.70 0.61 0.50 2.17 1.10

Mean for subset

3.7

6.6

7.6

0.99

1.02

1.13

’ See Table 9 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

for C-l, see text.

52 TABLE

18

Mean deviations between experimental and predicted bubble-point vapour mole fractions d(y) (W) for systems of subset STK System a

d,(P)

b

pressures

d,(P)

(W) and

d(y)’

C-l

RKS

PR

C-l

RKS

PR

STK- I STK-2 STK-3 STK-4 STK-5 SIX-6 STK-7

2.4 5.2 5.4 3.1 3.4 2.6 3.5

2.2 6.6 6.0 6.8 3.2 5.6 4.3

3.9 8.6 6.8 9.3 2.8 5.5 6.8

0.81 1.36 3.92 1.88 1.60 0.95 1.12

0.42 1.10 3.96 I .67 0.88 0.79

0.59 1.42 4.00 2.07 1.24 0.80 1.01

Mean for subset

4.1

5.5

6.4

2.31

2.94

3.04

a See Table 10 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions z 0.01 and < 0.99.

1.36

for C-l, see text.

with c = d RT,/P,

(19)

where u is the volume given by the equation of state. This correction leaves the predicted equilibrium conditions unchanged.

TABLE

19

Mean deviations between experimental and predicted bubble-point pressures d,(P) vapour mole fractions d(y) (%) for systems of subset SY (Yarborough mixtures)

(%) and

C-l

RKS

PR

4y)C C-l

RKS

PR

SY-1 to SY-5 SY-6 SY-7 to SY-10 SY-11 and SY-12 SY-13 SY-14 to SY-17 SY-18 to SY-21 SY-22 to SY-33

3.4 6.2 9.9 4.8 6.6 9.1 5.0 5.9

6.8 5.6 3.7 8.0 9.8 4.9 5.2 5.9

6.0 5.6 -3.8 7.9 8.5 5.8 4.8 6.0

0.29 0.27 0.27 0.30 0.57 0.50 0.56 0.58

0.32 0.33 0.25 0.41 0.53 0.48 0.41 0.52

0.26 0.26 0.21 0.41 0.55 0.44 0.39 0.58

Mean for subset

6.3

6.1

6.5

0.47

0.42

0.42

System ’

d,(P)

b

a See Table 11 for description of systems. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

for C-l, see text.

53 TABLE

20

Mean deviations between compositions d(x,) (S) Subset a

d,(P,)

so s14 s17 SK SN

experimental

and

predicted

b

critical

pressures

d,(P,)

(a)

and

d(x,)

C-l

RKS

PR

C-l

RKS

PR

1.9 4.8 3.9 3.2 22.8

2.0 5.5 4.3 4.2 43.5

1.9 4.9 4.3 3.8 33.4

1.7 1.6 1.2 2.2 5.5

2.4 2.4 3.3 2.6 6.9

1.9 1.8 2.2 2.2 6.0

a See Tables 4-8 for description b RKS, Redlich-Kwong-Soave;

of systems. PR, Peng-Robinson;

for C-l, see text.

We determined optimal values of d for the first ten n-alkanes using liquid volumes given by accurate equations of state for temperatures in the range 0.50 < q < 0.95 and for pressures up to 400 atm. For C, to n-C, we used the Bender equation (Bender, 1971; Teja and Singh, 1977; Buhner et al., 1981), and for the other normal alkanes the equation of Nishiumi and Saito (1975). These values were correlated with the acentric factor o by the relation d = 0.02802 - 0.02787 o - 0.04421 exp( -6.46361 The results for volume prediction

TABLE

o)

(20)

for some compounds

using the corrected

21

Values of binary

interaction

Compound

parameters

used with the Redlich-Kwong-Soave

method

kij ’ Benzene

Carbon

Benzene Carbon dioxide Hydrogen sulphide Nitrogen Methane Ethane Propane i-Butane n-Butane n-Pentane n-Hexane n-Heptane n-Decane

0.000 0.079 [O.Ol]

0.079 0.000 0.097 - 0.033 0.092 0.135 0.130 [O.llS] 0.118 0.132 0.122 0.111 0.127

a Values in brackets

are estimated.

P.‘l 0.017 0.025 0.019 IO.01 IO.01

P.01 PO1 IO.01 WI

dioxide

Hydrogen [O.Ol] 0.097 0.000 [0.17] 0.079 0.092 0.078 0.055 [0.06] 0.068 [0.07] [0.07] 0.042

sulphide

Nitrogen

IO.11 - 0.033 [0.17] 0.000 0.032 [0.024] [0.056] [O.OS] [0.08] (0.0911 [0.126] [0.13] [0.089]

54 TABLE

22

Values of binary

interaction

Compound

parameters

used with the Peng-Robinson

method

k,, ’ Benzene

Benzene Carbon dioxide Hydrogen sulphide Nitrogen Methane Ethane Propane i-Butane n-Butane n-Pentane n-Hexane n-Heptane n-Decane a Values in brackets

Carbon dioxide

O.ooO 0.08 1 [O.Ol]

0.081 0.000 0.095

fO.11

-0.021

0.027 0.034 0.022

0.088 0.131 0.125 [0.112] 0.112 0.125 0.115

LO.01

WI WI FJ.01 WI W.01

0.100 0.107

Hydrogen sulphide

Nitrogen

IO.0 I] 0.095

LO.11 -0.021 [O.lS] 0.000 0.035 [0.028] [0.060] [0.08] [O.OS] [0.095] [0.130] [O.1301 [0.106]

,::y 0.077 0.092 0.076 0.049 [0.06] 0.061 IO.071 IO.071 0.033

are estimated.

C-l method (eqns. (I8)-(20)) are presented in Table 25. In all cases the proposed method gives results superior to those obtained using the Redlich-Kwong-Soave or Peng-Robinson equation.

TABLE 23 Values of binary

interaction

Compound

parameters

ki, ’ Benzene

Benzene Carbon dioxide Hydrogen sulphide Nitrogen Methane Ethane Propane i-Butane n-Butane n-Pentane n-Hexane n-Heptane n-Decane ’ Values in brackets

used with the C-I method

Carbon

o.oOfJ 0.071 ]O.Ol]

[O.ll - 0.007 0.020 0.017

IO.01 IO.01 FJ.01 IO.01 IO.01 WY are estimated.

-

dioxide

0.071 0.000 0.094 0.089 0.059 0.105 0.115 0.104

0.109 0.117 0.099 0.107 0.114

Hydrogen [O.Ol] 0.094 ]::Yj 0.042 0.087 0.090 0.050 [O.OS] 0.075 [0.07] IO.071 0.056

sulphide

Nitrogen

LO.11 - 0.089 [0.12] 0.000 0.035 [0.014] [0.026] [0.02] (0.021 [0.018] [0.073] [0.054] -0.016

55 TABLE 24 Mean deviations between experimental and predicted bubble-point pressures d,(P) (W) and vapour mole fractions d(y) (W) for the methane-n-nonane system (S17-3) a

T(K)

d,(P) C-l

b

4v)c RKS

PR

C-l

RKS

PR

223.2 248.2 273.2 298.2 323.2 348.2 373.2 423.2

4.9 1.9 3.2 3.8 2.0 3.1 0.3 2.9

31.6 24.3 20.9 14.8 11.6 9.3 10.3 5.7

42.1 29.9 23.8 17.3 13.6 10.6 10.9 5.2

0.74 0.37 0.64 0.44 0.11 0.73

1.01 0.78 0.51 0.70 0.09 0.77

0.91 0.64 0.28 0.34 0.11 0.82

Mean for system

2.8

15.0

17.4

0.54

0.65

0.54

a See Table 6 for reference. b RKS, Redlich-Kwong-Soave; PR, Peng-Robinson; ’ Vapour mole fractions > 0.01 and < 0.99.

for C-l, see text.

Binary mixtures containing saturated liquid and vapour We based our comparison$ and vapour on measurements TABLE

for binary mixtures containing saturated liquid reported by Shipman and Kohn (1966) for the

25

Mean relative compounds Compound

’

Methane [I] Methane (21 Methane [3] Propane [l] n-Nonane [4] n-Decane [l] n-Decane [5] Carbon dioxide [6] Hydrogen sulphide (71

deviations

d,(v)

(I)

between

experimental

and predicted

volumes

for pure

Temperature range (K)

Pressure range (atm.)

Number of points

C-l

Method b RKS

PR

278-511 102-180 273-623 278-511 311-511 311-511 298-358

I-680 4-335 16-400 I-680 2-677 14-680 I-962

184 21 155 184 75 154 55

0.59 1.26 0.54 2.06 1.80 I .49 1.34

1.67 1.30 1.93 4.93 15.19 14.91 17.02

2.61 10.11 1.95 4.46 2.93 2.7 I 4.67

220-420

I-800

280

1.30

3.00

2.00

278-444

I-680

202

1.59

3.02

3.74

a References for experimental values: [l] Sage and Berry (1971); 121 Goodwin and Prydz (1972); (3) Douslin et al. (1964); [4] Carmichael et al. (1953); [5] Snyder and Winnick (1970); [6] Angus et al, (1973); (71 Reamer et al. (1950). b RKS, Redhch-Kwong-Soave; PR, Peng-Robinson; for C-l, see text.

56

methane-n-nonane binary system. The comparisons were made using two different methods: (a) without vapour-liquid equilibrium estimations: volumes were calculated using experimental values for temperature and pressure, and for liquid and vapour compositions; (b) with vapour-liquid equilibrium estimations: here experimental temperatures and liquid compositions were used to calculate liquid volumes as a function of estimated pressure, and saturated-vapour volumes for the estimated equilibrium compositions under the estimated pressure. These estimations were performed using k,, values of 0.0495 for the Redlich-KwongSoave and 0.0530 for the Peng-Robinson equation. The best volume correction was obtained with quadratic combination rules for the parameter c in the C-l equation: c=

1 ’

cij =

cxixicij

(21)

J

( cicjp2

(22)

TABLE 26 Mean relative deviations d,(o) (R;) between experimental and predicted saturated liquid volumes for the methane-n-nonane system, using data from Shipman and Kohn (1966): A, without equilibrium calculations; B, with equilibrium calculations Number of points

Pressure range (atm.)

Method a

223.2

10

10-100

248.2

10

10-100

273.2

22

IO-315

298.2

22

10-315

323.2

22

10-315

348.2

22

10-310

373.2

10

10-100

423.2

10

10-100

A B A B A B A B A B A B A B A B

T 6)

’ RKS, Redlich-Kwong-Soave; b Without volume correction. ’ With volume correction.

PR, Peng-Robinson;

C-lb

c-1c

RKS

PR

8.55 8.51 9.29 9.27 8.81 8.77 7.67 7.59 7.69 7.70 7.70 7.74 11.31 11.32 9.74 9.81

1.55 1.51

15.94 16.03 15.83 15.90 13.44 14.39 13.16 14.29 13.00 14.09 13.05 13.98 13.52 13.59 14.03 14.08

5.78 5.90 5.61 5.70 3.88 4.20 3.60 3.88 3.43 3.59 3.47 3.61 2.66 2.72 3.17 3.19

1.65 1.64 1.37 1.33 1.37 I .28 1.20 1.19 I .08 1.10 0.83 0.84 0.64 0.57

for C-l, see text.

TABLE 27 Mean relative deviations d,(o) (W) between experimental and predicted saturated vapour volumes for the methane-n-nonane system, using data from Shipman and Kohn (1966): A, without equilibrium calculations; B, with equilibrium calculations

T WI

Number of points

Pressure range (atm.)

Method a

223.2

10

IO-100

248.2

10

10-100

273.2

15

IO-200

298.2

12

IO-140

323.2

11

10-120

348.2

11

IO-120

373.2

10

10-100

423.2

10

10-100

A B A B A B A B A B A B A B A B

” RKS, Redlich-Kwong-Soave; h Without volume correction. ’ With volume correction.

PR, Peng-Robinson;

C-lb

C-l’

RKS

PR

1.81 7.67 1.20 1.32 1.14 2.07 0.69 5.11 0.58 3.97 0.46 5.34 0.53 0.28 5.68 7.99

1.81 1.67 1.20 I .32 1.14 2.07 0.69 5.11 0.58 3.97 0.46 5.34 0.53 0.28 5.68 7.99

1.62 4.87 0.80 0.91 1.97 1.48 1.09 5.30 0.86 3.69 0.82 4.92 1.74 I .47 5.39 6.89

2.93 2.16 2.76 2.93 2.14 5.32 1.95 7.3 1

1.66 ’ 6.81 1.44 8.3 1 0.35 1.88 5.21 8.30

for C-l, see text.

The results of the comparisons for liquid volumes are presented in Table 26; the C-l correction leads to great improvement over both the Redlich-Kwong-Soave and the Peng-Robinson results. Table 27 shows the results for the saturated-vapour volumes. The vapour being almost pure methane, the volume correction is excessively small. In this case the C-l method gives no improvement over the other two. CONCLUSIONS

The proposed C-l equation of state leads to good phase-behaviour prediction for alkane mixtures up to the critical state, even in the case of binary systems involving a light with a heavy alkane, for which other equations give poor results. For mixtures of alkanes with nonalkanes the results are generally good, and often a little better than those given by the Redlich-Kwong-Soave or Peng-Robinson equations. Using the volume translations defined by Peneloux, the method also gives

58

good estimates tures.

of the volumetric

properties

of pure compounds

and mix-

ACKNOWLEDGEMENTS

The authors thank the Direction Technique for financial support.

G&-r&ale de la Recherche

Scientifique

et

LIST OF SYMBOLS

a, b b,

6

c, d k

m, ml, m2, m3 P R rl’

f-2

f,,

F2

T Tr V d

X,Y =c =: ib % w

parameters in eqn. (1) reduced parameters defined by eqns. (I-5) and (I-6) parameters for volume correction (eqns. (18) and (19)) binary interaction coefficient parameters in the Soave correlation pressure (atm.) gas constant (82.0562 atm. cm3 mall’ K-‘) parameters in eqn. (1) reduced parameters defined by eqns. (I-7) and (I-8) absolute temperature (K) reduced temperature molar volume as given by eqn. (I) (cm3 mol- ‘) corrected molar volume liquid, vapour mole fractions experimental critical compressibility factor pseudocritical compressibility factor defined by eqn. (17) parameter in eqn. (II-l) parameters in eqn. (2) parameter (0.77 in this work) defining the family of the cubic equation of state Pitzer’s acentric factor

Subscripts

critical components

C

id APPENDIX

I: THE GENERAL

i, j CUBIC

The general cubic equation

EQUATION

OF STATE

of state is given by

P = RT/( u - b) - a/( u2 + /3u + 8)

0-l)

59

If r, and r, are the roots of the quadratic

in volume

l?+pv+6=0

(I-2)

eqn. (I-l) can be written in the form P=RT/(v-b)-a/(u-r,)(u-r,) Using the classical conditions @P/au),=,

= (a2P/avz)T_C

(I-31 at the critical point, i.e., = 0

(I-4)

a, b, r, and r2 (or 4, b, ,B and 8) can be obtained in terms of the critical properties T, and PC and fwo other parameters, Q, and Z; (Abbott, 1973). If the reduced variables 6, b, i, and ?z are introduced, as defined by ci = L?P,/( RT,)2

(I-5)

6 = bP,/RT,

(I-6)

P, = r,P,/RT,

(I-7)

f2 = r2PC/R T,

(I-8)

then Li,=CJ;

(I-91

& = z: + sz, - 1

(I-10)

P,, = z; - D,[OS - (a, - 0.75)“2]

(I-l 1)

?22c= z,* - nJo.5

(I-12)

+ (a, - o.75)1’2]

Z,* = P,K*/RT,

(I-13)

In this work we have omitted the subscript c for the parameters 6, P, and P2 (and 6, r, and r2), which are temperature-independent. Anticipating that for many applications the value of Zz (either as implied by eqn. (I-4) or as fixed by the user) may not be equal to the actual critical compressibility factor, we have used the designation Zz instead of Z,. When Zz is not equal to Z,, an apparent critical volume u,* can be defined by eqn. (I-13). From eqns. (I-9)-(1-13) it can be seen that a cubic equation of state which fulfills the critical conditions will be defined by only two parameters, li2, and Zz: 3, will characterize the family of the equation of state, and Zz a particular member of the family. For each family, it is possible to define a basic term which corresponds to an equation with the parameter S equal to zero : P = RT/( u - b) - a/( o2 + /3v)

(I-14)

60

For a given value of 52,, two values of Z,* make S equal to zero: they are given by z; = 8, [0.5 & ( G2,- o.75)“*]

(I-15)

In fact, only the root with the minus sign has physical meaning This solution corresponds to r, = 0. So eqn. (I-3) becomes

(Z,* < 0.375).

P=RT/(W-f+l/u(u-r,)

(I-16)

The values of the characteristic state are presented in Table I. APPENDIX

II: CORRELATIONS

The C-I equation a = 0.456533a(

parameters

of some cubic equations

USED IN THE METHOD

of

BASED ON C-l

of state is defined by eqn. (I-3) with (II-l)

RT,)‘/P,

b = (Z; - 0.23) RT,/P,

(11-2)

r, = (Z; - 0.276106) RT,/P,

(11-3)

r, = (Z,* - 0.493894) RT,/P,

(11-4)

a= [I +m(1

- K”2)12

(11-5)

m = 0.381363 + 1.51188~ - 0.1993w2

(11-6)

z; = 0.319

(0 d 0.090)

(11-7)

Zz = 0.2910 + 0.4421 exp( -6.463610)

(o > 0.090)

(11-8)

The consistent

correction

for volumes (Peneloux

et al., 1982) is given by

u”= 0 + c cxixjci, i c,,

=

(11-g)

J

(II-IO)

(c,cy2

c, = d, RTJP,,

(11-11)

with di = 0.02802 - 0.02787~~ - 0.04421 exp( -6.46361~~)

(11-12)

if wi > 0.09; otherwise, di = 0. APPENDIX VALUES

III:

DEVIATIONS

BETWEEN

For the comparisons between used the following deviations.

CALCULATED

the calculated

AND

EXPERIMENTAL

and experimental

values we

61

Pressure

The percent root-mean-square

relative deviation,

given by

The percent mean relative deviation, given by

Composition

The percent mean deviation, 4.Y) = wwN,El&x,-Ycalcl

given by 0.01
Vohme

The percent mean relative deviation,

In these equations,

given by

NP, N, N,, NY and N, are the numbers

of points.

REFERENCES Abbott, MM., 1973. Cubic equations of state. Am. Inst. Chem. Eng. J., 19: 596-601. Akers, W.W., Burns, J.F. and Fairchild, W.R., 1954a. Low temperature phase equilibria: methane-propane system. Ind. Eng. Chem., 46: 2531-2535. Akers, W.W., Attwell, L.L. and Robinson, J.A., 1954b. Nitrogen-n-butane system. Ind. Eng. Chem., 46: 2539-2540. Ambrose, D. and Lawrenson, I.J., 1972. The vapour pressure of water. J. Chem. Thermodyn., 4: 755-761. Ambrose, D., Counsell, J.F. and Davenport, A.J., 1970. The use of Chebyshev polynomials for the representation of vapour pressures between the triple point and the critical point. J. Chem. Thermodyn., 2: 283-294. American Petroleum Institute, 1974. Selected Values of Properties of Hydrocarbons and Related Compounds. Res. Project 44, 23-2: K, KA, KB. American Petroleum Institute, Washington, DC.

62 Angus, S., Armstrong, B., De Reuck, K.M., Altunin, V.V., Gadetskii, O.G., Chapela, G.A. and Rowlinson, J.S., 1973. IUPAC Carbon Dioxide Tables. IUPAC Thermodynamic Tables Project Center. Pergamon Press, Oxford. Azarnoosh, A. and McKetta, J.J., 1963. Nitrogen-n-decane system in the two-phase region. J. Chem. Eng. Data, 8: 494-496. Baehr, H.D., Garnfost, H. and Poll& R., 1976. The vapour pressure of liquid ammonia: new measurements above 328 K and a rational vapour pressure equation. J. Chem. Thermodyn., 8: 113-119. Beaudouin, J.M. and Kohn, J.P., 1967. Multiphase and volumetric equilibria of the methane- n-decane binary system at temperatures between - 36 and 150°C. J. Chem. Eng. Data, 12: 189-191. Bender, E., 1971, Concerning the formulation of equations of state from which the saturation properties can be exactly calculated; illustrated with reference to methane. Kaltetech. Klimatisier., 23: 258-264. Besserer, G.J. and Robinson, D.B., 1973. Equilibrium phase properties of n-pentane-carbon dioxide system. J. Chem. Eng. Data, 18: 416-419. Besserer, G.J. and Robinson, D.B., 1975a. Equilibrium phase properties of i-butane-hydrogen sulfide system. J. Chem. Eng. Data, 20: 1l- 15. Besserer, G.J. and Robinson, D.B., 1975b. Equilibrium phase properties of nitrogen-hydrogen sulfide system. J. Chem. Eng. Data, 20: 157-161. Bierlein, J.A. and Kay, W.B., 1953. Phase equilibrium properties of system carbon dioxide-hydrogen sulfide. J. Chem. Eng. Data, 45: 618-624. Brewer, J., Rodewald, N. and Kurata, F., 1961. Phase equilibria for the propane-hydrogen sulfide system from the cricondentherm to the solid-liquid-vapor region. Am. Inst. Chem. Eng. J., 7: 13-16. Buhner, K., Maurer, G. and Bender, E., 1981. Pressure-enthalpy diagrams for methane, ethane, propane, ethylene and propylene. Cryogenics, 2 1: 157- 164. Butcher, K.L., Ramasubramanian, K.R. and Medani, M.S., 1972. Thermodynamic properties of the benzene and n-heptane system at elevated temperatures. J. Appl. Chem. Biotechnol., 22: 1139-1155. Carmichael, L.T., Sage, B.H. and Lacey, W.N., 1953. Volumetric behavior of n-nonane. Ind. Eng. Chem., 45: 2697-2699. Chu, T.C., Chen, R.J.J., Chappelear, P.S. and Kobayashi, R., 1976. Vapor-liquid equilibrium of methane-n-pentane system at low temperatures and high pressures. J. Chem. Eng. Data, 21: 41-44. Dastur, S.P. and Thodos, G., 1964. Vapour-liquid equilibrium studies for the ethane- n-pentane-n-heptane system at 150, 200, 300 and 350°F. Chem. Eng. Sci., 19: 9355952. Davalos, J., Anderson, W.R., Phelps, R.E. and Kidnay, A.J., 1976. Liquid-vapor equilibrium at 250.00 K for systems containing methane, ethane and carbon dioxide. J. Chem. Eng. Data, 21: 81-84. Douslin, D.R., Harrison, R.H., Moare, R.T. and McCullaugh, J.P., 1964. P-V-T relations for methane. J. Chem. Eng. Data, 9: 358-363. Elbishlawi, M. and Spencer, J.R., 1951. Equilibrium relations of two methane-aromatic binary systems at 150’F. Ind. Eng. Chem., 43: 1811-1815. Elliot, D.G., Chen, R.J.J., Chappelear, P.S. and Kobayashi, R., 1974. Vapor-liquid equilibrium of methane-n-butane system at low temperatures and high pressures. J. Chem. Eng. Data, 19: 71-76. Fredenslund, A. and Mollerup, J., 1974. Measurement and prediction of equilibrium ratios for the C,H,-CO, system. J. Chem. Sot., Faraday Trans. 1, 70: 1653-1660.

63 Freze, R. and Chevalier, J.L., 1978. Une equation d’ttat de Clausius modifike reprksentant 1’Ctat liquide. Can. J. Chem. Eng., 56: 375-381. Glanville, J.W., Sage; B.H. and Lacey, W.N., 1950. Volumetric and phase behavior of propane-benzene system. Ind. Eng. Chem., 42: 508-513. Goodwin, R.D. and Prydz, R., 1972. Densities of compressed liquid methane and the equation of state. J. Res., Natl. Bur. Stand. (U.S.), 76A: 81-101. Graboski, MS. and Daubert, T.E., 1978. A modified Soave equation of state for phase equilibrium calculations. 1. Hydrocarbon systems. Ind. Eng. Chem., Proc. Des. Dev., 17: 443-448. Grauso, L., Fredenslund, A. and Mollerup, J., 1977. Vapour-liquid equilibrium data for the systems C,H,-N,, C,H,-Nz, C,Hs-N, and C,H,-N,. Fluid Phase Equilibria. 1: 13-26. Hamam, S.E.M. and Lu, B.C.Y., 1974. Vapor-liquid equilibrium in the ethane-carbon dioxide system. Can. J. Chem. Eng., 52: 283-286. Haman, S.E.M. and Lu, B.C.Y., 1976a. Isothermal vapor-liquid equilibria in the binary system propane-carbon dioxide. J. Chem. Eng. Data, 21: 200-204. Haman, S.E.M. and Lu, B.C.Y., 1976b. Phase equilibria for the system propane-ethane-carbon dioxide. Can. J. Chem. Eng., 54: 333-336. Harmens, A., 1977. A cubic equation of state for the prediclion of N,-Ar-0, phase equilibrium. Cryogenics, 17: 5 19-522. Im, U.K. and &rata, F., 1971. Phase equilibrium of carbon dioxide and light paraffins in the presence of solid carbon dioxide. J. Chem. Eng. Data, 16: 295-299. Kahre, L.C., 1974. Low-temperature K data for methane-n-butane. J. Chem. Eng. Data, 19: 67-71. Kalra, H., Robinson, D.B. and Krishnan, T.R., 1977a. The equilibrium properties of the ethane-hydrogen sulfide system at subambient temperatures. J. Chem. Eng. Data, 22: 85-88. Kalra, H., Robinson, D.B. and Besserer, G.J., 1977b. The equilibrium phase properties of the nitrogen-n-pentane system. J. Chem. Eng. Data, 22: 215-218. Kalra, H., Kurata, H., Robinson, D.B. and Ng, H.J., 1978. Equilibrium phase properties of the carbon dioxide-n-heptane system. J. Chem. Eng. Data, 23: 317-321. Kennedy, J.T. and Thodos, G., 1960. Reduced density correlation for carbon dioxide: gaseous and liquid state. J. Chem. Eng. Data, 5: 293-297. Kidnay, A.J., Miller, R.C., Parrish, W.R. and Him, M.J., 1975. Liquid-vapor phase equilibria in the N,-CH, system from 130 to 180 K. Cryogenics, 15: 531-540. Kohn, J.P. and Bradish, W.F., 1964. Multiphase and volumetric equilibria of the methane-noctane system at temperatures between - 110 and 150°C. J. Chem. Eng. Data, 9: 5-8. Kreglewski, A. and Kay, W.B., 1969. Critical constants of conformed mixtures. J. Phys. Chem., 73: 3359-3366. Lehigh, W.R. and McKetta, J.J., 1966. Vapor-liquid equilibrium in the ethane-nbutane-nitrogen system. J. Chem. Eng. Data, 11: 180-182. Mehra, V.S. and Thodos, G., 1965a. Vapor-liquid equilibrium in the ethane-n-butane system. J. Chem. Eng. Data, 10: 307-309. Mehra, V.S. and Thodos, G., 1965b. Vapor-liquid equilibrium in the ethane-n-heptane system. J. Chem. Eng. Data, 10: 211-214. Mehra, VS. and Thodos, G., 1966. Vapor-liquid equilibrium constants for the ethane-nbutane-n-heptane system at 150, 200 and 250°F. J. Chem. Eng. Data, 1 I: 365-372. Mehra, V.S. and Thodos, G., 1968. Vapor-liquid equilibrium constants for the ethane-nbutane-n-heptane system at 300 and 350°F. J. Chem. Eng. Data, 13: 155-160.

64

Miisovsky, J. and Wichterle, I., 1975. Vapor-liquid equilibria in the ethane-propane system at low pressures. Collect. Czech. Chem. Commun., 40: 365-370. Muirbrook, N.K. and Prausnitz, J.M., 1965. Multicomponent vapor-liquid equilibria at high pressures. 1. Experimental study of the nitrogen-oxygen-carbon dioxide system at 0°C. Am. Inst. Chem. Eng. J., 11: 1092- 1102. Nagahama, K., Konishi, H., Ho&no, D. and Hirata, hf., 1974. Binary vapor-liquid equilibria of carbon dioxide-light hydrocarbons at low temperature. J. Chem. Eng. Jpn.. 7: 323-328. Nishiumi, H. and Saito, S., 1975. An improved generalized BWR equation of state applicable to low reduced temperatures. J. Chem. Eng. Jpn., 8: 356-360. Ohgaki, K. and Katayama, T., 1976. Isothermal vapor-liquid equilibrium data for binary systems containing carbon dioxide at high pressures. J. Chem. Eng. Data, 21: 53-55. Ohgaki, K., Sano, F. and Katayama, T., 1976. Isothermal vapor-liquid equilibrium data for binary systems containing ethane at high pressures. J. Chem. Eng. Data, 21: 55-58. Olds, R.H., Reamer, H.H., Sage, B.H. and Lacey, W.N., 1949. Phase equilibria in hydrocarbon systems: the n-butane-carbon dioxide system. Ind. Eng. Chem., 41: 475-482. Peneloux, A., Rauzy, E. and Freze, R., 1982. A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilibria, 8: 7-23. Peng, D.Y. and Robinson, D.B., 1976. A new two-constant equation of state. Ind. Eng. Chem. Fundam., 15: 59-64. Peters, S. and Eicke, H.F., 1970. Das phasengleichgewicht in den systemen N, - n-heptan, N, -2,2,4_trimethylpentan und Na-methylcyclohexan bei hoheren drucken und temperaturen. Ber. Bunsenges. Phys. Chem., 74: 190-194. Poettman, F.H. and Katz, D.L., 1945. Phase behavior of binary CO,-paraffin systems. Ind. Eng. Chem., 39: 5161-5162. Poston, R.S. and McKetta, J.J., 1966. Vapor-liquid equilibrium in the nitrogen-n-hexane system. J. Chem. Eng. Data, 11: 362-365. Price, A.R. and Kobayashi, R., 1959. Low temperature vapor-liquid equilibrium in light hydrocarbon mixtures: methane-e&me-propane system. J. Chem. Eng. Data, 4: 40-52. Reamer, H.H. and Sage, B.H., 1962. Volumetric and phase behavior of the ethane- n-decane system. J. Chem. Eng. Data, 7: 161-168. Reamer, H.H. and Sage, B.H., 1963. Volumetric and phase behavior of the n-decane-carbon dioxide system. J. Chem. Eng. Data, 8: SOS-51 3. Reamer, H.H. and Sage, B.H., 1966. Volumetric and phase behavior of the propane- n-decane system. J. Chem. Eng. Data, 11: 17-24. Reamer, H.H., Olds, R.H., Sage, B.H. and Lacey, W.N., 1943. Phase equilibria in hydrocarbon systems: the methane-n-decane system. Ind. Eng. Chem., 37: 822-828. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1950. Volumetric behavior of hydrogen sulfide. Ind. Eng. Chem., 42: 140-143. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1951a. Volumetric and phase behavior of the methane-hydrogen sulfide system. Ind. Eng. Chem., 43: 976-981. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1951b. Volumetric and phase behavior of the methane- n-butane- n-decane system. Ind. Eng. Chem., 43: 1436-1444. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1953a. Volumetric and phase behavior of n-pentane-hydrogen sulfide system. Ind. Eng. Chem., 45: 1805-1809. Reamer, H.H., Selleck, F.T., Sage, B.H. and Lacey, W.N., 1953b. Volumetric and phase behavior of n-decane-hydrogen sulfide system. Ind. Eng. Chem., 45: 1810-1812. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1956. Volumetric and phase behavior of the methane-n-heptane system. Ind. Eng. Chem., Eng. Data Ser., 1: 29-42.

65 Reamer, H.H., Sage, B.H. and Lacey, W.N., 1960. Volumetric and phase behavior of ethane-n-pentane system. J. Chem. Eng. Data, 5: 44-48. Roberts, L.R., Wang, R.H., Azamoosh, A. and McKetta, J.J., 1962. Methane-n-butane system in the two-phase region. J. Chem. Eng. Data, 7: 484-485. Rodrigues, A.B.J., McCaffrey, D.S. and Kohn, J.P., 1968. Heterogeneous phase and volumetric equilibrium in the ethane- n-octane system. J. Chem. Eng. Data, 13: 164- 168. Roof, J.G. and Baron, J.D., 1967. Critical loci of binary mixtures of propane with methane, carbon dioxide and nitrogen. J. Chem. Eng. Data, 12: 292-293. Sage, B.H. and Berry, V.M., 1971. Phase Equilibria in Hydrocarbon Systems: Behavior of the Methane-Propane-n-Decane System. Monogr., Res. Project 37, American Petroleum Inslitute, Washington, DC. Sage, B.H., Hicks, B.L. and Lacey, W.N., 1940. Phase equilibria in hydrocarbon systems: the methane-n-butane system in the two-phase region. Ind. Eng. Chem., 34: 6158-6160. Sage, B.H., Reamer, H.H., Olds, R.H. and Lacey, W.N., 1942. Volumetric and phase behavior of methane-n-pentane system. Ind. Eng. Chem., 34: 1108- 1117. Shipman, L.M. and Kohn, J.P., 1966. Heterogeneous phase and volumetric equilibrium in the methane-n-nonane system. J. Chem. Eng. Data, 11: 176-180. Snyder, P.S. and Winnick, J., 1970. The pressure, volume and temperature properties of liquid n-alkanes at elevated pressures. Proc. 5th Symp. on Thermophysical Properties, Am. Sot. Mech. Eng., ASME, New York, pp. 115- 129. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197-1201. Sobocinski, D.P. and Kurata, F., 1959. Heterogeneous phase equilibria of the hydrogen sulfide-carbon dioxide system. Am. Inst. Chem. Eng. J., 5: 545-55 1. Somait, F.A. and Kidnay, A.J., 1978. Liquid-vapor equilibria at 270.00 K for systems containing nitrogen, methane and carbon dioxide. J. Chem. Eng. Data, 23: 301-305. Stryjek, R., Chappelear, P.S. and Kobayashi, R., 1974a. Low temperature vapor-liquid equilibria of nitrogen-methane system. J. Chem. Eng. Data, 19: 334-339. Stryjek, R., Chappelear, P.S. and Kobayashi, R., 1974b. Low temperature vapor-liquid equilibria of nitrogen-ethane system. J. Chem. Eng. Data, 19: 340-343. Teja, A.S. and Singh, A., 1977. Equations of state for ethane, propane and n-butane. Cryogenics, 17: 591-596. Usdin, E. and McAuliffe, J.C., 1976. A one-parameter family of equations of state. Chem. Eng. Sci., 31: 1077-1084. Van Horn, L.D. and Kobayashi, R., 1967. Vapor-liquid equilibria of light hydrocarbons at low temperatures and elevated pressures in hydrocarbon solvents. J. Chem. Eng. Data, 12: 294-303. Veirosta, J. and Wichterle, I., 1974. The propane-pentane system at high pressures. Collect. Czech. Chem. Commun., 39: 1246- 1248. Wagner, W., 1973. New vapour pressure measurements for argon and nitrogen and a new method for establishing rational vapour pressure equations. Cryogenics, 13: 470-482. Wang, R.H. and McKetta, J.J., 1964. Vapor-liquid equilibrium of the methane-nbutane-carbon dioxide system at low temperatures and elevated pressures. J. Chem. Eng. Data, 9: 30-35. West, J.R., 1948. Thermodynamic properties of hydrogen sulfide. Chem. Eng. Progr., 44: 287-292. Wichterle, I. and Kobayashi, R., 1972a. Vapor-liquid equilibrium of methane-ethane system at low temperatures and high pressures. J. Chem. Eng. Data, 17: 9-12. Wichterle, I. and Kobayashi, R., 1972b. Vapor-liquid equilibrium of methane-propane system at low temperatures and high pressures. J. Chem. Eng. Data, 17: 4-9.

66 Wichterle, 1. and Kobayashi, R., 1972~. Vapor-liquid equilibrium of methane-ethane-propane system at low temperatures and high pressures. J. Chem. Bng. Data, 17: 13-18. Wiese, H.C., Jacobs, J. and Sage, B.H., 1970a. Phase equilibria in hydrocarbon systems: phase behavior in the methane-propane-n-butane system. J. Chem. Eng. Data, 15: 82-91. Wiese, H.C., Reamer, H.H. and Sage, B.H., 1970b. Phase equilibria in hydrocarbon systems: phase behavior in the methane-propanen-decane system. J. Chem. Eng. Data, 15: 75-82. Yarborough, L., 1972. Vapor-liquid equilibrium data for multi-component mixtures containing hydrocarbon and nonhydrocarbon components. J. Chem. Eng. Data, 17: 1299133 (microfilm edition). Zenner, G.H. and Dana, L.I., 1963. Liquid-vapor equilibrium compositions of carbon dioxide-oxygen-nitrogen mixtures. Chem. Eng. Progr., Symp. Ser., 59: 36-41.