PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Fundamental Equations of State Based
of Hybrid Data
Frankfurt, 12. 03. 2013
Gábor Rutkai
Monika Thol
Roland Span
Rolf Lustig
Jadran Vrabec
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
R. Span, “Multiparameter Equations of State”, Springer, Berlin (2000)
good knowledge in entire fluid region : ~10 substances
For pure chemical substances…
satisfactory knowledge : <100 substances
For mixtures…
the experimental data availability is much worse
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Equations of state for CO2 (Span and Wagner, 1996)
0 ResF, , ,
R T
7
0 0 0 0 0
1 2 3 i i
i 4
, ln a a a ln a ln 1 exp n
i i i i i
7 34t d t d cRes
i i
i 1 i 8
, a a exp
i i
39t d 2 2
i i i i i
i 35
a exp ( ) ( )
Ideal part:
Residual part:
Helmholtz Energy: T = 216 … 1100 K, p = 0 … 800 MPa
i
42b 2 2
i i i
i 40
a exp C ( 1) D ( 1)
Total:
187 Parameters
5 013 exp. Data
τ =Tc / T δ = ρ / ρc
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Derivatives of the Helmholtz energy
nm
nm
n m n m
A
resART
p011
resres
T
AARTp
020121
resres AART
p11011
resideal AART
E1010
resresideal AAART
H0110101
pressure
internal energy
enthalpy
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
residealv AAR
C2020
resres
resresresidealp
AA
AAAA
R
C
0201
2
11012020
21
1
resid
resres
resres
AA
AAAA
RT
Mw
2020
2
1101
0201
2 121
resresresidresres
resresres
AAAAAA
AAAjtR
02012020
2
1101
110201
211
isochoric heat cap.
speed of sound
Joule-Thomson coeff.
isobaric heat cap.
Derivatives of the Helmholtz energy
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Simulation framework
TF /
nm
nm n m
R. Lustig., Mol. Phys., 110, 3041 (2012).
R. Lustig., Mol. Sim., 37, 457 (2011).
R. Lustig., J. Chem. Phys., 100, 3060 (1994).
A single NVT ensemble
simulation per state
point yields:
ms2 (www.ms-2.de)
S. Deublein et al., Comp. Phys. Comm., 182, 2350 (2011).
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Hydrogen sulfide model
*T. Kristof, J. Liszi, J. Phys. Chem. B, 101, 5480 (1997).
Rigid 1CLJ + 4 point charges
united-atom model*
• Optimized based to VLE data only
• How does it perform in other regions…?
H H
S
M
qH qH
qM
qS
σS
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Hydrogen sulfide
• 50 state points x 9 properties
= 450 measurements
• 1 point takes 10 h (4 cores)
• no user interaction required
• carried out on a cluster (1 day)
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
EoS: E.W. Lemmon, R. Span, J. Chem. Eng. Data, 51, 785 (2006).
„internal energy“
„pressure“
„compressibility“
„isochoric heat capacity“ „thermal pressure
coefficient“
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Overall performance of force fields
13 fluids tested:
Ar, Kr, Xe, O2, N2, HCl, CO2, H2S, SO2, NH3,
CH3OH (methanol), C6H12 (cyclohexane),
C6H18OSi2 (hexamethyldisiloxane)
Molecular models (force fields) perform almost always excellent in the entire
fluid region.
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
simulation
experiment
DA
TA
SE
T
improve
existing EOS
simulation
experiment (VLE only)
yield reasonably good EOS
Target substances: Ar, Kr, Xe, H2S, cyclohexane, methanol
(VLE almost always available) consider all available data
homogeneous region homogeneous region
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Penoncello (rec. by NIST) this work
% D
evia
tio
n
Temperature / K
Temperature / K
Isochoric heat vapacity (cyclohexane)
Speed of Sound (cyclohexane)
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
this work Span-Wagner (rec. By NIST)
CO2
Pressure / MPa Pressure / MPa
% D
evia
tio
n (
iso
ba
ric h
ea
t ca
p.)
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
250
300
350
400
450
500
550
600
650
700
750
0 5000 10000 15000 20000 25000 30000
Density / mol m-3
Te
mp
era
ture
/ K EOS*
EoS: L. Sun and J.F. Ely, Fluid Phase Equilib., 222-223, 107 (2004).
i i
i i i
6t dRe s
i
i 1
8t d c
i
j k 1
, a
a exp
Calculation of VLE without the Gibbs Ensemble or similar methods
H2S
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
250
300
350
400
450
500
550
600
650
700
750
0 5000 10000 15000 20000 25000 30000
Density / mol m-3
Te
mp
era
ture
/ K EOS*
How to predict VLE without the Gibbs Ensemble or similar methods
H2S
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
250
300
350
400
450
500
550
600
650
700
750
0 5000 10000 15000 20000 25000 30000
Density / mol m-3
Te
mp
era
ture
/ K
How to predict VLE without the Gibbs Ensemble or similar methods
H2S
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
250
300
350
400
450
500
550
600
650
700
750
0 5000 10000 15000 20000 25000 30000
Density / mol m-3
Te
mp
era
ture
/ K
How to predict VLE without the Gibbs Ensemble or similar methods
H2S
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
How to predict VLE without the Gibbs Ensemble or similar methods
250
300
350
400
450
500
550
600
650
700
750
0 5000 10000 15000 20000 25000 30000
Density / mol m-3
Te
mp
era
ture
/ K
H2S
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
-2
-1
0
1
2
300 320 340 360 380
-2
-1
0
1
2
-4
-2
0
2
4
% D
evia
tio
n
Temperature / K
Sat.
Liquid
Density
Vapor
Pressure
Enthalpy of
Vaporization
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Molecular simulation data sets
• Particularly useful for EOS development (cost effective, fast)
EOS based on simulation data sets
• Simulation (homogeneous region) + VLE measurements → good EOS
• Offer an alternative VLE calculation
Outlook
• Force fields may be optimized via EOS using EOS
• Mixtures may be tackled as never before
Summary
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
simulation
experiment
improve existing
EOS
Yields reasonably
good EOS
May yield accurate VLE
for the potential model
almost always available
experiment (VLE only)
simulation simulation
DA
TA
SE
T
EO
S f
itti
ng
complex simple
i i i i i
k lt d t d cRes
i i
i 1 j k 1
, a a exp ....
slow, best overall representation Fast, ~30%
PROF. DR.-ING. HABIL. JADRAN VRABEC ThET
Equations of state for CO2 (Span and Wagner, 1996)
0 ResF, , ,
R T
7
0 0 0 0 0
1 2 3 i i
i 4
, ln a a a ln a ln 1 exp n
i i i i i
7 34t d t d cRes
i i
i 1 i 8
, a a exp
i i
39t d 2 2
i i i i i
i 35
a exp ( ) ( )
Ideal part:
Residual part:
Helmholtz Energy: T = 216 … 1100 K, p = 0 … 800 MPa
i
42b 2 2
i i i
i 40
a exp C ( 1) D ( 1)
Total:
49 Parameters
153 Exponents
5 013 exp. Data
τ =Tc / T δ = ρ / ρc