m
m
I Sodaty of PotrobumEn@wra I
SPE 26667
A Simple Cubic Equation of State for Hydrocarbons and
Other Compounds
Mohsen Mohsen-Nia, U, of Illinois;HamiciMrxidaress,Amir-KabirU,; and (3A, Mansoori,*
U. of illinois
“SP~ Member
E
C$fWhl1SS3,
SOMY
This paper wae wcpued
of Petroloum Englneara,
for preaemalien
1.s,
m tfw SSlfr Annual T4cfmical ConfSlenCO and Exhlblllon 01 the Soclsly of P9ttOImim Englnoers held In Houslon, Texas, 3--S Octobe? 1993.
Thla papaf wan eekctad fof praaematkm by an SPE Program Commltfee followlng revlow of Information mntalrrad In M abatrac! submitted by the author(s). Contents Of Iho paper,
M pfaeamad, havo m baen rovkwed by tM Soahty 01 Palralcum Eriulnaera and NO sutijeal to cor?eaflon by the author(s). The material, m preeemed, does not nacaaaar::
rellect
MY*
of t~ *fy
of P@*um
E%ffIsw$, M ~ffk.m. W membam. pepera Pr*MntWf at SPE mawngs are Wbjact to publication ravlsw by Edit. rlal Committals of the Society
of Pef@awn Englrrme. Pemtbabn10VWYb reatrkfad teen #mrecf 01 M me than SW wwd:. Illuslrwkms may rwI bo c@led. The sbatracl ahwld centuln consplcuoua acknowledgment
of Wwra srid by whom the paper la pfeaemed. Write Llbfarlan, SPE, P,O. Sax S33S3S, Rkhwdaon, 7!( 76CWMS3S, U.fi.A. Telex, 1S3245 SPEUT.
ABSTRACT
INTRODUCTION
A new simple two-constant cubic equations of state for
hydrocarbons, hydrocarbon mixtures, and other nonassociating compounds is introduced. This equation of
state is based on the statistical mechanical information
available for the repulsive thermodynamic functions and
the phenomenological knowledge of the attractive
intermolecular potential tail contributions to the
thermodynamic properties. This new two-constantparameter cubic equation is in the following form:
and ati =0.48748 R%diw /P@ and ~ii =0.064662RTdi/P~i
It is shown that this equation of state is more accurate
than the Redlich-Kwong equation, which has been
considered to be the best two-constant-parameter cubic
equation of state. This new equation is appiied for
calculation of thermodynamic properties of hydrocarbons
and other non-associating fluids and fiuid mixtures of
interest in the oil and gas industries.
References and illustrations at the end of paper.
651
The success of oil and gas recovery, production, and
processing operations is related to the availability of
accurate and simple equations of state for prediction of
thermodynamic
properties
of fiuids.
Recent
developments in modeling the phase behavior of
reservoir fluids indicate that more wwiousconsiderations
must be given to fundamental approaches to produce
simple and accurate equations of state*-9. Although,
there has been considerable progress made in the
development of equations of state based on the statistical
mechanics, the cubic equations of state are stili wideiy
preferred in industrial calculations. Equations such as
the van der Waals, Rediich-Kwong, Heyen and their
modifications are extensively used in petroleum and
naturai gas engineering caiculationsl-’.
In this report, we first present a simplified, but accurate,
equation for the positive (repulsive) term of equations of
state. men we introduce a new version of cubic equations
of state using the simplified repulsive term deveioped in
part L A new cubic equation of state which contains two
constant parameters is then introduced. The resulting
equation of state is used for property prediction of a
variety of hydrocarbons, other fluids, and mixturee of
practical interest. This equation of state is shown to be a
simple cubic equation with superior accuracy than the
RK equation which has been considered to be the most
accurate two-constant-parameter cubic equations of state.
-t
2
A SIMPLE CUBIC EQUATION OF STATE FOR HYDROCARBONS AND OTHER COMPOUNDS
THE NEW REPULSIVETERM OF EQUATIONS
OFSTATE
The importance of short-range repulsive forces in an
equation of state for determining the thermodynamic
properties of fluids is well established by the works of
numerous investigators.2’by’l’”’G.This is because more
accurate expression for the repulsive term of an equation
of state results in a better agreement with the
experimental data. However, the major weakness of the
industrial equations of state is due to their inaccurate
repulsive term,
z (W) = 1%/RT = v/(v-b)
(1)
.
An accurate equation for representing the repulsive effect
is the MCSL equation which reduces to the following
form for pure fluids%
[64v%16vzb+4vb2
-b3]/[4v-b]3
(2)
This equation can replace the repulsive term, v/(v-b), in
the cubic equations. However, due to the fact that after
this modification the resulting equation of state was not
going to be cubic any more Scott’4 introduced the
following equation for the repulsive term of pure fluid
equations of state.
z ~w) = Pv/RT=
(3)
0.5b)/(v -0.5 b)
(V +
Similarly, Kim, et al. (1986) presented another equation
for repulsive term of equations of statw
z (W) = R/RT
= (V
+
(4)
0.77b)/(v -0.42 b).
Neither of the above two equations are accurate enough
to replace eq.(2)as it is demonstrated in Figure L In order
to produce a simple, but accurate, repulsive term for cubic
equations of state, we consider a number of representative
cubic equations of state in a generalized from as shown in
Table 1.
m
SPE 26667
v~ln appearing in Table 1 is the smallest molar volume
for fluids which can be calculated based on different
equations of state (approximately the molar volume at
the triple-point) when the MCSL repulsive equation of
state is used to modify the above equations (replacing
v/(v-b) with eq. 2). Of course the MCSL equation of
state is valid for v/b values as low as 0.25. Considering
the fact that the v/b in the cubic equations of state
(excluding the van der Waals equation which, because of
its inaccuracy, is not used for practical calculations) do
not go below 0.7 we propose Lhe following repulsive
equation of state for v/b> 0.2
z ,,p = 1%/RT=
z ~rq)=Pv/RT=
~s
(V +
0.62b)/(v-O.47b)
(5)
This equation reproduces the MCSL equation for the
repulsive fluid with good accuracy for v/b >0.7 as it is
demonstrated in Figures 1 and 2. In Figure 1 the
percentage deviation of calculating the repulsive
compressibility factor from the exact MCSL equation as
calculated by eq. (5) is compared with the Scott and Kim,
et.al models. According to tlis figure eq. (5) is in better
agreement with the MCSL equation than the other
simplified repulsive equations for v/b >0.7.
In Figure 2 the repulsive excess Helmholtz free energy
(over the ideal gas) ,
(A- Aig)/RT= - 2.319Ln(l - 0.47b/v)
(6)
as derived from eq. (5) is compared with the similar
calculations using the Scott and Kim, et,al models.
According to this figure eq. (6) is in better agreement
with the accurate MCSL equation of state than the other
simplified repulsive models.
We have extended the applicability of equations (5) and
(6) to repulsive mixtures by writing them in the
following forms
z ~[rql = (v + 0.62bJ/(v
- 0.47bJ
Parameters of a generalized equation of state
(A~- AU,t8)/RT= -2.319 I@ - 0.47bJv)
Z = Pv/RT = Z(q) -(av/F4T)
/ [T&(v2+&-dbc)]
and using the following mixing and combming rules,
(7)
(8)
accordingto differentcubicequations
b~= (1/4)(3~ ~ xixjbij+ ~ Xibii)
Equationof State
~
B
d?
Vud,,h
VanderWads
Heyen
Peng-Robimn
Redlich-Kwong
o
0
0
0.581
Obtcl
o
al
blc
0.806
0.833
1/2
b
o
0.893
ij
where;
(9)
i
bfi= (bU~j3+ b~f3;/8
Eqs. (7) and (8),joined with eq. (9), are tested extensively
for discrete as well as polydisperse repulsive mixtures
and it is shown to be in very good agreement with the
652
M. MOHSEN-NIA, H. MODDARESS AND
WE 26667
accurate, but complicated, MCSL11repulsive mixture
equation for v/b > 0.7,
We have calculated the
thermodynamic properties of equimolar binary mixtures
(X1=X2=l/2)with (bzJb11)1j3 = 1.5,2.0,2.5, and 3.0. The
results of these calculations are compared with the
accurate MCSL repulsive mixture equation of state as
reported in Figures 3 and 4. The maximum percentage
deviation
of the binary
repulsive
mixture
compressibility factor and excess Helmholtz free energy
for molecular diameter ratios of (bJbl#J3
= 1.5 to 3 is
calculated to be 3% for v/b >0.7, which is in the range of
accuracy of MCSLequation.
In order to test further the accuracy of the proposed
repulsive term of the mixture equation for multicomponent mixtures we have used it to calculate
thermodynamic properties of polydisperse (continuous)
repulsive systems through different distribution
functions f(r). Following the method of Salacuse and
StelllGwe can express eq.(9) for the polydisperse systems
as the following
bm= (1/16) (7%+ ~mJ
(lo)
Where, ml, ~ and m~ are the first, second and third
moments of distribution function, respectively, defiied
as:
(11)
ri f(r) dr
~=
[o
For the normal distribution:
f~(r) = (1/2rr@t12exp[-(r- i)2/2q ]
(12)
the moments are given by
rq=i + (2q/lr)l~
q= F* +q + 2(231/n)*/t
(13)
(14)
~ =~3 + 2(2/2r)~l~3J2+ 3q ~ +3~2(2q /rc)112
(15)
where q and ~ are the variance and mean, respectively.
Therefore, the mixture b~/v, eq. (10) for the normal
distribution is as the following
bin/v = 4(@#)
[1+ (~)(~~
rn
+ ( 1.875x + 2.25)q + (~(a)’n]
lrP
?C
(16)
853
G.A.MANSCXXU
3
We can calculate the compressibility factor and
Helmholtz free energy of this polydisperse repulsive
mixture by joining eq. (16)with eq’s (7) and (8),
For the Schultz distribution:
fs(r) = -J-(@)s
r
+ 1 # [-(Q ) r]
r
(17)
where ~ is the first moment of the distribution, and s is
related to variance of f.(r) by,
y2
q(r) = var(r) = ~
5+1
(18)
we can derive the following expressions for the b~/v of
the Schultz polydisperse repulsive mixture
b#v=4 ( pNOp)(@+ 2)(8s + 15),
6V
8(s + 1)2
(19)
‘l’henby replacing this equation in eq’s (7) and (8) we can
calculate the compressibility factor and Hehnholtz free
energy of the Schultz polydisperse repulsive fluids.
We have calculated the thermodynamic properties of
polydisperse repulsive mixtures obeying the normal and
schultz distributions. The results of these calculations
are compared with the accurate polydisperse MCSL
repulsive equation of state*l in Figures 5 to 8. According to
these figures the deviations of the multi-component
repulsive mixture compressibility factor and Hehnholtz
free energy for Shultz (with ~=1 & q =1/3,1/6,1/11) and
normal (with ~=1 & h=O.1,0.3,0,5)distributions are quite
low and acceptable for v/b >0.7 range of interest in
industrial equations of state.
We have also made calculations for values of ~ other
than unity with similar results as to Figures 5-8 which
are not be reported here due to space limitation.
At this stage we propose the following modified
equation of state
Z= Pv/RT
,,
= (v+O.62b)/(v-O.47b)-(av/RT)/T
‘(vz+.h-ci!bc)
(2o)
The difference between this equation and the
generalized equation reported in Table 1 is the
replacement of the new repulsive compressibility factor
given by eq (5). Provided the proper numerical values of
E, ~, m-idd? are replaced in this equation one can
reproduce the new versions of the equations of state
.}
4
4
SPE 24667
A SIMPLECUBICEQUATION OF STATE FOR HYDR0flARBON5
ANDOTHERCOMPOUNDS
volume and contains only two constant parameters. By
applying the critical point constrains, we determine
parameters a and ~ as the following
reported in TabIe 1. It should be pointed out that this
change of the repulsive term will not keep the PengRobinson and Heyen equations in cubic forms anymore.
By applying
the critical
point
constraints:
a =0,48748R2TCS’2
/F’C and b =0.064662RTC/PC
(23)
(~P/ilv)~C=(ii2P/~v2)~C=0
(for all equations); ~d Z = ZC
The critical compressibility factor ZC=0.333will remain
(for the three-parameter equations like Heyen) we can the same as the RK equation of state. In order to
derive expressions for parameters of the new version of demonstrate the validity of the repulsive term of this
equations of state with respect to the critical properties
equation of state we have calculated v~#b as reported
as reported in Table 2.
in Table 3 for different fluids. According to this table for
ali the fluids studied vm,#b values are in the allowable
Xhkl@ Parameters of eq. (20) with respect to the
range of v/b >0.7 where the repulsive part of eq.(22)
critical properties,
accurately represents the MCSL repulsive equatkm of
state.
2,= F’CvC/RTC
y%:
H
Calculation of the minimum v/b ratio
a/RTCvCb/vC c/vc New
Original
for different fluids according to Eq.(23)
Fluid
Van der Waals
1.429
Heyen
1.94$
Fkng-Robinson ~.%l
Redlich-Kwong 1.513
~ese
Mu o
0.371” 0.997’
0.370 0
0341 0
0.364
0.373
0.27
0.27
0.272
0.3U7
0.314
0s333
Ethane
n-Butane
n-Propane
Oxygen
Isobutane
Ethylene
NzOd
HS
valuesare based on ZC=027.
wdb
0.76
0.76
0.73
0.80
0.73
0.80
0.89
0.86
According to this table, except for the Heyen equation for
0.73
Nk3
which ZCis an input parameter, the ZCvalues calculated
based on the h@VV
version of the other equations of state
are closer to the experimental ZCvalues than the ZC’S Extension of equation (22) to mixtures will take the
fol[owing form: calculated by the original form of the equations of state.
(~ ~ aij)/RT
Z=~=v+l.31918M.
- I j
(24)
T1/2(v+ ~ @
THE NEW TWO-CONSTANT-PARAMETER CUBIC
V-&m
1
EQUATION OF STATE
By replacing d =0 in eq. (20) it will become a cubic
equation in the following form
Z= (v -t 0.62b)/(v - 0.47b)- (a/RT)/fi(v
+ ~).
and
(21)
l%e choices of E and ~ are still in our disposaL For the
sake of simplicity we choose ~=1/2, as proposed by
Redlich and Kwongl”, and 4=0.47b as appears in the
denominator of the repulsive term of q. (21). As a result
we introduce a new cubic equation of state in the
following form
z=
(v + 1.13191)/(v -8) - (a/RT)/T1i2(v +8)
(22)
Equation (22) is a cubic equation of state in terms of
b,j=(5,,1/3.+&j;/3)3/8 and
aij= (1- ki,)(ai,aj~.
In eq. (24) we have applied the new mixing rule for
repulsive term (eq. 9), For the attractive term, we have
used the usual cubtc equation rnixiig rules in order to be
able to observe the corrections in the repulsive term of
this equation for mixture calculations. It should be
pointed out that there are also other possibilities for the
mixing rules of the attractive term of this equation of
state17. This new equation of state is used to predict
properties of pure fluids and mixtures as it k reported
below.
654
..
iiE26667
M. MOHSEN-NIA, H. MODDARESS AND GA. MANSOORI
RESULTS AND DISCUSSION
Cubic equations of state may be classified into two
categories: (i) Those with two constant parameters like
the van der Waals, Berthe!ot, Dleterici, and RedlichKwong (RK) equations. (ii) Those with more than two
parameters (or two or more temperature-dependent
parameters) and they include the Peng-Robinson, Heyen
and SoaVeequations 1-7.Among the first category of cubic
equations of state the RK equation is still considered to
be the most accurate equation of all. As a result in order
to test the accuracy of eq. (22), which is also of the first
category of equations, for prediction of the properties of
pure fluids and mixtures it is compared here with the RK
equation. Table 4 shows the comparisons of the result of
density calculations using eq. (22) with the RK equation
and the experimental data. According to this table the
result of eq. (22) are generally superior to the RK
equation for 17out of 19pure fluids tested.
In order to test the applicability of eq. (24) for mixtures,
we have used it for calculation of the properties of a
number of systems. In all the calculations reported here
for eq. (24), and for the RK equation, we have chosen the
value of the unlike-interaction parameters, kij’s,equal to
zero. This choice of kij=O is made in order not to
complicate the comparisons of different equations of
state with the fitting of their ki~s. However, for more
accurate calculations of mixture properties this
parameter may befitted to some mixture data.
On Figures 9 and 10 the density calculations for two
different binary high-pressure (500 bar) liquid mixtures
(n-octane + ndodecane and n-decane + n-tetradecane) are
reported and are compared with the results of the RK
equation. According to these figures predictions by eq.
(24) are better than the RK equation.
As it was mentioned earlier , in all the calculations
reported here, the unlike interaction parameters, k,~s,
are assumed to be zero. By choosing appropriate non-zero
kij values the proposed equation of state would
accurately correlate and predict properties of the
mixtures reported in thk paper. The Icjjdata bank for
this equation is not reported here due to space limitation.
The proposed equation of state maybe made even more
accurate by choosing its parameters (a & b) to be
dependent on temperature as it is usually done in other
industrial equations of statel-’.
ACKNOWLEDGEMENTS
This research is supported in part by the National
ScienceFoundation, Grant No. CTS-9108595.
NOMENCLATURE
A=
a
b
c
fz(R)
K,i
NO
P=
R=
r
F=
for mixtures. On Figure 11 the volubility of methane innhexane is reported versus temperature at two different
pressures of 40.5 and 81.1 bars. According to this figure
also the predictions by eq. (24) are closer to the
experimental data than the 12Kequation.
In Figures 12 to 14 equilibrium pressure-composition
diagrams for methane + n-decane (at 344.26 K), carbon
dioxide + n-decane (at 344.26 K), and for nitrogen +
carbon dioxide system (at T = 270.0 K) are reportwi
According to these three figures the predictions by eq.
(24) are far better than the predictions by the RK
equation.
=
=
=
=
=
.
=
T=
x
z=
We also have performed volubility and VLE calculations
5
s
=
=
Hehnholtz free energy
attractive constant in equation of state
repulsive constant in equation of state
third parameter in equations of state
distribution function
binary interaction parameter
Avogadro number
pressure
Universal gas constant
random variable
the mean value and first moment in
normal and Schultz distribution function
respectively
absolute temperature
mole fraction
compressibility factor
a parameter in Schultz distribution
function, defined in eq.(19)
Greek letters
=
density
P
z=
3.1415927
variance of normal distribution function
l’1=
Subscripts
=
c
i,j
=
ij
=
ig
=
m=
N=
critical property
component indices
property of i-j interaction
ideal gas property
mixture properly
normal distribution
Schultz dist~bution
..
6
A SIMPLE CUBICEQUATION OF STATEFOR HYDROCARBONSAND OTHER COMPOUNDS
,.
SW 26667
REFERENCES
1, Eclmister, W.C. and Lee, B. I.: Applied Hydrocarbon
Thermodynamics, Gulf Pub. Co., Houston, TX,1986.
2, Elliot, J. and Daubert, T.: “Revised Procedure for
Phase Equilibrium Calculations with Soave Equation of
State; I &EC Proc. Design & Develop., 1985, vol 23, pp.
743-748,
12. De Santis, R., Gironi F., and MarreIli, L,: “VaporLiquid Equilibrium from a Hard Sphere Equation of
State: Ind. Eng, Chem. Fundamentals., 1976,15,183.
13. Kim, H., Lin, H.M., and Cimo, K.C,: “Cubic Chain of
Rotators Equation of State,” Ind. Eng. Chem.
Fundamentals., 1976,Vol. 25, No. 1,75-83.
Du, P.C. and Mansoori, G.A.: “Continuous Mixture
Computational Algorithm of Reservoir Fluid Phase
Behavior Applicable for Compositional Reservoir
Simulation” SPE Paper # 15953,Proceedings of the 1986
Eastern Regional Meeting of SPE and Proceedings of the
9th SPE Symposium on Reservoir Simulation, Society of
Petroleum Engineers, Richardson, TX, 1986.
14. Scott, R.L.: Physical Chemistry, an Advanced
Treatise, Henderson, D., cd., Vol. 8A, Chapt. 1,
Academic Press, New York, 1971.
4. Benmekki, E.H. and Mansoori, G.A.: “Minimum
Miscibility Pressure Prediction with Equations of State;
SPE Reservoir Engineerin&, May 1988, 559-564.
16. Salacus, J.J. and Stell, G.: “Polydisperse System
Statistical Thermodynamic,
with Applications to
Several Models Including Hard and Permeable Spheres:
J. Chem. Phys., Oct. 1982, 77(7),3714-3725.
3.
5. Chorn, L.G*and Mansoori, G.A. (Editors),: CT+Fraction
Characterization Advances in Thermodynamics, Vol. 1,
Taylor & Francis Pub. Co., New York, N.Y., 1988.
6. Mansoori, G.A. and Savidge, J.L.: “Predicting
Retrograde Phenomena and Miscibility using Equations
of State~’SPE Paper No. 19809, Proceedings of the 1989
Annual SPE Meetin& Society of Petroleum Enginews,
Richardson, Texas, 1989.
7. Ahmed, T.: “Hydrocarbon Phase Behavior”, Gulf Pub.
Co., Houston, TX,1989.
8. Kawanaka, S. Park, S.J. and Mansoori, G.A.: “Organic
Deposition from Reservoir Fluids” SI’E Reservoir
Engineering Journal, May 1991,185-192
9. Escobedo, J. and Mansoori, G.A. “Heavy Organic
Deposition and Plugging of Wells (Analysis of Mexico’s
Experience),:” Proceedings of the II LAPEC, SPE Paper #
23696, Society of PetroIeum Engineers, Richardson, TX
March, 1992.
10. Redlich, O., and Kwong, J.N.S.: “On Thermodynamic
of Solutions V: Chem. Rev., 1949,44,233.
11. Mansoori, G.A., Carnahan, N.F., Starling, K.E., and
Leland, T.W.: “Equilibrium Thermodynamic Properties
of the Mxture of Hard Sphere: J. Chem, Phys., 1971,54,
1523; see also Carnahan, N.F., and Starling, K.E.:
“Equation of state for nonattracting spheres: J. Chem.
Phys., 1969,51,635-636.
15. Haile, J.M. and Mansoori, G. A.: Molecular-Based
Study of Fluids, Advances in Chemistry Series, Vol. 204,
ACS, Washington, D.C., 1983.
17. Mansoori, G.A.: “Mixing Rules for Cubic Equation of
State; ACS Symposium Series 300, 1986, Part 15, 314330,, Washington, D. C.
18. Takagi, T,, and Teranishi, H.: “ Ultrasonic Speed and
Thermodynamics for Binary Solutions of n-Alkanes
Under High Pressures:’ Fluid Phase Equilibria, 1985,20,
315,
19. Shim, J. and Kohn, J.P.: “Multiphase and Volumetric
Equilibria of Methane-n-Hexane Binary System at
Temperatures Between 110°and 150°C~J. Chem. & Eng.
Data, 1962,7,3-8.
20. Reamer, H.H., and Sage, B.H.: “Phase Equilibrium in
Hydrocarbon Systems, Volumetric and Phase Behavior
of the n-Decane+Carbon dioxide System} J. Chem. Eng.
Data, 1963, 8,508-513.
21. Reamer, H,H., Fiskin, J.M., and Sage, B.H.: “Phase
Equilibria in Hydrocarbon Systems, Phase Behavior in
the Methane-n-Butane-n-Decane System,” Ind. Eng.
Chem., 1949, 41,2871-2875.
22. Somait, F.A. and Kidnay, A.: “Liquid-Vapor
Equilibria at 270 K for Systems containing Nitrogen,
Methane and Carbon dioxide: J. Chem. Eng. & Data,
1978,23,301-305.
656
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for density
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SPE26667
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1.7
3.4
2.6
3.1
3.9
2.8
3.3
2.8
2.2
2.6
2.5
3.2
3.7
4.7
6.4
6,2
3.1
26
1
.8”-
3.
Z
2
2.
/
1-
If&
q
%,
-2
%,
%,,,
%,%,
.
,/
../
,/
,
#/,,/’
,,**
Eq~
Eq(d)
A@’”
/
*P”
“’’’’’’”%,, ,,,,,,,, ,*#J” \+
e
1,6
s,
#_
..-
-’%,
-1-
,“”~
,<,,.’”
,#,,*
SCOT
Mm ttt.a
,,,,,,,
SCOTT
4+
o
am
01
am
CM
a25
aa
a26
Cs
c#-
I
tl
ac6
al
a\6
Rrmiwd
Rfxiuctsd Denslly
a2
QZEI
OensfW
a3
a3s
I 4
Ffgura 2. ‘211c excasa Helmholtz free enm& awmflng to different approxlmatfonz
utd the exact MCSL fmrd ephere equ.et mr of stao? versus reduced density, fr/(4v), for
the pura hard sphete fluld.-
Ff&arr 1. TM prrrmtsge
davfaffon of comprrmlbillty
fmlor from the ●xact MCSL
frstd.zphrrs
eqrratfon of smte venue the rrducad dmd~.
WvL
for th~ Pure hatd
spbare-flutd.
-
4>
I.e
s
1.6
ji
2.
1.4
;
●
●
**
1-
I
./’
Q
-q
....
&?/dll
)
,/’
a 1.6
~,,a,’s.~$:
::)
i
#“’
...*-
ah
-1
am
a4
da
at
Reduced Density
Fl$um 4- Tfm excess Helmholtr free ene~
●ccording to Eq. (9) and the exact MCSL
qrmf{on of etate vema the reduced denelty, (btl+b#W
for
hard sphere mixture ●
Ffguas S - l%e pertcnfage dwiatlon of cmopmmlbllity fmtor from the easct tnfxtww
MCSL bard sphrrs equatfon vmmu the Qhrtum zaduc6rf density, th,t+b#W),
md
ferdfffermtt
mtrlccufsrdbetetett
etim
fvrbkmry
mlxterazof
the bhrary mixture
hzrrf sphere flufde.
hard sphere fluid With (b~bj)”a
than 3 the two theorfea would match even better
W7
~.
For %fit~”s
thanthfa reported care.
ta~lo~ ltia
-= Ver. =
o
a60
am
al
Calff
a2
t126
Reduced DensItv
LL3
I
cl=
a4
Reduced lXrnrJty
Flgrcnr 5. The percmrtagt devkatlon of compressibility
factor from the exsct mixture
MCSL hard sphere cquatfon of state for the polydl!iperae hard sphere rrrlxhcrc with
normal
value
dlstribul[on,
vecatra the mixture
of F = 1 and dlfferznt
vmlances.
reduced
density,
(nf5)N@mY
Figure
hard ephere
and for mean
9
,8
.4
N ..:
Wu.=l]a
-z ,,,,,8,
..*
/
-.-d
free cnerffy according
to t?q. (9) and the exact MCf6L
of attte vcrvua the reduced
hard sphere mixture
dcnalty,
with normal dlstrlbuNon
(n/6)N@mv
for
with mean valae of
less than 0.5 the two ?heorlca
would
match
—
MCX%(W
2
+
=1/~
../
,$
‘M&wr.=1/11)
,9”
*P”
E;(EMvw.=1/q
56”
1.6
s
9’”
E:(Q,@=l/11)
~
,,?’”
,,,9’ * *
s,
/’
%
*,
Vat.=m
.-..
equation
i = 1 and variance = 0.5, For varfances
even better than thle reported caae.
dktrlbutlon la dcffnedby eq.(18).
:
#-*— ..%-1
\
J
mixture
the polydlsperae
moment, my of the normal
The third
6. The excece Helmholtz
Q ;/
~,,,y
,9”?
,9” *
,p’ *
+
.+
+
*
+
‘%
o
0
Laoo
al
als
Li2
Q26
ci3
cA3a
Reduced Density
Flgurc 7- The percentage deviation of hzrd.epher mixture compree$lblllty
factor
from the exact mixture MCSL hard sphere equation of state for the polydlsprcse hsrd
sphere
mixture
with
(d61N#h+3)($*2)/1$+1):,
Shultz
distrlbutlon,
versus
the mixture
and for mean value of ~s 1 and dlfferant
reduced
density,
varlancea.
Figure 8. The excaas hard.sphere m{xttwa HelmhoIk free energy accordlnB to @q. (
and the exact MCSL hard sphere mixture equatfon of state vecaus tht reduc{
density,
(ti6)N#(s+3)ls+2)/lt+l)t,
distclbutlon
for the polydlsper$e
hard sphere mixture
with mean vafue of ~ E 1 and variance
with Shul
= l/S & lfll.
10
#
==
0- .... ... .. .. ........... .. . .. .......
:
c
4
m.
v
= -10=
—
e
o
=
,g
g 43 0
—R(
0.0
. .. . .. . .
. .. ...... ..... .... . .
...”
.
z
= -1o0
=
NEW
.
. . . .
. .
.
. . .
.
. . . . . .
. .
,“...””
—
9
‘s
g -20-
. . . . . . ...4
NEW
—FK
is
●
-30
o-
#
0.2
,
0.4
~
3
#
0.6
0.6
!
1.0
-30~
0.0
0.2
0.4
~
0.6
0.6
f .0
Ffgrme 9. Tire calculated percentage devlatlon of (n-octane + ndodecane)
Ilquld
mixture density fmm the experimental data [18L accordhrjj to the present and RK
miaturc cqu&rM
of state O@N, vemus n.doclecane mole fraction at T.298.lS K arrd
Figure 10. The calculated percentage devlaNon of Iccdectarre + n.tctradecane)
Ilqald
mixture density fmm the cxparfmentd
date [18L accordhrs to the present and RK
mixture equatlona of state fki~O), venue n-tctradcmrre
mole fractfnn ●t TD29R15 K
P=SOObar.
and P=5W bar.
SPi26667
!+----1.0
P m 40.5*.
P = 81.1 biw
z
NEW
NEW
—
—
.“, ,..”...,,..
-*-sI*---1
FO
F
●
m.
Et@
T .344,26
o
NEW
120
m
—
K
o
100
●
80
/
... ... ............. .. . ........ . ........................ ...
o
80
1
1!””’”’”””’’””’”
0.0
200
~so Temperature
K 300
Figure 11. Pccdlctiom
of the euhcbill
of methane
pce$ent and RK mlxtute equations o!/ -tetc (ktrO),
10 ct-hcxsne accordln$
40
o
‘ .[,
.,.
0
~
20
0.s
0:4
0:2
to the
vecmm temperature St two
different pretmcres.The experimental data ceported00 this graph am taken from
1.0
(X1), (Y1)
Ret [19L
of the equilibcham prcssum-composition
dia~m for [czrbon
dioxide+ n-decane)systemaccordingto Me presentand ~ ●
qUa@M Of ~ta?e mJ@)#
FlgcIra KC. Prediction
at 34L26 K. The expccimcntd
data mpoficd
On fhis ~ph
I
I
i
0. I
0,2
0.s
are !ak~n fmm Raf.
[201.
I
w
l—
300-
+
0.8
Elcp.
0
o
2oo-
100*
0
[
0.0
oo~
EXO
I
0.4
Q
[
0.s
(xl ),(Y1)
1.0
FI ure 14 . Prediction
of the equiIibrhcm
pressure.compu.sltion
dfcgram
for
(n ! trogen + carbon dloxfde) system according to the present and RK eqccatione of
state Cc,r=), at T M 270 K. The experlmentaf data reported on this graph me teken
(X1),(Y1)
WCC
13Predlctfon
of the equilibrium
premurr.composition
diagram for
(methoe
+ ndecacre) systcm accordici~ to thr pra$mct and RK equations of Stat*
~
), ● 344.26 K. llw expecfmmrtaf data cepoctcd on this graph are tafwn from Ref.
from Ref. [22f.
El!?
664