1993 20 years of RK EoS
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1993 20 years of RK EoS

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It can be seen
also that the criticaL isotherm is best reproduced with a c value
corresponding to a critical compressibility equal to, or slightly
lower than L/3, i.e. with c values very similar to those determined
from low-temperature liquid densities. That justifies the common
assumption of a constant c va1ue, but prevents one from expecting
miracles in the critical area, which is the weak point of aII cubic
c) Parameter mixinq rules
From theoretical considerations, a linear mixing rule for b
and a quadratic one for ilart can be derived. A linear mixing rule
involves no binary parameter, but for the attractive parameter, in
a few cases only (e.9., mixtures of hydrocarbons of the same type),
the generalized expression can be applied:
a- (Ex ifr)'
resulting from the more general quadratic rule:
a=EEx.x .a.LJ I J rJ
when letting-
a. .=ú.-IJ I f
More generaIJ.y, even for
by hydrocarbons, dr empirical
mean must be applied:
regular mixtures such as those formed
correction term over the geometric
cl- .=l1l l-k..)r'a.a.Ll' 1 l (1s)
which, in principle, has to be determined by correlation of
experimental VLE data. Fortunately, Urj values can be considered as
independent of the temperature and are almost constant for each
class of compounds (e.9., all COr-alkane pairs), so that their
values can be estimated from similar systems, when experimental
data are lacking.
The above mixing rules are quite adequate with mixtures of
hydrocarbons and sIÍghtly polar compounds, such as carbon dioxide
and hydrogen sulfide, but with systems containing polar components
they give only a first-order approximation and must be replaced by
other, more sophisticated rules, involving at Ieast two empirical
parameters for each binary pair.
Theoretical considerations prevent assuming other than
quadratic mixÍng rules for a and b, in order to keep a quadratic
mixing rule of the second virial coefficients (B=b-a/RT); so one
proposal bras to use a quadratic rule for b also, thus introducing a
second binary constant. UnfortunateJ.y the two binary constants in a
and b are correlated and for strongly non-ideal systems they tend
to assume absurd val-ues. Furthermore, it can be seen that, in order
to have a correct mixing rule for higher virial coefficients also,
a linear mixing rule of b would be required.
Another solution satisfying a quadratic mixing rule of the
second virÍal coeffieient is given by the so-calIed
density-dependent mixing rules ( Lr-idecke, Prausnitz, 1985) ,(MolIerup,1985), where a quadratic rule is applied for the
attractive parameter when the density tends to zero, and a
non-guadratic one at non-zero densÍties. A major drawback of this
solution is that, at least with mixtures, the resulting equation of
state is no longer cubic.
Most commonly, authors applied a Iinear rule for b, focusing
on the attractive parameter to give the required flexibility (and
forgetting the quadratic rule).
One proposal (Kabadi,Danner,L985), arising from the attempt to
reproduce the mutual solubility of water and hydrocarbons, was to
use Iinearly composition-dependent Orj values, which results in
different k. 
values in the two liquid phases and a cubic mixing1J
ruLe f or ttatt .
Another, very attractive proposal, which opened a way to the
rich fields of the liquid state theories, was that by Huron and
Vidal (Huron,Vidal ,L979). By the simple, reasonable assumption
that, at infinite pressure, the excess Gibbs energy tends to a
finite limit, a linear mixing rule of b was confirmed and the
following rule was derived:
a/b=Ex . a, /b, -g:/ 1n/=Ex, ( a, /b, -IrV:/ 1n2 ) (l_6)
Eq.(16) in fact is useLess in determining the interaction
parameters, but it allows to defÍne the analytical form of the
mixing rules, by simply assuming tor gl one of the models available
from Iiquid-state theory.
Besides, connecting the nixing rules to an excess Gibbs energy
makes it possible even to predict nI O" applying the principle of
group contributions (just like in the well known UNIFAC method). In
the Iast years, two predictÍve models (Tochigi et aI.,1990),
(Fredenslund et a1.,1991) have been presented, based on the SRK EOS
and a UNIFAC-type f@ model.
Referring to the infinite-pressure excess properties avoids
any reference to a physical state and allows the treatment of
supercritical components aIso, which was impossible with all the
methods based on liquid-phase activity coefficients. If a table of
infinite-pressure group contributions is developed, it can be thus
extended to substances like hydrogen, nitrogen, methane, carbon
dioxide, etc. , what r^ras impossibi le with UNIFAC and simi Iar
A similar, but not identical approach has been followed with
the Peng-Robinson EOS by Abdoul et al. (1991-), and with the R-K and
P-R EOS by Lermite and Vidal ( 1-992 ) .
Another attractive solution was proposed by Michelsen
(Michelsen, 1990ab), (Michelsen,Dahl, 1990), to predict high-pressure
V-L equilibria, when no experimental data are available. It uses T
values generated by a group- contributÍon method like UNIFAC, to
defÍne the mixing rules of the SRK equation of state and so predict
directly VLE K values. The method appears simpler and more
efficient than previous versions (MoIIerup, L986), (Soave,1986),
(Gupte et a1.,1986); although restricted theoretically to mixtures
of subcritical compounds (or better, to components with reduced
temperatures lower than about 0.9) it can be easily extended to the
light gases. Basing on Michelsen's work, Gmehling and coworkers
(HoIderbaum, Gmeh1ing, 1991) are no\^r extending the UNIFAC group
contribution table to Iight compounds such as methane, carbon
dioxide and even hydrogen.
Fig.6 shows the phase envelope of the propane-benzene system
at 444.3 K (beyond propane's critical temperature), ds calculated
by Michelsenrs method, with UNIFAC f values: experimental VLE data
hrere correctly reproduced up to the critical point, over 60 bars.
o 05 1
\./ I ^ , ^ ^ ^ ^ ^ \
ljf uudf ìe /
Fig.6. VLE of propane-benzene system
Crosses:experimental- SoIid lines:calc., Michelsen method
In spite of the development of high-speed computers, enabling
the use of more complex (and more accurate) equations of state,
cubic equations, and in particular the R-K equation in its many
versions, seem still far from having concluded their rday. Not
particularly accurate, but easy to solve and program, moderate in
computer requirements, easy to adjust, they remain the work-horse
of students, researchers and designers.
Further developments are stilI needed anyway, in particular to
improve their predictive ability; some of them are outlined here:
a) improve the vapor pressure prediction in the low temperature
i'essrre t ar
range: expressions used normally for o(T*) are usually accurate
in a pressure range from, say, I/IOO bar upwards, but the
heaviest components of a mixture can have much Lower saturation
pressures and any error in their evaluation can affect seriously
the calculated dew-point conditions. It is necessary to develop
new ot functions which can be extrapolated with confidence to
much Lower reduced temperatures than presently.
improve the treatment of heavy hydrocarbons: first, by making
their parameters in the a function predictable from available
properties. On such matter, a good work is being carried on at
the University of MarseiIIe for the Peng-Robinson EOS (Carrier
et aI.,1988; Rogalski et al., l-990). Second, methods for the
estimatÍon of pseudocomponent binary interaction constants
should keep into account their higher or lower aromatic nature,
by the use of parameters bound to it (Iike, e.9., the blatson K
improve high-temperature, high-pressure behaviour: current ot(T*)
functions are