Vapor—liquid equilibria at high temperatures and pressures in binary mixtures containing H2, CH4 and CO2 with high boiling hydrocarbons: experimental equipment and results

155

Vapor- Liquid Equilibria at High Temperatures and Pressures in Binary Mixtures Containing H 2 , CH 4 and CO 2 with High Boiling Hydrocarbons: Experimental Equipment and Results Dampf- Fliissigkeitsgleichgewichte bei hohen Temperaturen und Driicken in Zwei-Stoffgemischen aus H 2 , CH4 und CO 2 mit hoch siedenden Kohlenwasserstoffen: Apparatur und experimentelle Ergebnisse UWE WATERLING,* DAQING ZHENG** and HELMUT KNAPP Institute of Thermodynamics and Reaction Engineering, Technical University of Berlin, Berlin (F.R.G.) (Received August 31, 1990; in final form November 7, 1990)

Abstract An apparatus was designed, built and operated with the purpose of studying vapor-liquid equilibria at temperatures 450 < T < 650 K and pressures 1 < p < 35 MPa or critical pressure. The liquid in the high pressure equilibrium cell was agitated by a magnetically driven mixer. Samples can be taken from the vapor and the liquid phase. The composition was determined by a combination of gravimetric and volumetric measurements. Test measurements were performed on binary systems investigated previously by others in order to check the reproducibility and accuracy of the measurements.

Kurzfassung Eine Apparatur wurde entworfen, gebaut und in Betrieb genommen mit der Absicht, Fliissigkeits-DampfGleichgewichte zu untersuchen im Bereich hoher Temperaturen 450 < T < 650 K und hoher Driicke 1 < p < 35 MPa oder kritischer Druck. Die Fliissigkeit in der Hochdruckzelle wurde mit einem Riihrer gemischt. Aus cler fliissigen oder dampff6rmigen Phase k6nnen durch Kapillaren Proben entnommen werden. Die Zusammensetzung wird durch eine Kombination gravimetrischer und volumetrischer Messungen bestimmt. Es wurden biniire Gemische aus H 2 , CH 4 und CO 2 mit Heptan, Toluol und Tetralin untersucht in einem Druckund Temperaturbereich, in dem bereits ver6ffentlichte Ergebnisse verfiigbar waren. Durch Vergleich der Ergebnisse konnte die Reproduzierbarkeit und die Genauigkeit der Messungen iiberpriift werden.

Synopse Zweifellos konnen die Motoren von Land-, Seeund Luftfahrzeugen am einfachsten, wirtschaftlichsten und wirkungsvollsten mit jiussigen aus Erdol gewonnenen Brennstoffen betrieben werden. Es ist zu erwarten, dajJ die Menschen auch dann, wenn die Olvorrate verbraucht sein werden, den dringenden Wunsch verspuren, sich moglichst schnell uber grojJe Entfernungen hin und her bewegen zu konnen. Es gibt mehrere Szenarios zur Versorgung der Motoren mit

*Present address: Hoechst AG, 6230 Frankfurt-on-Main 80, F.R.G. **Permanent address: University of Petroleum, Beijing, China. 0255-2701/91/$3.50

Energie: durch verjiussigte Kohle, durch Methanol, durch jiussigen Wasserstoff oder durch elektrische Energie. Sowohl wirtschaftlich als auch energetisch ist verjiussigte Kohle der giinstigste Energietrager. Deshalb ist es wichtig, unser 'Know-how' uber Kohleverjiussigung zu bewahren und zu verbessern und deshalb wurde von der Bergbauforschung GmDH eine Pi/otanlage und von der Ruhrkohle-AG und der VEBAOel AG eine 200 Tonnen pro Tag Produktionsanlage gebaut und betrieben auf der Basis des historischen Bergius - Pier- Verfahrens. Bei der Auswertung der Betriebsergebnisse zeigte es sich, dajJ es vorteilhaft ware, im Labor zusatzlich in speziellen Apparaturen Phasengleichgewichte unter einfacheren, genau definierten Bedingungen zu untersuchen. Zur Berechnung der Vorgange in den Warmeubertragern und Abscheidern,

Chern. Eng. Process., 29 (1991) 155-164

© Elsevier Sequoia/Printed in The Netherlands

156 in denen das gasformige Produkt aus dem HydrierReaktor abgekiihlt, teilweise kondensiert und in Fliissig- und DampJfraktionen getrennt werden soli, wird Information benotigt iiber die kalorischen Eigenschaften des Gemisches und iiber die FliissigkeitsDampf-Gleichgewichte. PVTX-Zustandsgleichungen enthalten aile Informationen, um mit Hilfe von exakten Gleichungen den EinflufJ des Druckes auf die Enthalpie (h - hOy und auf das chemische Potential bzw. die Fugazitat berechnen zu konnen. Die in den Zustandsgleichungen fiir reine StoJfe enthaltenen stoJfspezijischen Koefjizienten konnen bestimmt werden durch Anpassung an experimentelle PVT-Daten und Dampfdruckkurven. Die gemischspezijischen binaren Koefjizienten in den Mischungs- und Kombinations-regeln konnen durch Anpassung an experimentelle VLE-Daten ermittelt werden. Zur Untersuchung von Phasengleichgewichten bei hohen Temperaturen und hohen Driicken wurde eine Apparatur entwickelt, gebaut und betrieben. Als Ergebnis einer eingehenden Literaturrecherche wurde ein Programm vorgeschlagen zur experimentellen Untersuchung typischer binarer Gemische aus tiefsiedenden StoJfen (H2 , CH4 , CO 2 ) und hochsiedenden KohlenwasserstoJfen (vgl. Tabelle 1). 1m Bericht wird die Konstruktion und die Betriebsweise der Versuchsapparatur beschrieben (Bilder ]-3, Tabelle 2). Die Ergebnisse der Testmessungen an den Systemen C0 2 1 Toluol, N 2 ln-Heptan und CO 2 I Tetralin werden vorgestellt (Bilder 4-6, Tabellen 4-6) und mit veroJfentlichten MefJwerten verglichen.

Introduction There is no question that the human species has the urgent desire to travel in passenger cars or boats or airplanes. These vehicles are most efficiently driven by engines burning liquid fuels. Sooner or later the oil resources will be depleted (even if the drivers would be willing to lose a little prestige and save fuel by using 30 kW diesel engines in their passenger cars, consuming only I gallon per 60 miles). Assuming the human species then still desires to drive automobiles there are various alternatives: the engines could be powered by liquefied coal, by electric power, or by hydrogen. Liquefied coal is considerably more efficient with respect to energy and that also means more economical. Therefore, we should be prepared to maintain and improve our knowledge about coal liquefaction processes. For this purpose Ruhrkohle-AG and VEBAGel AG built and operated a 200 tonne per day coal liquefaction plant in Bottrop, F.R.G., based on the historical Bergius-Pier process. Cooperating with this joint venture, the Institute of Thermodynamics and Reaction Engineering, Berlin, studied the possibilities of developing a basis for process design. It became obvious that the operating data taken at the pilot plant were not sufficient for the development or improvement of correlations to

calculate the equilibrium conditions in the heat exchangers and separators where the product stream from the hydrogenation reactor is cooled down, partially condensed and separated into various fractions. It was recommended that experiments should be peformed in a specially designed high pressure, high temperature equilibrium cell and information obtained first for binary mixtures under conditions where thermodynamic equilibrium can be reached and where no chemical reactions take place.

Definition of the problem The product leaving the reactor where coal is hydrogenated at high pressure (25-35 MPa) and at high temperature (350-450°C) consists mainly of the surplus recycle hydrogen (60-80 mol %), aliphatic, naphthenic and aromatic hydrocarbons from a wide boiling range (from CH 4 up to C 30 ), some hydrocarbons with oxygen, nitrogen, sulfur, etc., water vapor and inorganic substances. In order to be able to do process calculations, the engineer requires to know the distribution coefficients of all components between the coexisting phases and the calorific properties of the mixtures. The mixtures are so complex that we cannot expect success in developing correlations describing such mixtures. It is therefore reasonable to begin with correlations for mixtures consisting of only a few of the major components. The usual proven procedure is based on knowledge of volumetric and calorific properties: (1) an equation of state describing PVTX properties; and (2) the heat capacity of the standard state, c~(T).

With an equation of state the effect of pressure on the properties of fluid mixtures can be calculated in accordance with rigorous rules, that is, the following properties can be determined: (1) the fugacity coefficients of components in mixtures, cPi; (2) the enthalpy departures, h - hO. The characteristic parameters in equations of state describing the pure components can be found by fitting to experimental volumetric data including the vapor pressure curve. The characteristic parameters describing the mixtures can be determined with mixing and combination rules. These rules, specifically the combination rules, contain empirical binary parameters that can be found by fitting to experimental vapor-liquid equilibrium (VLE) data. A successful selection of mixing and combination rules must be based first on experimental results of binary mixtures in the temperature and pressure range of interest. It was, therefore, decided that an apparatus suitable for such measurements should be built. In this report, we describe the experimental equipment, the instrumentation, and the analysis procedure; we first present the experimental results that can be compared with published information.

157

Experimental program

requires the installation of complicated recycle pumps.

Previously published information was collected by a literature search. The VLE of a great number of mixtures containing H 2 , CH 4 and CO 2 , and hydrocarbons such as hexane, heptane, octane, decane, hexadecane, 2,2,4-trimethylpentane, cyclohexane, methylcyclohexane, bicyclohexyl, benzene, toluene, tetrahydronaphthalene (Tetralin), diphenylmethane, l-methylnaphthalene, 9,10-dihydrophenanthrene, mcresol, m-xylene, quinoline, thionaphthene and some of the gas-oil and coal-oil mixtures, have been studied at high temperatures and pressures (see ref. 1 for an overview, ref. 2 for a review and refs. 3-44). Based on the conclusions from the literature search, a new experimental program was suggested (see Table 1). In this report, results from the investigation of the first three binary systems including the comparison with the published data will be presented. There are three principal methods used to operate equilibrium cells. Method I. The vapor and liquid fractions are pumped at constant flow through a mixing chamber where equilibrium is established at a controlled temperature and pressure. Then the liquid and vapor phases are separated and analysed chemically. Method II. The vapor and/or the liquid fraction are recirculated through the high pressure equilibrium cell. Samples can be withdrawn from the recycle streams and analyzed chemically. Method III. A high pressure equilibrium cell is filled with the components. The liquid in the cell is stirred until equilibrium is established. Samples of the vapor and liquid fractions are withdrawn through capillaries and analyzed chemically. The third method was chosen, as the first method needs a large supply of materials and the second

Experimental The entire apparatus consists of three sections: the supply system for the feed, the high pressure equilibrium cell, and the sampling system. Whenever samples are removed from the vapor or liquid phase in the cell, the formation of two phases is unavoidable due to the changes in temperature and/or pressure. Special attention must, therefore, be paid to the sampling system in order to ensure reliable chemical analyses. A schematic flow diagram of the supply system for the feed, the thermostat and the equilibrium cell is shown in Fig. 1. The cell, 1, is located in the stainless steel box, 0, insulated with rock wool. The electric heater, 6, is regulated by a temperature controller, TIC, while the air is circulated by a ventilator, 7. The liquid in the cell is agitated by a mixer, 2, that is driven via a magnetic clutch by an electric motor, 3. Samples can be withdrawn from the cell through capillaries, 4. The cell can be filled with gaseous and liquid substances through the supply lines,S. Gaseous substances such as hydrogen, methane or carbon dioxide can be taken from storage bottle 12 to fill high pressure vessel 10, where they are cooled or liquefied with liquid nitrogen. High pressure vessel 11 serves as a standby volume to avoid extreme pressures. After closing valve VIS the pressure in storage vessel 10 can be increased by warming the vessel to room temperature. The gaseous substances can then be transferred through the quick connection 9 and feedline 5 into the equilibrium cell. In addition, the feed gas can be compressed from the gas bottle, 12,

TABLE 1. Experimenal program Component

T(K)

P(MPa)

(I)

(2)

Min-Max

Min-Max

H2 H2 H2 H2 H2 H2 H2 CH 4 CH 4 CH 4 CH 4 CH 4 CO 2 CO 2 CO 2 CO 2 CO 2

Quinoline m-Cresol Tetralin Diphenylmethane Tetradecane SR oil" Tetralin/Quinoline Quinoline m-Cresol Tetralin Diphenylmethane Tetradecane Quinoline m-Cresol Tetralin Diphenylmethane Tetradecane

500-600 600-667 550-600 600-670 480-600 602 550 600-666 600 600 600 517-600 600-666 600-611 544-600 600 490-600

1-29 1-30 1-32 1-33 1-32 1-33 3-32 1-31 1-30 1-22 1-33 1-32 1-23 3-20 1-27 1-28 1-24

"SR oil = Shell refined oil.

Experimental points

17 19 46 16 49 12 32 35 II 19 25 48 33 20 67 18 53

Fig. I. Schematic flow diagram of the equilibrium cell, thermostat and feed supply system: I, high pressure equilibrium cell; 2, mixer; 3, electric motor with magnetic coupling; 4, capillaries for sampling; 5, filling line; 6, heaters; 7, ventilator; 8, rupture disc; 9, high pressure fast connection; 10, cryogenic pump with liquid nitrogen Dewar; II, gas storage vessel; 12, gas bottle; 13, high pressure reciprocating compressor with pneumatic actuator; IS, dosimetric precision displacement pump; 16, liquid storage containers; 17, vacuum pump.

directly into the equilibrium cell with the reciprocating compressor, 13. Liquid substances are transferred from storage bottles 16 with a plunger type of dosimetric pump, 15, through the lower feedline, 5, into the cell. The entire system or various subsections can be evacuated by the two-stage vacuum pump, 17, into the vent system. Figure 2 shows the mechanical design of the high pressure equilibrium cell. It is built of nickel alloy steel and designed for a pressure of 40 MPa at 770 K. Tightness of the cell was accomplished by pressing the edge of the lid against the conical surface of the cell. After equilibrium is reached (1-2 hours), the mixer in the cell is stopped. First the liquid sample is taken, as the pressure changes very little during sampling. Afterwards, the vapor sample is withdrawn within approximately 3 minutes. Samples from the cell can be transferred through the heated capillary tubing, 4, to the sampling system (Fig. 3). The line can be purged through the shutoff valves V5A and V5B through the cold trap into the vent system. The vapor and liquid systems are identical; however, depending on the vapor-liquid ratio of the separated sample, the volumes of the calibrated

vessels 20A and 20B can be changed. After opening the shutoff valves VIA and VlB, the sample is admitted with needle valves V2A and V2B. Immediately after expansion, vapor will either flash off from the liquid sample or liquid will condense from the vapor sample. The liquid fraction is collected in the cold traps l8A and 18B. The vapor accumulates in the storage vessels 20A and 20B. Withdrawal is stopped after the pressure in the previously evacuated sample system has increased to approximately atmospheric pressure. As the volatilities of the two components are extremely different, the high boiling liquid hydrocarbon will all be collected in the liquid traps 18A and 18B (the partial pressure of the hydrocarbon is so low that the amount of hydrocarbon in the vapor phase can be neglected). The low boiling gas will be collected in the available volume of the sampling system (the amount dissolved in the liquid hydrocarbon can be neglected). The mass of the liquid fraction, m L , is measured on precision scales. The volume of the sampling system V S was calibrated accurately. The temperature T and pressure pare measured and the amount of light component nLC and of heavy component nHC can be determined

159

.

x

~~

\

Fig. 2. Mechanical design drawing of the equilibrium cell.

where M is the molar mass, R the gas constant, z the compressibility factor, and vL is the specific volume of the heavy component.

ments or the system caused by calibration, fluctuation, sample handling, etc., are indicated. The substances, suppliers and purities, as specified by the suppliers, are shown in Table 3. In addition, samples were taken and analyzed by gas chromatography to check the specified purity. After the experiment had been completed, liquid samples were taken from the equilibrium cell to check for possible chemical decomposition of the hydrocarbons.

Instrumentation

Results

The instruments, the methods of measurement and the manufacturers are listed in Table 2. In addition, the sensitivity and accuracy of the instru-

The T, p, X, Y data determined in the experiments are listed in Tables 4-6. The K values and relative volatilities 1X 12 are also listed. The experimental

according to the following equations: nHc =mLjMHc

p(V S _

mLv L)

( 1)

nLC

=

zRT

(2)

Xi

or Yi

= nJ'Lnj

(3)

160 vapor sample

liquid sample

(4)

120Al

sample lines

( 2081

heater--

Y2A

19A

YIA

Y18

18A

188

198

YS8

YSA

vent

Fig. 3. Schematic flow diagram of the sample handling system: 18A, 18B, separators for liquid fraction; 19A, 19B, cold traps; 20A, 20B, gas storage vessels with calibrated volumes.

TABLE 2. Sensitivity and inaccuracy of instruments and systems

TABLE 3. Purity and supplier of feed gas and liquid

Instrument Method Manufacturer

Sensitivity Accuracy

Substance

Purity

Supplier

T-sensor PT 100 Q Electric resistance Digital multimeter Prema

0.1 K 0.2K

Nitrogen Carbon dioxide n-Heptane Toluene Tetralin

99.95% 99.5% > 99.7% >99% >98%

Linde Rommenh611er Fluka Riedel de Haen Henkel

Pressure transducer Electric capacity

0.01 MPa 0.05 MPa

Precision scale Electric force compensation Sartorius

0.0001 g 0.0005 g

TABLE 4. Vapor-liquid equilibrium data for the carbon dioxide( 1)-toluene(2) system at T = 476.27

± 0.2 K

x, (mol/mol)

y, (mol/mol)

K, (y, /x,)

K2

1X l2

(MPa)

(Y2/x 2 )

(K] /K2 )

1.71 2.29 2.84 3.60 5.76 7.44 8.99 12.53 14.07 14.72

0.030 0.053 0.067 0.095 0.178 0.219 0.274 0.408 0.463 0.528

0.418 0.541 0.624 0.686 0.761 0.788 0.795 0.785 0.765 0.726

13.8 10.3 9.32 7.22 4.28 3.60 2.90 1.92 1.65 1.38

0.601 0.484 0.403 0.347 0.291 0.272 0.282 0.364 0.438 0.580

23.0 21.3 23.1 20.8 14.7 13.2 10.3 5.27 3.76 2.38

p

161 TABLE 5. Vapor-liquid equilibrium data for the nitrogen( I) -heptane(2) system at T

453.15

=

± 0.2 K

XI (mol/mol)

YI

a l2

(mol/mol)

KI (YI/XI)

K2

(MPa)

(Y2/x 2 )

(K I /K2 )

1.52 2.1 I 3.72 5.63 7.15 9.35 11.96 16.20 20.69 23.06 25.09 27.66

0.Dl8 0.030 0.062 0.098 0.131 0.178 0.229 0.312 0.41 I 0.447 0.521 0.580

0.507 0.656 0.809 0.850 0.870 0.888 0.895 0.892 0.876 0.868 0.846 0.812

28.8 22.0 13.0 8.63 6.66 5.00 3.92 2.86 2.13 1.94 1.63 1.40

0.501 0.355 0.203 0.167 0.150 0.137 0.136 0.157 0.210 0.239 0.321 0.447

57.5 62.0 64.0 51.7 44.4 36.5 28.8 18.2 10.1 8.1 I 5.08 3.13

p

TABLE 6. Vapor-liquid equilibrium data for the carbon dioxide( I) - Tetralin(2) system at T p

(MPa)

XI (mol/mol)

1.54 3.05 4.09 5.14 5.70 8.16 9.92 10.66 11.29 13.37 15.27 17.12 19.41 2I.I4 22.50 26.60

0.020 0.055 0.078 0.103 0.126 0.175 0.212 0.229 0.241 0.284 0.331 0.371 0.425 0.468 0.500 0.645

KI (YI/X I )

K2

a l2

(Y2/x 2 )

(KdK2 )

0.611 0.804 0.839 0.873 0.886 0.907 0.915 0.915 0.909 0.913 0.912 0.907 0.897 0.889 0.878 0.803

30.1 14.6 10.7 8.47 7.03 5.18 4.32 4.00 3.77 3.21 2.75 2.44 2.11 1.90 1.76 1.25

0.397 0.207 0.175 0.142 0.131 0.1 13 0.108 0.111 0.119 0.122 0.132 0.147 0.179 0.208 0.244 0.554

75.8 70.5 6I.I 59.6 53.7 45.8 40.0 36.0 31.7 26.3 20.8 16.6 11.8 9.13 7.21 2.26

e

CO 2

6

2

n

10 I

"" ~

c

V

V

0

V

V

0 ~

e

"'0

6

o'()

CD

6

..

~

E :J

6

10

:;

~

2

0 .D 10 e

...:c It

V

II

:J
0

0

14

12

0

~

0

0

V

..... ..... 0

V

V

476.27 477.04

0

II

a

16

T• K 0 V

">..

544. I 5 ± 0.2 K

(mol/mol)

YI

10 2

.....

=

0

a':

0

V

CD

~

~

l!J

6

V

0

0 V

V

V

0

0

"-'

8
'10 'lOQ

4

2

Cjie 3

4 5

0

IY

(7

2

2

0

VJ~

2

3

4 5

0

0

0 ~

0 V

a

0.0

0.2

C-iie(2J Fig. 4. K -p and p -X, Y diagrams of a carbon dioxide(l) -toluene(2) mixture at T

0.4

0.6

0.6

1.0

Concentration "I end YI' Mol/Mol

Pressure, MP.

=

CO 2 (1)

476.27 K: 0, this work; \l, Ng and Robinson [30].

162 N2

V

~~

2

"

~ >.

0

V1

•

t;) _

10 I

[!IF!Jf

8

6

.,z

...c II

4

Y

~~

2

0 ~

100

e

8 6 4

1;

2

II

0 0

:l aw

'.

3

~

Q..

.. ...L.. :l

C7HI6 2

~

,0

Y

." O.

~

.,p. <;0

~ vov 0.40

v5

0.60

Concen~re~ion

C7HI6

Y

~ vO 0.20

Pressure, MPa

•

0

~

0 0.00

3. 5

~

~

WJ

9

3

Y

00

I

12

6

~

Y ~

15

• Y.'

4 5

0

18

2

3

.V

<;;00

vO

W

24

~ 8 B ~~ofl.

8 6

.~

27

:E:

,

w

21

!2l

uri

• • ••

30

II

c::o

i3""

:l

453.15 453.15 453.15 453.15 455.35

;

f'Jv

33

T• K 0

1.00

0.80

"I end YI' mol/moL

N2 (1 J

12J

Fig. 5. K -p and p-x, y diagrams of a nitrogen(l)-heptane(2) mixture: 0, this work; 'V, Figuiere et al. [45]; D, Brunner et al. [46], ., Peter and Eicke [47]; ~,Akers et al. [48].

CO 2

T , 0 V

x

~

2

II

o

27

8

0

v

II

0

4

i:

~o

cii

..... .L ~

00 %0 0

16 15

v

2

3

0

v~

0

Pressure.

0

~ ov ov

~

3

Cl aHI2 2

4 5

~

l
l

om>OO

0

0

6

0

0

0

12

0

0

0

0 0

9

0

0

0

0

V

0 0

0

21

v9,

2

0

0

@()

101 8 6

0

0

24

>-

,;,,... ~

30

K

544.31 543.55

3. 5

MP.

G'

cJ1 0 0.00

(!I)

0.20

0.40

Concentre~ ion

0.60

V

0.80

1.00

xI end YI' mol/mel

C1aHl2 12J

CCl 2 (1)

Fig. 6. K -p and p - x, y diagrams of a carbon dioxide(l) - Tetralin(2) mixture: 0, this work; 'V, Sebastian et al. [49].

points are plotted in p - x, y and log K -log p diagrams in Figs. 4-6. For comparison, the diagrams also show experimental points taken by other authors. Summary

A static equilibrium cell was built and operated in order to investigate liquid-vapor equilibria in binary

systems consisting of light components such as hydrogen, methane and carbon dioxide and of high boiling hydrocarbons such as Tetralin, diphenylmethane, tetradecane, etc. The equipment was tested by comparing the experimental results with previously published results. Good agreement could be established. The results of the measurements and those of the correlating procedures will be published later.

163

Acknowledgements The authors express their gratitude and appreciation to DFG (Deutsche Forschungsgemeinschaft) for financial support of Uwe Waterling, to the Ministry of the Petroleum Industry (PRC) for financial support of Daqing Zheng, to the craftsmen Lothar Kroll and Klaus Gatzmann for manufacturing and maintaining the equipment, and to the student helper Halvart Koppen for assisting in the experimental work.

Nomenclature enthalpy, J mol- I equilibrium coefficient molar mass, g mol- 1 mass of liquid, g number of moles pressure, MPa gas constant temperature, K volume of gas, cm 3 = Vim, specific volume, cm 3 g-l mole fraction of component in vapor phase mole fraction of component in liquid phase compressibility factor

h K M

m n p

R T V

v x y z

relative volatility fugacity coefficient

IX

4>

Subscripts i

HC LC

component i heavy component light component

Superscripts

V L

vapor phase liquid phase

References

2 3

4

5

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