Fundamental Equations of State Based of Hybrid Data

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


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


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


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


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


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).


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


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)


EoS: E.W. Lemmon, R. Span, J. Chem. Eng. Data, 51, 785 (2006).

„internal energy“

„pressure“

„compressibility“

„isochoric heat capacity“ „thermal pressure

coefficient“


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.


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


Penoncello (rec. by NIST) this work

% D

evia

tio

n

Temperature / K

Temperature / K

Isochoric heat vapacity (cyclohexane)

Speed of Sound (cyclohexane)


this work Span-Wagner (rec. By NIST)

CO2

Pressure / MPa Pressure / MPa

% D

evia

tio

n (

iso

ba

ric h

ea

t ca

p.)


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


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


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


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



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


-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


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


Thank you for listening!

This project is funded by


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%


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

Documents

Fundamental Equations of State Based of Hybrid Data