C?temLs/Enginwinp ScienceVol. 37,No. 3, pp.

463-473,1982
Printi in GreatBritain.

A NEW CUBIC EQUATION OF STATE FOR FLUIDS AND

FLUID MIXTURES

NAVIN C. PATEL
Department of Chemical Engineering, Loughborough University of Technology, Lou&borough, LEll 3TU,
England

and
AMYN S. TEJA*
School of ChemicalEngineering,GeorgiaInstitute of Technology, Atlanta, GA 30332, U.S.A.

(Received28 February 1981)

Abstract-A new cubic equation of state for pure fluids is presented in this work. The new equation requires the
critical temperature and pressure, as well as two additional parameters to characterize each particular fluid.These
parameters have been evaluated by minimizing deviations in saturated liquid densities while simultaneously
satisfying the equality of fugacities along the saturation curve. Thus, good predictions of volumetricproperties in
the liquid region are obtained, while accuracy in vapour-liquidequilibriumcalculationsis maintained.Parameters
for polar as well as nonpolar fluids are presented in this paper. In the case of nonpolar fluids, the two parameters
required can be correlated with the acentric factor. No such relationship with independently measured quantities
could be found for polar fluids. It is shown that the new equation reproduces many of the good features of the
Soave and Peng-Robinson equations of state for nonpolar fluids, whilst overcoming some of the limitations of
these equations for polar fluids. Applications of the equation of state to the correlation of phase equilibria are
demonstrated.

INTRODUCTION curately reproduce the experimental saturated liquid

Since the time of van der Waals, many equations of state volume at a particular temperature. Schmidt and
have been proposed for the representation of the volu- Wenzel(41, in particular showed that the optimum value
metric properties of pure liquids. These equations have of this substance dependent critical compressibility was
ranged in complexity from simple expressions containing not, in general, equal to the experimental critical com-
two or three constants to complicated forms containing pressibility of the fluid of interest.
more than fifty constants. Although the many-constant The present work is an extension of the works of
equations have been utilized for precise representation Soave, Peng and Robinson and of Schmidt and Wenzel.
of volumetric data, they are not generally preferred far The equation of state proposed here uses, in addition to
phase equilibuium calculations and in process simulation the critical temperature T, and critical pressure PC, two
studies, partly because they require excessive computer substance dependent parameters J and F as input
time and partly because it is difficult to obtain general- parameters. For non-polar fluids, these parameters can
ized forms of these equations suitable for mixture cal- be related to the acentric factor o, so that with suitable
culations. In many situations, therefore, the use of sim- assumptions, the equation reduces to those of Soave,
ple cubic equations of state represents a satisfactory Peng and Robinson and Schmidt and Wenzel. The new
compromise between accuracy and speed of com- equation thus reproduces many of the good features of
putation. these three equations and, in addition, it can be applied
Probably the most successful cubic equations for to polar fluids such as water, ammonia and the alcohols.
phase equilibuium calculations have been those proposed The extension of the equation to mixtures is also
by Soave[l] and Peng and Robinson[2]. Both the Soave demonstrated below.
(RKS) and the Peng-Robinson (PR) equations assume a
particular (fixed) value of the critical compressibility THENEWEQUATlONOFSI'ATE
factor and, as a result, the predicted densities of the The equation of state proposed in this work has the
saturated liquids and the predicted critical volumes differ following form:
considerably from their experimental values (especially
for substances whose critical compressibilities are p=RT_ U[U (1)
significantly different from the values assumed by these u-6 U(u+b)tC(u-6)
equations). Fuller131 and, more recently, Schmidt and
Wenzel[4] among others, introduced a substance depen- where R is the universal gas constant, “II” is a function
dent critical compressibility which allowed them to ac- of temperature and b and c are constants. The form of
the cubic equation chosen is not new, similar forms
*Authorto whom correspondence should be addressed. having been chosen earlier by Harmens[5] and

463
464 N. C. PATELand A. S.TEIA

Mollerup[6] among others. By making certain assump- give accurate results for hydrogen and they recommend:
tions, two well-known cubic equations of state can be
obtained from eqn (1). When c = b, eqn (1) reduces to (2= C, exp (-CzTR). (12)
the Peng-Robinson expression and when c =O, it
reduces to the Redlich-Kwong or Soave equations. A similar function was recommended by Heyen[ 121
Acceptable prediction of both low and high pressure
behavior requires at the very least that the critical com- a =exp[C(l -TR”)]. (13)
pressibility factor implied by the equation of state be
treated as an empirical parameter, different in general We have used both eqns (13)and eqn (11)in our equation
from the experimental value of Z,[7-91. It is also well- of state, although it should be noted that eqn (13) con-
known that the predicted value of the critical com- tains one more constant than eqn (11). For the 38 sub-
pressibility factor (denoted by lc below) is not an im- stances studied in this work, we found that eqn (13)
portant indicator of the overall performance of any offered no advantages over eqn (11). We therefore
equation of state[lOl. For these reasons, the new equa- recommend the use of eqn (ll), with F being treated as
tion of state was constrained to satisfy the following an empirical parameter. The equation proposed by us
conditions: therefore contains two parameters & and F, in addition to
T, and PC.
JP
(av>=o 1;
EVALUATIONOF &AND F
The following trial and error procedure was adopted
C?P for evaluating & and F. Initially, & was set equal to
(TP) =o To
(3)
0.307 or l.lZc, whichever value being closer to the
experimenta critical compressibility Z,. Using eqns (8-
p,v,_
RT,- L (4)
lo), values of a., &., and R, were then calculated. Using
these values of !‘I,, f& and n,, a value of a was obtained
at each temperature along the saturation curve such that
Instead of letting & have a value equal to the experi- the equilibrium condition:
mental value of the critical compressibility factor, an
arbitrary value was chosen. Thus, & was treated as an fL”=f” (14)
empirical parameter, our treatment so far being identical
to that of Schmidt and Wenzel[4]. It should be noted that was satisfied at each point. (The expression for the
if & = 0.3074,eqn (1). together with constraints (2) and fugacity is given in the Appendix). F was then calculated
(3), reduces to the Peng-Robinson equation, Similarly, if by a least squares fit of eqn (1I). The following sum was
[= = 0.3333, eqn. (1) reduces to the Soave or Redlich- minimized:
Kwong equations.
Application of constraints (2-4) to eqn. (I) yields:

aVl =fI,(R’T,2/P,) dTR1 (3

pi = I - TR,i”2 (16)
b = &(RT,/PJ (6)
Where TR.iis the value of TR at the ith data point and
c = D,(RTc/Pc) (71 (I,,~,; is the value of a which satisfies eqn (14) at that
point.
where The condition for a minimum of eqn (15) leads to a
&=I-3j, (8) cubic equation in F which can be solved analytically and
the smallest positive root was taken in the subsequent
n, =3{;t3(1-2yC)n,tn,‘t1-3& (9) calculations. Using the values if R,, fib, fl, and F,
saturated liquid densities were calculated and compared
and 0, is the smallest positive root[4] of the cubic: with experimental values obtained from the literature,
the average absolute deviation at the chosen value of L
ah’ •t (2 - 3[c)&2 t 35,2& - k’ = 0 (10) being noted. The value of & was then changed by 0.001
and new values of O,, fib, & and F were obtained by
For a[TR], we chose the same function of reduced solving eqns (8)-(11). The average absolute deviation in
temperature as that used by Soave and Peng and Robin- saturated liquid densities was obtained and noted. This
son. It is given by: procedure was repeated until a minimum occurred in the
value of the average absolute deviation. After the first
u = [I + F( 1- T,“2)]2. (111 iteration, it became clear whether the value of & should
be increased or decreased.
Both Soave and Peng and Robinson correlated the slope The optium values of & and F correspond to the
F to the acentric factor o of a substance. Recently, minimum deviation in saturated liquid densities and the
Grahowski and Daubert[ll] found that eqd (II) failed to equilibrium condition of equality of fugacities. The pro-
 A newcubic equation of statefor fluidsand fluid mixtures 465

posed equation using optimum values of & and F is The new equation gives lower average deviations in
therefore expected to give good predictions of liquid both the vapor and liquid phases than the P-R equation.
phase densities and vapor-Iiquid equilibria (Schmidt and For polar components and heavy hydrocarbons, the new
Wenzel adopted a very similar approach, except Z;*and equation gave consistently better predictions than the
eqn (11) for nonpolar fluids were determined from a P-R equation and the R-K equation. Deviations between
single vapor pressure and density point. They did not experimental and calculated saturated liquid densities are
extend their procedure to polar fluids such as water and the plotted against reduced temperature for n-eicosane and
alcohols or to long molecules such as eicosane). Optimum ammonia in Figs. I and 2. Except in the region close to
values of & and F for 38 pure fluids (including polar the critical point, the new equation gives accurate pre-
substances) are given in Table 1. Experimental saturation dictions of saturated liquid densities.
pressures and saturated liquid densities were used to Agreement in the critical region (0.9 < TR< 1.0) can be
obtain & and F. References to all the data used can be improved if & is assumed to be a linear function of
found in Ref. [IS]. temperature in this region. The value of & changes from
the calculated value &’ to the experimental value Z, as
TR changes from 0.9 to 1.0, so that no additional
PUBE FLUtD CALCULATIONS parameters ate required if a linear function of tem-
The proposed equation of state was used to calculate perature is assumed. Thus, for 0.9 < TR< 1.0
densities, vapor pressures, enthalpy departures and
entropy departures of pure fluids. Calculated values were ie’=k-IO(L-Z,)(TR-0.9). (17)
compared with experimental values (when available) and
with values obtained using other equations of state. According to this equation. &’= Z, when TR= 1.0, so
that the experimental value of the critical compressibility
Densities is reproduced. This resulted in improvement in the
Saturated liquid and vapor densities for the 38 com- average deviations in saturated liquid densities from
ponents studied in this work were calculated using the 11.56%to 3&i%, although the average absolute deviation
new equation, the Peng-Robinson equation, the Heyen in saturated vapor densities increased from 2.32% to
equation, the Redlich-Kwong equation and the 5.37%. No cubic equation gives accurate predictions of
BWRS114 equation. Average absolute deviations in both these properties in the critical region[41. Accuracy
these properties are given in Tables 2 and 3. in the representation of saturated liquid densities in the

Table 1, Parameters of the new equation of state constants F, C, 1 correspond to eqns (11)and (13)

0.328 0.450751 0.524130

Nikgen 0.329 0.516798 0.673567
oxygen 0.327 O.L167035 0.5ri5990
Merhane 0.324 0.455336 0.52632”
Lrhane 0.317 0.561567 0.708265
Ethvlene 0.318 0.55U369 0.64223fI
0.317 0.64804Q 0.76327b
0.324 0.6613C!5 0.750730
Cl.310 O.bb11179 O.b59b02
0.309 0.67838? 0.831715
0.315 0.683133 0.77563:
0.315 O.bq6423 0.7112573
0.306 0.7't6~i70 0.851904
0.314 0.741095 0.854607
0.305 0.801E05
lb 0.30: 0.86@858
17 0.301 0.9185U
16 O.3Cl 0.982750
19 Cl.297 1.021910 1.2Q9.74:
20 0.297 1.OBWlb l-29107”
21 0.294 1.115585 1.339255
22 0.29' I.179982 1.31938$
23 0.291 1.188785 1.427823
211 0.283 1.297054 1.354358
25 0.276 1.276058 1.53673?
26 0.277 1."09b71 1.741225
27 Carbon dioxide 0.309 0.707727 0.865W
28 Carbon monoxide 0.328 0.53506C 0.678260
29 Sulfur-dioxide 0.307 0.754966 0.8714Pb
30 nydrogen-sulfid‘ 0.320 0.583165 o.es5551
31 Wafer 0.2bO 0.689803 0.=*7%6
32 Ammonia 0.282 0.62709P 1.425500
33 Benzene 0.310 Cl.704657 0.880633
34 Methanol 0.272 0.972708 0.939455
35 0.3po 1.230395 1.221152
36 0.303 1.241347 1.80624s
37 Cl.304 1:1997e7 3.503ElU
38 0.311 1.242855 2.811893
 N. C. PATELand A. S. TEZJA

Table 2. Comparison of saturated liquid densities

New EC!.
NO.
Of Tr SC= cc c;=ftTFl Hey en P-R R-K
BWRI
AA1
:omponent Points Range AAD (8) AAD (8) AAD' (%) AAD (%) AA? ($1 C%!

34 0.556-0.981 4.27 2.45 1.80 8.77 5.Ob

32 0.501-0.983 4.10 2.32 1.20 8.88 4.7E
36 0.530-0.983 4.30 2.31 1.99 8.16 5.74
&Lane 32 0.524-0.976 4.59 3.03 1.52 7.66 5.72
:thane 28 0.563-0.982 6.95 3.63 2.65 7.12 13.32
:thylene 24 0.550-0.983 4.82 2.72 1.46 6.07 II.79
'r0pa"e 33 0.570-0.984 5.58 2.M 1.63 5.98 14.51
=ropyle"e 27 0.563-O.Q7& 4.35 2.27 5.99 7.17 a,?7
kG?tyl.?ne 21 0.62U-0.972 4.91 1.83 2.lto 5.30 21.91
I-Butane 28 0.627-0.960 5.57 3.39 2.26 5.61 19.36
i-Butane 29 0.578-0.980 2.29 2.61 3.82 3.UO 15.64
l-B”tell-2 26 0.651-0.966 3.80 1.45 1.19 4.52 lE.EB
I-PSlta”e 30 0.638-0.981 3.Ub 3.11 1.22 4.00 20.1?
i-Pentane 21 0.579-0.965 4.83 2.24 1.Q1 5.43 23.M
,vHexane 32 0.613-0.97u 3.11 2.2" 0.62 3.12 2U~O?
n-Heptane 23 0.637-0.987 4.44 2.04 3.23 4.60 29.56
n-octane 23 0.586-0.977 3.31 1.34 3.9E 6.55 31.40
n-ND"ane 27 0.525.0.7bO 0.53 0.53 3.62
o-Decane 27 0.536~0.771 0.62 0.62 5.Ri -
"-U"deCa"e 27 0.545-0.781 0.63 0.63 5.81 -
n-~odecane 27 0.576-0.790 0.70 0.70 7.u7
"-Widecane 27 0.563-0.799 0.70 0.70 2.41 6.81
n-Tetradecane 27 0.569-0.806 0.76 0.76 1.11 8.95
n-Heptadecane 21 0.592-0.832 0.94 0.94 1.76 14.62 -
n-Octadecane 27 0.598-0.839 1.07 1.07 18.89
n-Cicosane 27 0.615-0.850 1.04 1.04 18.42
Carbon dioxide 44 0.712-0.987 4.48 1.78 5.70 5.22 23.30
Carbon monoxide 21 0.542-0.960 3.81 2.10 2.70 8.48 LI.97
Sulfur dioxide 33 0.593-0.989 11.46 1.71 1.70 4.49 35.lU
Rydrogen sulfide 29 0.561-0.982 3.47 1.57 0.83 5.84 10.29
water 29 0.546~,J.Q78 3.99 1.57 2.69 26.33 64.86
Ammonia 22 0.698-0.986 3.21 2.4'1 6.14 15.90 4i.83
Benzene 39 0.553-0.988 3.87 1.69 0.77 4*05 20.95
Hethanal 22 0.588-0.695 0.51 0.51 0.55 20.05
Ethanol 26 0.575-0.719 0.51 0.51 0.96 4.33 -
Propan-l-01 30 0.536-0:1x 0.66 0.66 7.09 3.06 -
Butan-l-al 23 0.609-0.730 0.28 0.28 10.09 2.18
Pentan-l-cl 30 0.542-0.736 0.53 0.53 7.17 1.32 -

Table 3. Comparison of saturated vapor densities

New EQ.

LC = 5 C'= f[T ]
Heye" P-R R-K BURS
component AA" (;, :A, c%f AAD (%) AAD ($1 AAD ($1 AAD (%)

Argon 0.36 1.15 4.32 1."2 0.33

Nitrogen 0.55 1.32 3.32 1.20 0.75
Oxygen 0.70 1.23 3.58 0.91 0.44
Methane O.E8 1.50 4.20 1.66 0.51 0.58
Ethane 0.99 2.12 4.03 1.38 1.97 0.20
Erhylene 0.71 1.60 3.72 1.04
Propane 0.8C 2.;2 4.03 1.23
Propylene 2.43 2.71 3.10 1.68
Acetylene 2.Or 2.11 2.86 1.87
n-Butane 0.'3 1.62 3.98 0.78
i-Butane 1.30 2.30 4.39 1.73
1LButen.e l.kl 2.31 U.84 1.71
n-Fentane O.iQ 1.78 5.27 0.62
i-Fentane O.bG 1.99 ".Cl 1.13
n-Hexane 3.87 1.70 3.ua 0.88
n-Heptane C.67 1.32 1.54 0.63
"-Octane 2.LE 3.08 3.85 2.86
Carbon dioxide 0.71 2.13 E.iD 2.00
Carbon monoxide 3.;1 3.24 3.39 3.lr5
Sulfur dioxide 1.34 i.fi" 3.73 1.11
Hydrogen sulfide 2.58 3.36 6.37 3.38
water 1.26 2.36 4.47 3.76
Alnonia k.iiO ?.(IC 3.64 6.81
Ber.zere 2.:e 3.27 4.80 7.23

average (t 1 1.44 2.14 4.14 1.89 3.35 0.26

 A new cubic equation of state for fluids and fluid mixtures 461

, the simultaneous solution of the following equations:

CI-EICOSANE
0. w
pL:pV=p (18)
f”=f” (19)

where L and V refer to the liquid and vapor, respectively.

-10. Except for n-propanol, n-butanol and n-pentanol, the
3
overall average deviation between calculated and
experimental vapor pressure for the 38 substances was
2
z
found to be 0.86%. For the three alcohols, the deviations
P-R EQWlION
8 were of the order of 5%.
_---__
a? -
I’
-2O_ , Enthalpy and entropy departures
/
/
Bnthalpy and entropy departures for saturated liquids
5 .6 .7 B .9
calculated from the new equation were compared with
REDUCED TEMPERRTURE values calculated using the BWRS equation for 8 light
hydrocarbons. The BWRS equation predicts the enthalpy
Fig. 1. Comparisons of calculated and experimental saturated and entropy departures for these fluids within experi-
liquid densities of a-eicosane. mental error (deviations less than 1%). Values from the
new equation compare very favorably with the BWRS
equation for these fluids, as shown in Table 4.

GENERALnATION OF EQUATION OF STATE

PARAMETERS& AND F
One way of extending the equation of state to new
substances is to generalize the equation of state con-
stants. In order to apply the new equation of state to
-10. substances not studied in this work, the values of the
--__
-_.
parameters & and F have been correlated with the
E
L . acentric factor. The resulting correlations are given by
2 \
\
\ P-R EQUATIOI4
2
0
-2o-
\ F = 0.452413t 1.309820-0.295937~ (20)
R \
\ & = 0.329032- 0.076799~t 0.0211947~‘. (20
\

-me I However, as expected, the generalized equations apply

.e .9 1.
to nonpolar substances only (3 parameter CSP). The
REWCEO TEJIPERATUQE values of F and & for water, ammonia and the alcohols
did not lie on the curves predicted by eqns (20) and (21).
Fig. 2. Comparisons of calculated and experimental saturated The loss in accuracy in predicting compressibilities of
liquid densities of ammonia. nonpolar fluids using the generalized constants was less
than 1% and, often, less than 0.1%.
critical region always leads to a loss of accuracy in the
representation of saturated vapor densities and vice USEOF AN EXPONFNTIALFUNCTIONFOR a
versa. Nevertheless, the use of eqn (17), represents a It has been pointed out by many authorslll-141 that
good compromise. the temperature function for q used by Soave and Peng
Overall, the eleven constant BWRS equation gives and Robinson does not reproduce the correct tem-
excellent predictions of both saturated liquid and vapor perature behavior of the constant “a” at high tem-
densities. However, the use of this equation is restricted peratures. This is mainly because the function becomes
to normal alkanes from methane to n-octane, for which zero at finite TR and then starts to rise with temperature.
the equation of state constants have been reportedU21. The approach of real gas behavior to that of an ideal gas
Densities in regions other than the saturation region at high temperatures requires that a -0 as TR+~.
have also been calculated. Overall, predictions using the Heyen therefore proposed eqn (13) which has the
new equation are better than those obtained using other required characteristics but contains one additional con-
cubic equations of state (Average deviations of 2.4% stant. Constants C and n have been evaluated for the 38
compared with 3.3% for the PR equation for 3175 data substances and are given in Table 1. Both eqn (11) and
points including available data for methane through n- eqn (13) are plotted for methane and n-decane in Figs. 3
decane). and 4. The functions are almost identical upto TR = 12
for methane and T, - 2.0 for n-decane and yield almost
Vapor pressures identical values of densities upto these conditions. Since
The vapor pressures of pure fluids were calculated by higher temperatures are unlikely to be encountered for
468 N. C. PATEL
and A. S. TWA

Table4 Enthalpyand entropydeparturesof saturatedliquid

No. *H AS
of Tr Qanpr New Fq. New 69.
S"b*t.WCe Points AAn (") AA" (58)

Methane 37 0.524-0.476 2.11 1.07

Ethane 28 0.563-0.9R2 2.53 1.16

Propane 33 0.570-0.9RL 1.52 1.21

n-Butane 28 0.627-0.980 2.41 1.2L1

n-Pentme 30 0.635-0.181 1.70 0.92

n-wxanc 32 0.613-0.974 2.55 1.44

n-lteptane 23 0.637-0.987 1.53 O.R5

n-octane 74 0.586-0.972 1.13 0.79

Overall average (%) 2.06 1.09

these substances (especially in phase equilibrium cal-

-EQN- (IO
culations), it appears that the use of the exponential
_-.”
EQN.(f3) function for n does not lead to much improvement (except
for substances such as hydrogen which are normally at
high reduced temperatures).

COMMENTSON THEVALUES OF 5, AND F

d
The new equation of state proposed in this work
requires two parameters lC and F, in addition to T, and
PC, for each pure substance. For nonpolar substances,
however, & and F can be correlated with the acentric
factor O, so that only three constants (T,, PC and O) are
needed for such fluids.
04 . , , , , , , , ,
0 10 20 For all substances considered, Z; was found to be
greater than Z,. This was also found to be true by
REDUCED TEWERAlVRE
Schmidt and Wenzel using a similar equation of state.
Fig. 3. (I as a function of the reduced temperaturefor methane. For light nonpolar substance (w = 0, & = D.329),the new
equation is comparable to the S-R-K equation and for
components whose acentric factors are close to 0.3 (O =
0.3, & = 0.307) the proposed equation is comparable to
the P-R equation. Thus, characteristics of both the SRK
and PR equation are implicit in the new equation.
- EQN.<ll)
However, the application of the new equation extends lo
- - - EQN.~~~) heavy hydrocarbons (upto n-eicosane) and polar sub-
stances.

EXTENSIONTO MXTIRES
Equation (1) can be used for the calculation of mixture
properties if the constants a,b,c are replaced by the
mixture constants u,, b,, c,,, as follows:

(22)

b, = &b, (23)
1

RECUCED TEMPERATURE

Fig.4. a as a function of the reduced temperature for n-decane. c, = 2 xic,. (24)

 A newcubic equation of state for fluidsand fluid mixtures 469

The choice of this model is completely arbitrary, the only practice, however, this is not strictly true and .$ should
justification being the success with which analogous be determined at conditions of interest. In our work,
equations have been used with other equations of state values of & were obtained at a single temperature for
such as the Soave and the Peng-Robinson equations. It is each binary pair-the temperature being carefully selec-
also possible to reduce eqn (1) to the Soave and the ted to lie in the middle range of reported experimental
Peng-Robinson equation by setting c, = 0 and c,,, = b, data. It should be added that the temperature depen-
respectively as in the case of pure fluids. This would not dence of & was found to be small, and the optimum
be possible if a different mixture model (in particular, a values of & may be used to predict VLE at other
different combining rule for c,,,) is used. temperatures.
The cross-interaction term a, in eqn (22) evaluated
using the following mixing rule: VLBRESULTS
Optimum value of .$ had been evaluated using the
aij = &j(lZ&) “’ (25) Soave, the Peng-Robinson and the proposed equation
and are given Table 5. The three equations were used for
where & is a binary interaction coefficientwhich must be VLE calculations for 32 binary systems containing the
evaluated from experimental data. Compared with light hydrocarbons, carbon dioxide and hydrogen sulfide.
vapor-liquid equilibrium (VLE) predictions, the sen- The new equation was also used for calculations involv-
sitivity of predicted bulk mixture properties such as ing an additional 20 systems containing the heavy
density and enthalpy to the value of <ii is small. There- hydrocarbons, water and the alcohols. The results are
fore it is common practice to use binary VLE data for shown in Table 6.
the determination of 6, values. In principle, no further Our results for the different groups of binary systems
information is required to predict the properties of ter- studied in this work can be summarized as follows:
nary and higher systems.
(a) Light hydrocarbon binaries (upto n-decane)
In this study, the optimum value of .$ for each binary
pair was obtained by minimizing the absolute average In general, the three equations of state correlate data
deviation in the bubble point pressures at selected tem-
for these systems equally well. Moreover, except for
peratures. The determination of the bubble point pres-binaries containing methane, values of &j for these sys-
sure or vapor-liquid equilibrium in general, requires that
tems were found to be close to 1.0. This is true for all
the following equalities be satisfied: three equations and supports Soave’s conclusion that no
binary interaction coefficients are needed for VLE cal-
f,” = fi” (i = 1,2,. * . .m) (26) culations involving light hydrocarbon binaries not con-
taining methane. For systems containing methane, the
where fi denotes the fugacity of component i and the new equation gives & values which are closer to unity
superscripts V and L denote the vapour and liquid than those obtained using the Soave or Reng-Robinson
phases respectively. An equation for the fugacity of equations. In general, the Soave and Peng-Robinson
component i is given in the Appendix. equations give very similar values of & for such sys-
tems.
EVALUATIONOFTHEBINARYlNTERACTlONCOEFFlClEN't
A number of criteria may be chosen for evaluating the (b) CO,-light hydrocarbon, and H&light hydrocarbon
optimum value of &. Among these criteria are [ Ill: binaries
(1) minimization of deviations in bubble point pres- The optimum values of tii obtained from the three
sures; equations are slightly different from each other, but are
(2) minimization of deviations in Rash volumes. usually in the range 0.84-0.90for systems containing COZ
Grabowski and Daubert[lll used both criteria with the and 0.91-l .O for systems containing H,S.
Soave equation and found that convergence problems
were encountered with the second criterion for close- (c) Light hydrocarbon-heavy hydrocarbon binaries
boiling mixtures because the correct value of 5, was These systems include binaries such as methane-n-
needed in advance to “find” the hvo phase region. The eicosane, ethane-n-eicosane etc. Only the new equation
application of the first criterion, on the other hand, gave acceptable deviations in bubble point pressures for
produced no convergence problems and led to good these systems. Most of the optimum & values were
results for both flash and bubble point calculations. found to lie in the range l.W.08.
In this study, the optimum value of & was obtained by
minimizing the absolute average deviation in the pressure (d) Alcohol-water binaries
at a selected temperature for each binary mixture. The Only the new equation of state was used to calculate
absolute average deviation is defined as: tij for these binaries. The methanol-water system could
be correlated with an optimum value of & of 1.083and
the ethanol-water binary could be correlated with a value
of 1.075.
Typical predicted P vs x curves for COz-n-butane,
Generally, & is assumed to be independent of tem- methane-n-eicosane and water-methanol are shown in
perature, pressure, density and composition[ll, 16). In Figs. 5-7. Predicted P vs x curves for light hydrocarbon
470 N. C. PATEL
and A. S. TWA

Table 5. Optimumvalues of & and bubble point correlationusing three different equations of state

6 a.15 0. “O‘b
:: s R. 11 0.0189
3. 0.63 0.0117
4. 9 4.97 “.“OZI
5. 9 0.56 0.0149
6. 0.46 D.0023
7. 0.14 0.41
8. 2.19 a. 0093 2.44
9. 1.02 a.acoP 1.12
10. 1.15 0.0040 1.21
11. 0.97 0.0060 1.26
12. 0.18 II, on51 0.33
13. 0.12 o.cm14 0.18
14. 1.41 “.024L 1.31
15. 0.96 a.““72 0.60
16. 0.49 a.0144 0.51
17. O.hO O.“lhR 0.80
18. 0.25 0.0029 0.41
19. 0.18 0.0013 0.43
20. 9 0.96 O.O"," I.86
21. 9 0.39 0.0013 0.76
22. 8 3.19 Cl.0629 3.27
23. 10 I.17 O.""," 1.18
24. 10 0.50 0.0122 0.50
25. 9 0.70 0.0050 0.55
26. 10 1.81 0. OOBZ 3.19
27. 9 1.11 0.0038 0.99
28. 9 L.70 “.02,1 2.43
29. B 3.00 0.0122 3.51
30. 9 0.78 0.0102 0.16
31. 8 0.90 0.0210 0.97
32. 9 2.92 0.0020 9.27

Table 6. Values of the deviations in bubble point pressures and vapour phase mode fractions with & = 1.0and
$ = tti, from equation (I)

NO.
of Temperature ‘ij = “O 'ij - 'ijapt
AP C%) bY1 E.. AP (0 AY1
NO. System Points K lloPt
-

1. Methane-propane 5 277.6 1.65 Il.0054 0.989 0.60 0.0048

2. Methane-benzene 7 338.8 2.45 0.006U 0.991 0.92 o.ao54

3. Methane-n-Eicosane 5 313.2 29.26 0.0000 l.ORO 1.09 0.0000

Q. Ethane-i-butane 8 311.8 3.57 0.0072 1.018 1.62 0.0038

5. Ethane-n-&cane 8 377.6 4.61 0.0455 1.019 0.60 0.0426

6. Ethane-n-dodecane 9 323.2 5.58 0.0009 1.016 2.26 0.0008

7. Ethsne-n-Eicosane 7 333.2 41.01 0.0002 1.070 9.15 0.0000

8. Ethylene-n-dodecane 8 298.2 7.97 0.0002 1.006 1.10 0.0000

9. Pmpane-bemene 10 377.6 3.5R 0.010, 0.988 1.86 0.0086

10. Propane-n-decan. 7 377.6 8.63 O.OOOB 1.024 0.79 0.0004

II. Nitrogen-methane 10 155.u 4.22 0.0299 0.968 0.71 0.0217

12. Nitrogen-propane 6 298.2 11.00 0.0269 0.926 3.RO 0.0161

13. Nitrogen-n-butane 6 311.0 10.83 0.01116 0.969 8.73 0.0151

IU. Nitrogen-n-hepfane 5 352.6 17.4 0.0032 0.911 3.30 0.0018

15. H2S-n-hepfane 5 352.6 10.97 0.0027 0.956 3.41 0.0018

16. HIS-n-decane 7 3'1

'+ .3 0.61 0.0004 0.999 0.53 0.0004

17. COj-n-derans 7 W~l.3 24.74 0."(107 a.903 2.14 n.oowJ

18. CO-pr0pan.Z B 273.2 7.11 Ct.0170 0.979 6.11 0.0151

19. “etka”ol-Yatel. 10 373.2 27.45 0.0750 1.083 1.75 0.0081

20. Ethanol-water 10 423.2 73.75 "."R$7 1.075 7."ll 0.0713

 A new cubic equation of state for fluids and fluid mixtures

systems using the Soave, the Peng-Robinson and the

new equation were identical. Multicomponent predic-
tions were also found to be comparable for systems of .a
light hydrocarbons.

DENSITIIB
Densities of five binary mixtures were calculated and
the results are shown in Table 7. Values of & were not

.2

a
.2 .4 .6 .a 1.0

Fig. 7. Vapor-liquidequilibria in the Methanol-water system at

373.2K.

used. Average deviations between calculated and

experimental densities were of the order of 2% and
compare favorably with deviations obtained using the
I Chaudron equation[17] with six constants per com-
0 .z .4 .b .8 1.0
ponent. A major advantage of using the new equation is
MOLE FRACTION that it gives accurate predictions of saturated liquid
Fig. 5. Vapor-liquid equilibria in the Co,-n-butane system at volumes as well as other properties of the equilibrium
213.2K. phases. Earlier, we showed that the new equation is
superior to the Soave and Peng-Robinson equations in
the representation of the saturated liquid densities of
pure substances. We found that this improvement

1
extends to mixture saturated liquid densities as well.
bo
Results for the n-butane-n-decane system are shown in
Fig. 8. As can be seen, the new equation is superior to
the Peng-Robinson equation.

CONCLUSION
This work demonstrates that the new equation of state
is capable of accurate and consistent predictions of the
thermodynamic properties of mixtures. The most inter-
esting feature of the new equation is its applicability to
I mixtures containing heavy hydrocarbons and polar sub-
b 1 .4 -6 .e f-0
stances, and the fact that it is cubic in volume and thus
HOLE FRKTtaJ easy to handle. It can reproduce with sufficient accuracy
Fig. 6. Vapor-liquid equilibria in the Methane-n-eicosane system at the liquid and vapour phase densities and yield very
313.2K. accurate VLE predictions. Comparisons have shown that

Table 7. Comparisonof density predictionsof binary mixtures

Ethawpropene 174 4X-858 15-10000 0.49% 2.n 1.88

Ethamn-pentam 152 498-920 200-10000 o.ao00 3.72 3.1,

Prqmr-n-pcntanc 41 619-B"" *o-i00 O.IbBO 0.67 1.27

Pmyne-n-pntane 28 Cm-709 *O-scI" 0.6511 ".hh 0.37

Propene.propane 268 470-858 15-10000 0.6289 1.92 1.79

H2S-n-pentane 123 498-800 zoo-1OOOC 0.6123 2.95 2.01

CESVol.37,~o.1-1
472 N. C. PATELand A. S. TEJA

-15 .20

MLUllE ( MS IWO?)

Fig. 8. Comparisonof saturated liquid volumes of n-butane-n-decane mixturesat 377.6K.

for VLE calculations, the new equation is as good as the Subscripts

Soave and Peng-Robinson equations for mixtures of c critical value
light hydrocarbons. For systems containing heavy talc calculated value
hydrocarbons and polar substances, the new equation is exp experimental value
superior to the Soave and Peng-Robinson equations. i, j component i, j
j ith data point
Acknowkdgement-NCP thanks the ScienceResearchCouncilfor R reduced value
the awardof a Studentship for the duration of this project. _
REFERJINCE’S
[I] Soave G., Chem.Engng Sci. 197227 1197.
[2] Peng D. Y. and Robinson D. B., Ind. Engng Chem. Fundls
NOTATION
1976 15 59.
a,6,c Constants in eqn (I) [3] Fuller G. G., Ind. Engng Chem. Fundls 1976 15254.
B = bP,lRT, [4] SchmidtG. and Wenzel H., Chem. Sci. 198035 1503.
[5] Harmens A. and Knapp H., Ind. Engng Chem. Fundls 1980
C constant in eqns (12 and (13) 19 291.
f fugacity [a] MollerupJ., as quotedin Ref. 5.
F parameter required in eqn (1) [7] Leland T. W., A.I.Ch.E, 1. 1%6 12 1227.
rn number of components IS] Abbott M. M., AJ.Ch.E. L 1973 19 5%.
191Abbott M. M., Equations of state in chemical engineering
number of moles
and research, Chao K. C. and Robinson R. L.. Vol. 128.
;: pressure ACS SymposiumSeries 1979.
R gas constant [lo] Lcland T. W. and ChappelearP. S., Ind. Engng Chem. 1968
T temperature 60(7) 15.
S entropy [II] GrabowskiM. S. and Daubert T. E., Ind. Engng Chem. Pm.
Des. Lku. 1978 17 443.
molar volume [12] Heyen G.. Proc. of the 2nd Inr. Con/. on Phase Equilibtiu
; total volume and Fluids Proverties in the Chemical Induftm,, Vol. 1.
mole fraction Dechema, Frankfurt 1980.
2 compressibility = Pu/RT 1131Starling K. E., Fluid Thermodynamic Pmverties of Light
Hydroiarbon Systems. Gulf, H&on 1973:
[14] Adler S. B., Spencer C. F. and Ozkardesh H., Phase &pi/-
Symbols ibria and Fluid Properties in the Chemical Industry, (Edited
temperature function, eqn (11) by Storvick T. and Sandier S. I.), Vol. 60, p. 150. ACS
a” temperature function, eqn (16) SymposiumSeries 1977.
[lS] Patei N. C., Ph.D. Thesis, Lo&borough Universify, 1980.
ic * value of P,u,/RT, calculated from eIqn (1) [16] Chueh P. L. and Prausnitz 3. M., Ind. Engng Chem. Fundls
parameter required in eqn (1) 1%7 6 492.
binary interaction coefficient [17] ChaudronJ., Asselineau t. and Renon H., Chetn. Engng Sci
constants in eqns (5-7) 1973 28 839.
 A new cubic equation of state for fluids and fluid mixtures 473

APPENW Entropy departure,

Derived properties using the new equation of state
Fugacity.

cp= conversion factor; Q, M, N are given above

Fugacity of component i in a mkture.

B = bPIRT
RTln($)=-RTln(Z-B)+RT($)-ph(@)
hf= (+J)&

+w+a{c,(3b+c)+b,(3c+b)}
2@-d ) 8d’ ’
N= bc+q]-“*
1
X~n(~)+*)
Q= (?+!$)A
Q=“+b+c
Enthalpy departure. 2

B = bP,IRT,
rp(H-H”)-RT(Z-I)-(TV-..a) [&ln($$)]

d= &,c+!!!..$$
p = conversion factor; Q, M, N are given above