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


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forced a so as to reproduce the vapor
pressure slope near the critical temperature, whose values were
estimated as a function of u:. He gave first a Iinear expression for
a(T*), which he Later (hJiIson, L966) modified by introducing a
multiplier LlTo*''2; vapor pressures calculated by such expressions
were very accurate at reduced temperatures approaching unity, but
they tended to diverge at the lower temperatures.
Zudkevitch and Joffe (Zudkevitch, Joffe, L97O) assumed both O.
and Q- to be temperature dependent, and determined the functionsD
na(TR), nb(T*) for single pure compounds, by imposing that
saturated-1Íquid and saturated-vapor fugacity coefficients
calculated at the experimental vapor pressure r^rere equal to those
estimated by a generalized correlation (Lyckman et aI.,1965) of the
kind:
log@s= ( togps )to'*. ( logpt )tn'
parameter. The common choice hras to refer
to its critical value, by multiplying it
ot(T*,<r) replacing I/&quot;î:'&quot; in eq . (2) , while
Tp oro ( TR,. )
=-n-R rR-Qb v*(v*+o')
rnfs= z.-L-ln( Z&quot;-nb) 
- 
(nalob) ln( I *frb/2&quot;) =-0 . 4o7o
349
the attractive parameter
by a correction factor
b was kept constant:
(3)
(4)
In a following paper (Joffe et aI.,1970), na(TR) and Ob(TR)
btere determined by imposing the saturated-Iiquid density value and
the isofugacity constraintr Of=Of, regardless of the values of the
single-phase fugacity coefficients and the saturated-vapor density.
Both methods failed at the critical temperature, since there
the values of Q.,QO are determined by the application of the
critical constraints, and hence both the values of r&quot; and Cs
calculated through the equation of state are defined:
(s)
(6)
SSV=VLV
The impossibility of forcing a two-parameter equation of state
to reproduce critical properties has been recognized definitely.
AII trials to improve the results at the criticat temperature have
been based on the use of a higher number of parameters. It has been
350
also seen that the inability of an equation of state to reproduce
saturated-phase volumetric properties does not affect its
phase-equilibrium performance; so most works have focused on the
correration and prediction of phase equilibria, negJ-ecting the
volumetric properties.
The works by Joffe and Zudkevitch did not provide any
generarized expression for the estimation of na(TR), nb(TR), which
had to be determined for each substance by correlation of
experimental vapor-pressure and liquid-density data.
The first generalized, wide-range predictive equations were
given in 1972 by thJ-s author (Soave,L972). tdhile nU hras assumed
constant, Qa(TR ) was determined from experimental vapor pressures
of non-polar substances and was expressed through a correction
factor o(T*) of the criticat value:
o. ( T* ) =a ( Tn )nu. ( o.&quot;= 0 . 427 48)
By an examination of the curves of
c.,; values (see Fig.4), it was seen that
a good approximation a Iinear function
(fis.5)
(7)
ot(T ) obtained for various
. R'
the square root of or was to
of the square root of T*
05 07 09
reduoed temperature
Sf,udre rccl of alpha
00 07 0B 09
square root or Tr
Fig.4. aJ-pha vs/ reduced temperature
fÍ9.5. The square root of alpha vsl the square root of T
351
6=1+mll-u5 )
' R',
The slope m r^ras assumed to be a
factor o and an analytical expression
(8)
function of the acentric
bras derived for it:
m=0 . 480+ 1 . 574t^r-0 .1.7 6r.o2
The 1972 version of the R-K equation hras the first
equation-of-state based method to be programmed on computers and
applied widely for the design of hydrocarbon-treatment plants. It
enabled the design of new process schemes and nehr working
conditions where the old, activity-coefficient based methods
failed. Before it, equations of state had found a limited
applicatÍon and were considered a theoretical peculiarity.
Its success proved the great potential of this kind of method
and marked the beginning of a flourishing of refinements and new
models, which lasts nowadays. It remains anlrway the first modern
equation of state, and this is the reason for the titte of this
paper.
FURTHER DEVELOPMENTS OF THE REDLICH-KT^IONG EQUATION
It is impossible to describe in one paper the refinements,
modifications and additions which in the last years have been made
to the original model. It is possÍble only to focus on a few
significant improvements, which have found wider application.
a) Pure-component vapor pressures
Eq.(8), although widely applied for cubic EOS's, is only an
approximation; fr is not completely rectitinear in fr*, and
furthermore the calculated vapor pressures are very sensitive to
any error in a at the lower temperatures. As a resuJ.t, calculated
vapor pressures tend to diverge from the experimental ones as the
temperature decreases. Besides, eq. (8) is not applicable to polar
compounds, particularly associating compounds Iike alkanoIs.
In order to extend the application to the low temperatures and
to polar substances, many Authors (Soave,1980), (Harmens,Knapp,
1980), (Mathias,Copeman,l-983), (Mathias, l-983), (Stryjek,Vera, 1986),
(e)
352
(MeJ-hem, 1989) have repraced eq. (8) by some two-parameter equation;
the two parameters must be determined by correlation of
experimental vapor pressures, which decreases the predictive
character of the equation of state and requires experimental vapor
pressure data. The last problem can be partly overcome by using for
a an equation which can be reduced to eq. (8) when necessary; it is
the case of (Soave,1991):
a=1+m( 1-T* ) +n( 1 -fi*f
an equation which becomes identical to eq.(8) when n=m(m+L)
that case m can be determined from <o according to eq. (9) ).
generally, fr,D are determined separately by correlation
experimental vapor pressures, when available.
( 10 )
(in
More
of
(11)
of this modification is that the values of
need not be changed: the effect is to
coefficients in the equilibrium phases by a
their ratio (the K values) are not affected.
original equations for the calculation of
b) Pure-component volumetric properties
It is well known that the R-K equation underestimates the
density of liquid phases by l-0-L5U (according to the value of the
acentric factor and the reduced temperature). That drawback has
been only reduced by using a different two-parameter equation of
state; e.9., the Peng-Robinson EOS yields higher Iiquid densities,
which are nearer to the average hydrocarbon densities, but are
stilr fÍxed. The sorution can onry tie in the use of a third
parameter.
An elegant proposal, which has been applied to other
two-parameter equations also, Ís the one by Peneloux and Rauzy
(PeneIoux,Rauzy,7982); they modified the R-K equation by
introducing a volume shift, not subjected to the critical
constraints and adjustabre for each compound to reproduce
experimental tiquid density values. rf c is the volume shift, the
R-K equation is rewritten as:
RT9--
v+c-b
a(T)(v+c) (v+c+b)
A great advantage
parameters a(T) and b
multiply the fugacity
common factor, so that
This allows use of the
353
parameters and K values, introducing the correction for density
calculations only.
The value of c depends on the substance considered and is
usually assumed constant. Irlhen determined f rom saturated liquid
densities, it varies only slightly with the temperature up to about
T*=0.8, so it is usual practice to determine it from the liquid
density at the boiling point or at T*=0.7 and, for non-po]ar
compounds, to predict Íts value from the value of the acentric
factor.
It is interesting to remark that, although it is possible to
use for each temperature the value which exactly reproduces the
saturated liquid density, that would give a perfect reproduction of
the critical density (the dream of equation-of-state builders!),
but a poor critical isotherm at aII other points.