A cubic equation of state for polar and other complex mixtures

Fluid Phase Equilibria, 29 (1986) 431-438 Elsevier Science Publishers B.V., Amsterdam

431 - Printed

in The Netherlands

A CUBIC EOUATION OF STATE FOR POLAR AND OTHER COMPLEX MIXTURES

JOSi 0. VALOERRAMA,

Chemical Enqineerinq Department. University & Minerals. Dhahran 31261, Saudi Arabia

of Petroleum

LUIS A. CISTERNAS,

Departamento de Inqenieria Quimica. (Chem. Enq. Course). Universidad de1 Norte. Casilla 1280, Antofagasta- Chile.

ABSTRACT A new version of Pate1 and Teja's equation cations

to vapor-liquid

equation

equilibrium

of state is presented,

of mixtures

containing

requires Tc, Pc and Zc as input parameters

dependent

function

and four generalized

Many systems containinq been treated, interaction

findinq

city criterion" equations

hydrocarbons,

excellent

parameters

parameters hydroqen,

results.

with appli-

polar fluids.

and contains

The

one temperature

(in terms of Zc). hydroqen

Regression

sulfide, etc. have

analysis

to estimate

binary

has been done by using an improved version of the "fuqa-

of Pauvonic

et al.

of state has been done.

Also comparison

with several other

In all cases the proposed

cubic

eouation works

better. INTRODUCTION Many (perhaps too many) equations

of state (EOS) of the van der Waals type

have been proposed

in the literature

dynamic properties

of pure fluids, and of vapor-liquid

mixtures.

Since the successful

for the prediction

modification

Redlich and Kwong in 1949, many researchers getting

better

for different

EOS for wider

are nowadays

Pate1 and Teja (1982), following

pressure,

is required,

such fluids and their mixtures.

0378-3812/86/$03.50

and by

studies of Fuller (1976) and Schmidt and they claimed

other EOS for simple fluids and which can be applied Special treatment

and temperature

done by on

available.

(1980), proposed a cubic equation which

mixtures.

equation

The ideas of Soave (1972) have been followed

many, and several modifications

Wenzel

to van der Waals's

(VLE) of

have made great improvements

ranqes of density,

type of fluids.

of physical and thermoequilibrium

however,

to obtain good results for

Here, we have followed

0 1986 Elsevier Science Publishers

to be as good as

to polar fluids and their

B.V.

these studies to pro-

432 pose a modified equations

version

of Pate1 and Teja's equation

respectively,

tional advantage

hereafter)

which has its good features

of being of general

valid for both polar and non-polar done using only one additional a wide variety DEVELOPMENT

applicability.

parameter.

with the addi-

Generalized

fluids are obtained

interaction

to as NEW and PT

correlations

and extension

to mixtures

Results are very good for

of fluids.

OF EQUATIONS

The equation

’ = +% where

(referred

we have used has the form given by Pate1 and Teja (1982):

(1)

V(V+b) : c(V-b)

"a" is a function

As suggested

of temperature,

and b and c are constants.

by Pate1 and Teja, the critical

by the equation

compressibility

factor

implied

of state (that is PcVc/RTc and not the real compressibility

factor) was treated as an empirical

parameter,

so eqn. (1) was constrained

to

satisfy these three conditions.

(s)=O;

($)=O;

and

PV s=

5,

at the critical

point

C

With these restrictions

a(Tc) = Ra R2 Tf/Pc

the terms a, b, and c are found as

;

b(Tc) = Rb R Tc/Pc

Pate1 and Teja used the acentric very involved equations equation parameter

for Rb).

factor as a third input parameter

for the calculation

simple polynomial

1.0634424

Zc+ 0.68289995

expressions

to obtain

0.180754784

Zc+ 0.061258949

Rc = 0.577500514-

1.898414283

Zc

a cubic

factor as a third input for Ra, Rb, and Rc.

Z;- 0.21044403

fib = 0.025987178+

For temperatures

(3)

of Ra, Rb and Rc (including

Here, we used the compressibility

and obtained

R, = 0.69368018-

c(l'c) = R, RTc/Pc

;

Z;+ 0.003752658

Z;

other than the critical we have used the expression

Z; (4)

sugges-

ted by Soave (1972) and also used by Pate1 and Teja: a(T) = a(Tc) ~1 (T)

and

@(i-) = [I + F(I - JTIT~)]*

where F = - 6.608 + 70.43 Zc - 159.0 Zf To obtain the correlations

(5) (6)

for Ra, Rb and Rc we have used the values of 5,

433 given by Pate1 and Teja to evaluate correlated

in terms of Zc.

Ra, Rb, and Rc, values which were then

For F we have also used the values given by Pate1

and Teja but we have also calculated better and more qeneral For mixtures a=CZxix ij

j a ij (1 - nij)

1J

APPLICATION

interaction

equation

For non-polar

found between improvements

mixinq

b = f xi bi

,

to obtain a

parameter

rules for a, b, and c.

and

,

and a..

1J

c = f xi ci

(7)

= (ai aj)& .

TO PURE FLUIDS

The proposed

hydrogen

F for several other comoounds for a(T).

we have used the conventional

where n.. is a binary

fluids.

correlation

has been applied

fluids

our equation

(mostly hydrocarbons)

and the original

were found for polar fluids

sulfide,

liquid density

etc.

to predict

PVT behavior

no major differences

of Pate1 and Teja.

such as water, ammonia,

Fig. 1 below shows the deviations

(p) of ammonia.

Deviations

of pure were

However, great methanol,

in the predicted

were calculated

as %C = 100

In Fig. 2 the compressibility factor of saturated methanol (Ptalc - Pexp )/P exp' as calculated by NEW is shown. Results are similar for other polar fluids.

15

NEW

$

c

Mechado & Street, 1983

G .l i2

10 5

.006 .6

.7

.a

.9

REDUCED TEMPERATURE Fig. 1. Deviations in the predicted liquid density of ammonia using the PT and NEW eqns.

350

400

450

TEMPERATURE (K) Fig. 2. Compressibilityfactor of saturated vapor and liquid methanol predicted by NEW.

434 APPLICATION

TO MIXTURES

Vapor-liquid

eouilibrium

proposed equations.

in binary mixtures

other EOS have been done.

Binary interaction

using PTxy data and a modified et al. (1981).

predicted using the

parameters

and comparisons

with

have been calculated

version of the "fugacity criterion"

of Pauvonic

The regression analysis was done by using the "maximum neigh-

borhood" method of Marquardt H2S Containing

has been

Several systems have been considered

as developed

in Reilly (1972).

Systems

Several H2S containing ,were considered

systems for which experimental

in this study.

Table 1 shows the deviations phase composition

VLE data were available

Some results are shown here. in predicting

bubble point pressure and vapor

for the H2S - nNonane system using several eouations parameters were used in each case.

Optimum values of the interaction

of state. Experi-

mental data were taken from Eakin and de Vaney (1974). While deviations

in predicting

0.7%) for all equations,

vapor phase

important

for bubble pressure calculations.

(PR; Peng and Robinson,

are low (average of

were found with the new equation

Improvements

with respect to Soave-Redlich-

Kwong (SRK; Soave, 1972), Soave-van-der-Waals Robinson

composition

improvements

(SVDW; Soave, 1984), and Peng-

1976) equations

are very important.

In Fig. 3, the Pxy diagram for the H2S - mXylene system is shown as predicted by the PT and the NEW equations. cases.

Bubble pressure calculations

Optimum values of the interaction

regression

analysis

of experimental

parameters were evaluated

VLE data.

through

At 478 K these parameters

nij = - 0.01093 for NEW and Bij = 0.00911 for PT. critical

were done in both

The PT equation

pressure for this system much lower than the experimental

dashed curve in Fig. 3 from P = 7 to the critical

are:

predicts value.

the The

point of the mixture was

extrapolated. In Fig. 4 a Px diagram for the H2S - HP0 system at two temperatures and 444.4 K) is shown.

While predictions

rate at lower temperatures, at higher temperatures. with composition

from PTxy data. Improvements

some deviations

of the order of 5 to 10% are found

We should notice that pressure

at higher temperatures

other interactions

Interaction

These are: nij =-0.10458

very rapidly

the polar and

oarameters

were calculated

(344.4 K) and nij = -0.02996

on the PT equation are not very important

of 2.47% for PT and of 1.71% for NEW).

increases

making more important

between HpS and HzO.

(344.4

with the NEW equation are very accu-

(444.4 K).

in the results (average

However, we should mention that conver-

435 qency

is much faster

computer

using NEW, some convergency

time is reduced

problems

. Deviations in the prediction of bubble pressure compositions for the HzS- nNonane system.

Table

Deviations

in bubble

pressure,

311.1

Range of P (MPa) 0.13 - 0.58

2.9

366.7

0.26 - 1.23

3.5

477.8

0.82 - 2.76

3.8

AVERAGE

0.13 - 2.76

3.4

T(K)

Deviations

are avoided,

and

in about 50%.

% DP = 100

I(Pex

- Pcalc)/Pex

1

PR

PT

NEW

1.9

3.8

2.2

2.1

6.7

5.8

2.4

1.1

2.5

2.9

2.9

1.9

3.7

4.2

2.5

1.7

SVDW

SRK

and vapor phase

in vapor phase Composition,

% DY = 100 1 (Y,, -Ycalc)/Yexp

311.1

0.13 - 0.58

0

0

0.1

0.1

0.1

366.7

0.20 - 1.23

0.7

0.8

0.8

0.8

0.9

477.8

0.82 - 2.76

1.4

1.3

1.2

1.2

1.2

AVERAGE

0.13 - 2.76

0.7

0.7

0.7

0.7

0.7

20

12

l

10

j

Selleck et al., 1952 NEW

16

14

2 2

4

L

I

.2

I

I

.4

.6

I

.a

MOLE FRACTION OF H2S Fig. 3. Vapor-liquid equilibrium in the H2S-mXylene system at 478K

I

.02

I

.04

.

.06

I

.oa

LIQUID MOLE FRACTION OF HzS Fig. 4. VLE in the HzS - Hz0 system at two temperatures

436 H2 - Containinq

Systems -

Several Hz-Hydrocarbon state.

Optimum

mental

VLE data.

systems were also studi'ed using five equations

parameters

were calculated

using reqression

Three systems are shown here: the binaries

Hz-prODane and the ternary Hz-ethylene-propane. meters were calculated For the HP-ethylene low deviations pressures

from experimental

system all equations

predict bubble

below

However,

10 MPa.

For HP-propane

critical

at higher pressures

pressures

the situation

for pressures

and NEW equations 20 MPa.Over

greater

para-

here.

pressure with very

up to 6 MPa.

for propane)

increase up to

greater

than 10 MPa.

Here the SVDW and the SRK equations The PR, TP,

below 3% up to 10 MPa, and of about 5% up to

several equations

well for pressures

deviations

than 3.5 and 7.0 MPa, respectively.

give deviations

predicts values with deviations

system Hz-ethylene-propane

using

(especially

earlier

are in all cases lower than experimental

is similar.

20 MPa the NEW equation

The ternary calculated

and

interaction

VLE data, as explained

Fig. 5 shows the results for pressures

diverge,

Hz-ethylene

The required

of

of experi-

(TP < 2.5%, SRK < 2%, SVDW < 1.5%, PR < l%, NEW < 0.5%) for

6% and the predicted values.

analysis

was also considered

of state.

All equations

Vapor phase compositions

for any pressure.

of about 7%.

and bubble pressure

perform

relatively

are not well predicted

The PR and the NEW equations

are

Fig. 6 shows some results.

the best.

c-

/ / 4

et al., 1975 -Predicted l

Him

\ ,_

\

.--Extrapolated /

-

NEW ---PT l Sagara et al., 1975

.05

.055

I .06

.065

LIQUID MOLE FRACTION OF H, Fig. 5. Bubble pressure predictions for the Hz - C2H4 system

.Y:,

.96

.97

.93

VAPOR MOLE FRACTION OF Hz Fig. 6. Vapor phase compositions in the Hz-C~H~-C,HE system

INTERACTION PARAMETERS 5 Propylene-ethylene 263 0.01797 ethylether-CO2

1 .2

I

1

.4

.6

313 0.02787

I .a

LIQUID MOLE FRACTION OF B Fig. 7. Px diagram for three systems as predicted by equation NEW

Other Systems Several other miscellaneous dicted bubble pressures ether-carbon

systems were also considered.

In Fig. 7 pre-

are shown for three systems: ethylene-propylene,

dioxide, and methyl acetate-carbon

dioxide.

Optimum

ethyl

interaction

parameters were used as shown in the same fiqure.

CONCLUSIONS The proposed equations

have shown to be appropriate

behavior of polar fluids and VLE of mixtures. some disadvantages and Chappelear,

as a third parameter

for predicting

PVT

Despite the fact that Zc presents

for a generalized

correlation

(Leland

1968) we have found that for polar fluids and for intermediate

to high pressures,

it works better than the acentric

this are found elsewhere The fugacity criterion

(Valderrama

and Cisternas,

to estimate

the binary interaction

proven to work well and to provide adequate Bubble pressure, within acceptable

factor.

More details on

1986).

parameters

parameter nij has

for VLE calculations.

dew pressure, and vapor and liquid compositions margin of deviations.

found using Patel-Teja

or other equations

These deviations

are predicted

are lower than those

of state.

ACKOWLEDGMENT The authors thank the University which made possible the preparation Conference. computer

of Petroleum

of this paper and its presentation

L.A.C. also thanks the Universidad

facilities

and support.

& Minerals for its support at the

de1 Norte for the use of its

438 REFERENCES Bae H., K. Naqahama, and M. Hirata, 1981. Measurement and correlation of high pressure vapor-liquid equilibria for the systems Ethylene-I Butene and Ethylene- Propylene. J. Chem. Eng. Japan, 14(l): 1-6. Eakin, B.E. and W.E. de Vaney, 1974. Vapor-liouid equilibria in HydrogenHydrogen Sulfide - C9 hydrocarbon system. Chem. Enq. Proqress Symp. Ser. 70 (140): 80-90. Fuller, G.G., 1976. A modified Redlich-Kwonq-Soave equation of state capable of representing the liouid state. Ind. Eng. Chem. Fundam., 15(4): 254-256. Hiza, M.J., C.K. Heck, and A.J. Kidnay, 1975. Liquid-vapor and solid-vapor equilibrium in the system Hydrogen-Ethylene. Chem. Enq. Progress Symp. Series, 64 (88): 57-65. Huang, S.S., and D.B. Robinson, 1984. Vapor-liquid equilibrium in selected aromatic binary systems: mxylene- Hydrogen Sulfide and Mesitylene-Hydrogen Sulfide. Fluid Phase Equil. 17: 373-382. Leland, T.W. and P.S. Chappelear, 1968. The Corresponding States Principle. A review of current theory and practice. Ind. Eng. Chem., 60(7): 15-43. Mechado, J.R. and W.B. Street, 1983. Equation of state and thermodynamic properties of liquid methanol from 298K to 489K and pressures to 1040 bar. J. Chem. Eng. Data, 28(2): 218-223. Ohgaki, K. and T. Katayama, 1975. Isothermal vapor-liquid equilibria for the systems Ethyl Ether-Carbon Dioxide and Methyl Acetate- Carbon Dioxide at high pressures. J. Chem. Eng. Data, 20(3): 264-267. Pauvonic, R., S. Javonovic, and A. Mihajlov, 1981. Rapid computation of binary interaction coefficients of an equation of state for vapor-liquid equilibrium calculations: Application of the RKS equation of state. Fluid Phase Equil., 6: 141-148. Patel, N.C. and A.S. Teja, 1982. A new cubic equetion of state for fluids and fluids mixtures, Chem. Eng. Sci., 77(3): 463-473. (Also, Chem. Enq. Comm., 1981 13: 39-53). Peng, D. and D.B. Robinson, 1976. A new two-constant equation of :::Late. Ind. Enq. Chem. Fundam. 15(l): 59-64. Redlich, 0. and J.N.S. Kwong, 1949. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev., 44: 234244. Reilly, M. (editor) 1972. Computer Programs for Chemical Enqineering Education. Vol. II - Kinetics. Sterlinq Swift Pub. Co., Manchaca- Texas, USA pp 276295. Sagara, H., S. Mihara, Y. Arai, and S. Saito, 1975. Vapor-liquid equilibria and Henry's constants for ternary systems containing hydrogen and light J. Chem. Eng. Japan, 8(2): 98-104. hydrocarbons. Schmidt G. and H. Wenzel, 1980. A modified van der Waals type equation of state. Chem. Eng. Sci., 35: 1503-1512. Selleck, F.T., L.T. Carmichael, and B.H. Sage. 1952. Phase behavior in the Hydrogen Sulfide- Water system. Ind. Eng. Chem. 44(g): 2219-2226. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of State. Chem. Eng. Sci., 27(6): 1197-1203. Soave, G., 1984. Improvement of the van der Waals equation of state. Chem. Enq. Sci., 39(2): 357-360. Valderrama, J.O. and L.A. Cisternas, 1986. On the choice of a third generalizing parameter for equations of state. Chem. Eng. Sci. (submitted for publication).