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Equations of State for the Calculation of Fluid-Phase Equilibria
Y.S. Wei and R.J.Sadus, AIChE J., 46, 169-191, 2000
Kim, Yong-SooThermodynamics & Properties Lab.
Korea University
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Introduction
Advantages of using Equation of State(EOS)
Wide ranges of temperature and pressure
Application of mixtures of diverse components
Various phase equilibria without any conceptual difficulties
This work
An overview of recent progress in EOS
Simple empirical EOS
Theoretically-based EOS
Relationships between different EOS
Role of molecular simulation data
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EOS for simple molecules
van der Waals EOS (vdW) (1873)
Hard-sphere (repulsive) + Attractive intermolecular interactions
A qualitative description of phase transitions
Inadequate to critical properties and phase equilibria
Requirement of modifications of attractive and repulsive terms
RTVa
bVVZ −−
=
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EOS for simple moleculesModification of Attractive term
Benedict-Webb-Rubbin EOS (1940)
Disadvantage
Requirement of plentiful, accurate PVT and VLE data for parmamter estimation
Difficulty of extension to mixtures
−
++
+
−+
−−+=
2223
52
2000
exp1
/1
VVVRTc
RTVa
RTVabRT
RTVTCARTB
Z
γγ
α
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EOS for simple moleculesModification of Attractive term
Redlich-Kwong EOS (1949)
Significant improvement over the vdW EOS
The impetus for many further empirical EOS
cc
cc
PRTbPTRa
bVRTa
bVVZ
/0867.0/4278.0
)(5.22
5.1
==
+−
−=
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EOS for simple moleculesModification of Attractive term
SRK (1972)
Prediction of phase behavior of mixtures in the critical region and improvement of accuracy of critical properties
cc
cc
c
pRTbm
TTm
pTR
Ta
bVRTTa
bVVZ
/08664.0 176.057.1480.0
114274.0)(
)()(
2
25.022
=−+=
−+
=
+−
−=
ωω
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EOS for simple moleculesModification of Attractive term
Peng-Robinson (1976)
Slight improvement of the predictions of liquid volumes
Superior to the VLE in hydrogen and nitrogen containing mixtures (Han et al., 1988)
[ ]
cc
cc
c
pRTbk
TTk
pTR
Ta
bVbbVVRTVTa
bVVZ
/07780.0 26922.05422.137464.0
1145724.0)(
)()()(
2
25.022
=−+=
−+
=
−++−
−=
ωω
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EOS for simple moleculesModification of Attractive term
The advantages of SRK and PR EOSs
Easy representation of the relation among temperature, pressure, and phase compositions in multicomponent systems
Only requirement of the critical properties and acentric factor
Little computing time
Overestimation of saturated liquid volumes.
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EOS for simple moleculesModification of Attractive term
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EOS for simple moleculesModification of Repulsive term
Reproducibility of complex phase transitions such as LLV equilibria.
molecules ofgeometry arbitrary for )parameter(ity nonspheric α
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EOS for simple moleculesModification of Repulsive term
Hard-sphere compressibility factors from different EOS with molecular simulation data
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EOS for simple moleculesCombining modification of both attractive and repulsive terms
Carnahan and Starling (1972)
The prediction of hydrocarbon densities and supercritical phase equilibria.
Chen and Kreglewski (1977)
The substitution of attractive term with the power series fit of MC data by Alder et al. (1972)
This attractive term is the inspiration for further development.
)()1(1
5.13
32
bVRTaZ
+−
−−++=
ηηηη
j
i
i
ijj kT
uDjZ
−
−−++= τ
ηη
ηηη3
32
)1(1
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EOS for simple moleculesCombining modification of both attractive and repulsive
termsShah et al. (1994)
Requirement of 3 properties of fluids : Tc, Vc, and acentric factor
Quartic equation, but it behaves like cubic equation.
Lin et al. (1996)
Extension to polar fluids.
Need of dipole moment
))(()( 0
02
0
1
0 αα
αα
α kVeVRTckaV
kVVk
kVVZ
−++
−−
+−
=
Repulsive Attractive
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EOS for Chain MoleculesPerturbed hard chain theory
Prigogine (1957)
Rotational and vibrational motions are depend on density
=> EOS and configurational properties are affected.
Nvr
NNf
N
N
qkTV
VNVQ )(
2exp
! ,3
−
Λ
= φ
)()()(),( int
1
3int TqV
Tqvqvrqc
fext
−
Λ
==
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EOS for Chain MoleculesPerturbed hard chain theory
Beret and Prausnitz (1975)
Development of PHCT EOS
More accurate expressions for repulsive and attractive partition functions
Meeting the ideal gas law at low densities
Deficiency in Prigogine’s theory
1
2 )( and )1(
)43(exp−
=
−
−=c
fextf V
VVqVV
ηηη
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EOS for Chain MoleculesPerturbed hard chain theory
Equation of State
Parameters :
Sucessful in calculating the various properties of fluids and phase equilibria
A practical limitations as a result of the use of Carnahan-Starling free-volume term and the Alder power series
Simplifying the PHCT EOS
= =
−−
+
−−+=
4
1 111
03
2
~1
~)()1(
41n
M
mnm
nm
TvmA
rvkTV
qcZ εηηη
ckqrv ,/,0 ε
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EOS for Chain MoleculesSimplified perturbed hard chain theory
Kim et al. (1986)
Parameters :
The SPHCT EOS retains the advantages of the PHCT EOS.
YVVYcVZ
cZ m*
*
3
2
)1(241
+−
−−+=
ηηη
ckqTVVT
TY / ,/ ,12
exp ***
ετη ==−
=
** ,, VTc
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EOS for Chain MoleculesHard-sphere chain theory
Wertheim’s thermodynamic perturbation theory (TPT)
The association site are replaced by covalent, chain-forming bonds.
Chapman et al. (1988) : Generalization of TPT
Zhs is Carnahan-Starling equation.
)(ln1)1(
∂
∂+−−=η
σηhs
hshc gmmZZ
3)1(22)(
ηησ
−−=hsg
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EOS for Associating FluidsStatistical associating fluid theory (SAFT)
Chapman et al. (1988, 1990)
NkTA
NkTA
NkTA
NkTA
NkTA assocchainsegideal
+++=
+
−−=
i j
ji
ij
seg
kTuDm
NkTA
τη
ηηη
3
2
)1(34
3)1(2/1ln)1(
ηη
−−−= m
NkTAchain
MX
XNkTAassoc
21
2ln +
−=
α
αα
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EOS for Associating FluidsStatistical associating fluid theory (SAFT)
Huang and Radosz (1990)
assocchaindisphs ZZZZZ ++++= 1
+
−−=+
i j
ji
ijdisphs
kTujDmZZ
τη
ηηη
3
2
)1(24
])2/1(1)[1()2/5()1(
2
ηηηη
−−−−= mZ chain
ρρ α
α α ∂∂
−= X
XZ assoc
211
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EOS for Associating FluidsStatistical associating fluid theory (SAFT)
Development of variable SAFT model
Simplified SAFT : Fu and Sandler (1995)
Galindo et al. (1996) : The expression of Boublik for the hard-sphere contribution
LJ-SAFT (Banaszak et al., 1994), VR-SAFT (Gil-Villegas et al., 1997), and so on.
YVVYVZ
mZs
mdisp*
*
+−=
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Comparing EOSInterrelationships between different EOS
New EOS
Modification of existing ones
Reuse of successful EOS to form a new EOS
The branches in next figure show different ways of representing intermolecular repulsion.
van der Waals, Carnahan-Starling, HCB, PHCT, and TPT
The precursor for the development of EOS
SRK in empirical EOS’s PHCT and SAFT in theoretical EOS’s
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Comparing EOSInterrelationships between different EOS
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Comparing EOSComparison with experiment
Experimental data
The ultimate test of the accuracy of an EOS
No absolute quantitative judgments about the relative merits of competing EOS
Why is an absolute quantitative judgments difficult?
EOS developers test their EOS against experimental data, but not offer an identical comparison with other EOS.
The accuracy of EOS is often dependent on highly optimized EOS parameters
EOS users adopt a favorite EOS with which they become expert in using.
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Comparing EOSComparison with experiment
The true value in using a theoretical EOS
Their improved ability to predict phase equilibria rather than merely correlate data.
Correlation of experimental data with PR/SRK at low pressure
=> Failure of the prediction of phase equilibria at high pressure Breakdown of vdW repulsion term
=> Using Carnahan-Starling or Guggenheim repulsion term
Ability of calculating full range of phase equilibria of mixtures.
Theoretical EOS’s, such as SAFT and PHCT are promising approaches.
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Comparing EOSComparison with molecular simulation data
Molecular simulation
Provision of exact data to test the accuracy of theory
Discrepancies between theory and MC
=>Failure of theory to represent the underlying model
Direct comparison of a theoretical model with experiment
=> No useful information
Direct comparison of a simulation-verified modelwith experiment
=>To indicate the strength or weakness of theory
Example => Show figure
Carnahan-Starling and Guggenheim equation is accurate !
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Comparing EOSComparison with molecular simulation data
Failure of accuracy in comparison of EOS with MC
Not merely due to the failure of theory to represent adequately the underlying model
Because of the limitations of theory to model the real molecules
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Conclusion
To meet the challenge posed by large and complicated molecules, EOS are being developed with an improved theoretical basis.
These new EOS are playing an expanding role in the calculation of various phase equilibria.
Molecular simulation have an ongoing and crucial role in the improvement of the accuracy of EOS