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A research partnership between Boise State University, Idaho National Laboratory, Idaho State University and University of Idaho. Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael Shaltry Pyroprocessing Technology Supathorn Phongikaroon Idaho National Laboratory Dept. of Chemical and Materials Engineering University of Idaho, Idaho Falls Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael Shaltry Pyroprocessing Technology Supathorn Phongikaroon Idaho National Laboratory Dept. of Chemical and Materials Engineering University of Idaho, Idaho Falls August 28, 2012 2012 IPRC Fontana, WI August 28, 2012 2012 IPRC Fontana, WI

Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

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Page 1: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

A research

partnership between

Boise State University,

Idaho National

Laboratory, Idaho

State University and

University of Idaho.

Center for Advanced Energy Studies

Electrochemical Zirconium RecoveryExperiments in Molten Salt System

Robert O. Hoover Michael F. Simpson

Michael Shaltry Pyroprocessing Technology

Supathorn Phongikaroon Idaho National Laboratory

Dept. of Chemical and Materials Engineering

University of Idaho, Idaho Falls

Electrochemical Zirconium RecoveryExperiments in Molten Salt System

Robert O. Hoover Michael F. Simpson

Michael Shaltry Pyroprocessing Technology

Supathorn Phongikaroon Idaho National Laboratory

Dept. of Chemical and Materials Engineering

University of Idaho, Idaho Falls

August 28, 2012

2012 IPRC

Fontana, WI

August 28, 2012

2012 IPRC

Fontana, WI

Page 2: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Outline

• Introduction

• Motivation and Goal

• Experimental Setup

• Experimental Procedures

• Summary & Future Work

2

Page 3: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Introduction

• The Experimental Breeder Reactor-II(EBR-II) was a metallic fueled,sodium cooled fast reactor operatedat Argonne National Laboratory-West (currently Idaho NationalLaboratory) from 1963-1994.

• This reactor was fueled with asodium-bonded, uranium-zirconiumalloy fuel.

Element Weight %1 Ion

U 80.596 III

Zr 10.805 IV

Na 2.160 I

Nd 0.930 III

Mo 0.771 III

[1] S.X. Li & M.F. Simpson, Journal of Minerals & Metallurgical Processing, 22(4), 192-198 (2005).3

• An electrochemical process was developed by ArgonneNational Laboratory to treat this stainless steel clad driver fuel.

• This electrochemical process is currently being used at IdahoNational Laboratory to treat the used EBR-II driver fuel.

Page 4: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

EBR-II Used Fuel Treatment Process

4

Page 5: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

• Cathode

U3+ + 3e- → U

Zrn+ + ne- → Zr

• Cathode

U3+ + 3e- → U

• Anode

U → U3+ + 3e-

Zr → Zrn+ + ne-

Mark-IV Electrorefiner (ER)

5

Page 6: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Motivation and Goal

• Zirconium constitutes a large amount of the EBR-II used driverfuel at greater than 10 wt%.

• Over time, zirconium metal tends to build up within the Mark-IV ER and must periodically be removed.

• In developing a process to electrochemically recover purezirconium, a knowledge of the Zr redox reactions that occur inthe ER is necessary.

• To better understand this process, modeling is essential.

• In the electrochemical modeling process, there are severalthermodynamic and electrochemical values that are required.

Standard Reduction Potential, E0

Diffusion Coefficient, D

Activity Coefficient, γ

6

Page 7: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

• Nernst Equation:

• What is the standard reduction potential, E0?

The standard potential is the equilibrium potential, E,of a given reduction reaction (i.e. Zr4+ + 4e- → Zr) at standard conditions.

• Pure substance at 1 atm pressure

– γ = 1 and Xs = 1

This is related to the Gibbs free energy change of thesame half-cell reaction.

s0 Xln

nF

RTEE

Standard Reduction Potential

00 nFEG

E Equilibrium potential

E0 Standard reduction potential

R Ideal gas constant

T Absolute temperature

n Number of transferred electrons

F Faraday’s constant

γ Activity coefficient

Xs Mole fraction at interface

DGi0 Gibbs energy change of reaction i

7

Page 8: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

• Mass Transfer:

• What is the diffusion coefficient?

Proportionality constant between molar flux and concentrationgradient.

• Fick’s 1st Law:

Follows the Arrhenius temperature dependence.

356.07.02ee

D

d0791.0

D

kdSh

Diffusion Coefficient

salts CCkAN

CDN

RT

HexpDD D

0

N Molar mass transfer

k Mass transfercoefficient

A Total electrode surfacearea

Cs Surface concentration

Csalt Bulk salt concentration

Sh Sherwood number

de Equivalent electrodediameter

ω Electrode rotation rate

ν Viscosity of salt

D0 Pre-exponential factor

DHD Activation energy for diffusion

8

Page 9: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Standard Reduction Potential, E0

(V vs. Ag/AgCl)Diffusion Coefficient,

D (cm2/s)Activity Coefficient, γ

Zr(IV)/Zr Zr(IV)/Zr(II) Zr(II)/Zr Zr(IV) Zr(II) Zr(IV) Zr(II)

[2] -1.22* --- -1.12 --- --- 9.6 10-6* 3.4 107

[3] -1.064* -1.121* -1.01* --- --- --- ---

[4] -0.838 --- -0.722 --- --- 3.081 10-3* 1.05 10-4*

[5] -1.064* -1.121* -1.007* --- --- --- ---

[6] -1.333* --- --- --- --- --- ---

[7] -1.1 --- --- --- --- --- ---

[8] -1.22 --- --- --- --- --- ---

[9] --- --- --- --- --- 4.48 10-3 1.9 10-4

[10] --- --- --- 1.13 10-5 --- --- ---

[11] 4.5310-6 – 1.1010-7

* Values reported are at 450 °C.

[2] R. Baboian, et al., J. Electrochem. Soc., 112.12 (1965).

[3] J.A. Plambeck, J. of Chemical and Engineering Data, 12(1) (1967).

[4] R. Ahluwalia, et al., Nucl. Tech., 126 (1999).

[5] A.J. Bard, “Encyclopedia of Electrochemistry of the Elements, Vol. X, FusedSalt Systems, p. 68, Marcel Dekker, Inc., New York (1976).

[6] M. Iizuka, et al., J. of Nuclear Materials, 297 (2001).

[7] T. Murakami and T. Kato, J. Electrochem. Soc., 155(&) (2008).

[8] A.V. Bychkov, Proceedings of the Workshop on Pyrochemical Separations,Avignon, France, March 14-16, 2000.

[9] R.K. Ahluwalia, et al., Nucl. Tech., 133 (2001).

[10] D. Yamada, et al., J. Alloys and Compounds, 444-445 (2007).

[11] C.H. Lee, et al., J. Electrochem. Soc., 159(8) (2012).

Available Literature Values

9

Page 10: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Experimental Setup

(a) (b) (c) (d)

(e)

(g)

(a) Al2O3 sheathed thermocouple- Monitored with Fluke 52II

(b) Tungsten working electrode(c) Ag/AgCl reference electrode

(5 mol%)(d) Glassy carbon counter electrode

lead

(e) Glassy carbon crucible/counterelectrode

(f) MgO secondary crucible(g) Eutectic LiCl/KCl salt containing

ZrCl4

(h) Furnace- Kerrlab with graphite crucible

(f)

(h)

10

Page 11: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Procedures

• Loading LiCl/KCl eutectic and ZrCl4

(1.0, 2.5, and 5.0) wt% ZrCl4

LiCl/KCl eutectic, 44/56 wt% (Sigma-Aldrich,99.99%)

ZrCl4 (Alfa Aesar, Reactor Grade, 99.5+%)

• Heating salt at 4 C/min to 500 C (±2) in the salt.

• Lowering electrodes into the molten salt.

Tungsten Working Electrode

• d = 2.0 mm (Alfa Aesar, 99.95%)

Glassy Carbon Counter Electrode Lead

• d = 3.0 mm (HTW-Germany)

Ag/AgCl Reference Electrode

• Ag wire, d =1.0 mm diameter (AcrosOrganics, 99.9%)

• 5 mol% AgCl in LiCl/KCl (Alfa Aesar,ultradry, 99.997%)

11

Page 12: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

• Samples taken for ICP-MS analysis.

Mixed:

• 1.0 wt%

• 2.5 wt%

• 5.0 wt%

Electrochemistry

• Cyclic Voltammetry

Scan range: 0 V to -2.4 V

Scan rate: 300 mV/s to 2.0 V/s

• Chronopotentiometry

Driving current: 70 mA to 300 mA

Analysis Results:

• (0.954 ± 0.117) wt%

• (2.49 ± 0.304) wt%

• (4.84 ± 0.585) wt%

12

Page 13: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Chloride Salt Ingots

Pure LiCl/KCl Eutectic 1.0 wt% ZrCl4

2.5 wt% ZrCl45.0 wt% ZrCl4

13

Page 14: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Cyclic Voltammetry (CV)

• A common electrochemical technique that can beused to determine the reactions that can occur in theelectrochemical cell.

• Potential is scanned through the range of interest and

current is measured.

Potential Range Scanned: 0 V to -2.4 V

• Li+ Reduction: -2.561 V [12]

• Cl- Oxidation: +1.065 V [12]

14

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0 10 20 30 40 50

Pot

enti

al,

E(V

vs.

Ag

/Ag

Cl)

Time (s)

Potential waveform for 100 mV/s.

[12] H.A. Laitinen and C.H. Liu, Journal of the American Chemical Society, 80(5), 1015-1020 (1958).

RT

FDnnFAC446.0

Ip

RT

FDnnFAC4958.0

Ip

2

EEE

c,pa,p

XlnnF

RTEE '0

• From the resulting current, reaction information can be determined.

Randles-Sevcik equation Delahay equation

Equilibrium potential

Apparent standard potential

Page 15: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

• An electrochemical technique that can be used to determine the diffusioncoefficient of ions within the electrolyte.

• A large driving current is applied and the resulting potential is measured.

• To maintain the applied current, the potential drops to a value at whichions of a given species are reduced.

• When the ion is fully reduced at the electrode surface the potentialdrops to a potential at which the next ion will reduce.

• This creates a plateau in the measured potential.

• The duration of this plateau, or transition time, τ, is related to diffusion coefficient, D, through the Sand equation.

• From the resulting potential response, transition time and diffusion canbe determined.

Chronopotentiometry (CP)

2

DnFCi

15

Page 16: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

Cu

rren

t,I

(A)

Potential, E (V)

LiCl/KCl 200 mV/s

300 mV/s

350 mV/s

400 mV/s

450 mV/s

500 mV/s

550 mV/s

600 mV/s

650 mV/s

700 mV/s

800 mV/s

900 mV/s

1.0 V/s

e2ZrZr

e3ZrZr2

4

24 Zre2Zr

Zre3Zr

Zre2Zr4

2

e4ZrZr 4

Zre4Zr

ZreZr4

e2ZrZr 42

Scan Rate

Cyclic Voltammogram (1.0 wt% ZrCl4)

Electrode Surface Area: 0.53 cm2

16

D

A

BC

E

F

Page 17: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.5 0.6 0.7 0.8 0.9 1

Pea

kC

urr

entD

ensi

ty,

i P(A

/cm

2)

√Scan Rate, ν1/2 (V/s)1/2

1.0 wt% ZrCl4

17

RT

FDnnFC4958.0ip

D

A

B

C

PeakDiffusion Coefficient, D (cm2/s)

n = 1 n = 2 n = 3 n = 4

A 9.26 10-5 1.16 10-5 3.43 10-6 1.45 10-6

B 1.50 10-4 1.88 10-5 5.57 10-6 2.35 10-6

C 1.52 10-5 1.90 10-6 5.62 10-7 2.37 10-7

D 1.19 10-3 1.48 10-4 4.39 10-5 1.85 10-5

A Zr4+ + 2e- → Zr2+

B Zr2+ + 2e- → Zr

Zr4+ + 3e- →Zr+

C Zr+ + e- → Zr

Zr4+ + 4e- → Zr

D Zr+ → Zr4+ + 3e-

Zr → Zr2+ + 2e-

Page 18: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

Cu

rren

t,I

(A)

Potential, E (V)

400 mV/s

500 mV/s

800 mV/s

900 mV/s

1.0 V/s

1.1 V/s

1.5 V/s

2.0 V/s

Scan Rate

Cyclic Voltammogram (2.5 wt% ZrCl4)

Electrode Surface Area: 0.72 cm2

18

D

A

B

C

E

Page 19: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

-0.7

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

-2.5 -2 -1.5 -1 -0.5 0

Cu

rren

t,I

(A)

Potential, E (V)

250 mV/s

350 mV/s

450 mV/s

550 mV/s

650 mV/s

Cyclic Voltammogram (5.0 wt% ZrCl4)

Electrode Surface Area: 0.69 cm2

Scan Rate

19

D

A

B

C

E

Page 20: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

Cu

rren

tD

ensi

ty,

i(A

/cm

2)

Potential, E (V vs Ag/AgCl)

1.0 wt%

2.5 wt%

5.0 wt%

Cyclic Voltammogram Comparison

ZrCl4 Content

20

Page 21: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Nernst Plot for Peaks B and D

R² = 0.9829

-1.09

-1.08

-1.07

-1.06

-1.05

-1.04

-1.03

-1.02

-1.01

-6.5 -6.0 -5.5 -5.0 -4.5 -4.0

Eq

uil

ibri

um

Po

ten

tia

l,E

B/D

(Vv

s.A

g/A

gC

l)

ln(X)

XlnnF

RTEE '0

• Apparent reduction potential:

E0 = -0.849 V (vs Ag/AgCl)

• Average number of electrons transferred in the reaction(s):

n = 1.71

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-2.5 -2 -1.5 -1 -0.5 0

Cu

rren

tD

ensi

ty,i

(A/c

m2)

Potential, E (V vs Ag/AgCl)

1.0 wt%

2.5 wt%

5.0 wt%

B

D

ZrCl4 Content

Xln0389.0849.0E

21

Page 22: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Chronopotentiometry (1.0 wt% ZrCl4)

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Po

ten

tia

l,E

(V)

Elapsed time, t (s)

70 mA

80 mA

90 mA

100 mA

150 mA

Applied DrivingCurrent

Electrode Surface Area: 0.53 cm2

22

Page 23: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Summary

• An experimental setup and process has been designed to testproperties relevant to the electrochemical recovery of zirconium inmolten salt.

• Cyclic voltammetry has been performed on (1.0, 2.5, and 5.0) wt%ZrCl4 in the molten LiCl/KCl eutectic salt at 500 C.

Cyclic voltammograms show complex behavior of zirconium inthe molten salt with presence of ZrCl4, ZrCl2, and ZrCl.

Range of diffusion coefficients in the LiCl/KCl eutectic wasdetermined.

• D = 2.37 10-7 – 1.48 10-4 cm2/s

Apparent standard reduction potential was determined for one pairof peaks.

• E0 = -0.849 V vs. Ag/AgCl

• Chronopotentiometry was performed and shows complex behavior.

23

Page 24: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Standard Reduction Potential, E0

(V vs. Ag/AgCl)Diffusion

Coefficient, D (m2/s)Activity Coefficient, γ

Zr(IV)/Zr Zr(IV)/Zr(II) Zr(II)/Zr Zr(IV) Zr(II) Zr(IV) Zr(II)

[2] -1.22* --- -1.12 --- --- 9.6 10-6* 3.4 107

[3] -1.064* -1.121* -1.01* --- --- --- ---

[4] -0.838 --- -0.722 --- --- 3.081 10-3* 1.05 10-4*

[5] -1.064* -1.121* -1.007* --- --- --- ---

[6] -1.333* --- --- --- --- --- ---

[7] -1.1 --- --- --- --- --- ---

[8] -1.22 --- --- --- --- --- ---

[9] --- --- --- --- --- 4.48 10-3 1.9 10-4

[10] --- --- --- 1.13 10-5 --- --- ---

[11] --- --- --- 1.1010-7 - 4.5310-6 --- ---

* Values reported are at 450 °C.

[2] R. Baboian, et al., J. Electrochem. Soc., 112.12 (1965).

[3] J.A. Plambeck, J. of Chemical and Engineering Data, 12(1) (1967).

[4] R. Ahluwalia, et al., Nucl. Tech., 126 (1999).

[5] A.J. Bard, “Encyclopedia of Electrochemistry of the Elements, Vol. X, Fused SaltSystems, p. 68, Marcel Dekker, Inc., New York (1976).

[6] M. Iizuka, et al., J. of Nuclear Materials, 297 (2001).

[7] T. Murakami and T. Kato, J. Electrochem. Soc., 155(&) (2008).

[8] A.V. Bychkov, Proceedings of the Workshop on Pyrochemical Separations, Avignon,France, March 14-16, 2000.

[9] R.K. Ahluwalia, et al., Nucl. Tech., 133 (2001).

[10] D. Yamada, et al., J. Alloys and Compounds, 444-445 (2007).

ThisWork

-0.849 (Effective) 2.3710-7 – 1.4810-4

24

Summary

Page 25: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Future Work

• Further analysis of chronopotentiometry data.

• Further work with ZrCl4 at additional concentrations.

• Zirconium electrodeposition experiments with transparent setupto analyze zirconium deposit morphology.

• Experiments to optimize/maximize zirconium recovery.

• Cathode material, operating temperature, applied potential.

• Electrochemistry experiments/analysis with UCl3 in LiCl/KCleutectic.

• Final phase of this work will explore the electrochemicalrecovery of zirconium in the presence of uranium.

25

Page 26: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Acknowledgements

• This work was performed as part of I-NERI Project 2010-001-Kin conjunction with Seoul National University and KoreaAtomic Energy Research Institute.

• Thanks to all those who have helped with this project includingDebbie Lacroix, Sean Martin, Ammon Williams, Josh Versey,Mike Pack, Cindy Hanson, and Dalsung Yoon.

26

Page 27: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Thanks!

Page 28: Center for Advanced Energy Studies...Center for Advanced Energy Studies Electrochemical Zirconium Recovery Experiments in Molten Salt System Robert O. Hoover Michael F. Simpson Michael

Activity Coefficient

• Nernst Equation

• What is the activity coefficient, γ?

A factor included in order to take account of deviations fromsolution ideality in the liquid phase.

It is related to the excess Gibbs energy, GE, the differencebetween the actual and ideal Gibbs energy of a solution.

It is defined as a ratio of the fugacity of the species in solutionand its mass fraction in solution times its pure species fugacity.

s0 Xln

nF

RTEE

lnRTGE

xf

[31] J.M. Smith, H.C. VanNess, & M.M. Abbott, Chemical Engineering Thermodynamics 6th Edition, p. 406, McGraw-Hill, Boston (2001).

GE Excess Gibbs energy

Fugacity in solution

x Mass fraction in solution

f Fugacity of pure species

S1