1

Thermodynamic modeling of some properties of electrolyte

solutions

Kaj Thomsen IVC-SEP, Department of Chemical

Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark.

E-mail: kth@kt.dtu.dk

3

Models for electrolytes

• Long range interactions

–Debye-Hückel electrostatic term

• Short range interactions

–Pitzer virial expansion in molality

–Electrolyte NRTL

–UNIQUAC

• Gas phase fugacity

–PR or SRK equation of state

4

Extended UNIQUAC

• Excess gibbs energy function

–Debye-Hückel term

–UNIQUAC term

• Activity coefficients and thermal properties are derived by standard methods known from classical thermodynamics

5

Standard states

• Water is the solvent

• Ions, non-electrolytes and gases are treated equally as solutes in water

0

1ln( ); 1

wxw w w w wRT x

* * *

0ln( ); 1

i

i

i i i i i xRT x

6

Gibbs energy of transfer

Kamps, A.P-S., Ind. & Eng. Chem. Res., 44(2005)201-225

7

Relative permittivity Relative permittivity of aqueous solutions

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100

Mass % solute

Re

lati

ve

pe

rm

itti

vit

y

.

NaCl, Hasted et al, 1948Ethanol, Åkerlöf, 1932

8

Conventional and ”Mixed solvent” approach

* *ln ln

"Mixed solvent" approach:

ln ln

i i i i

ideal excess

Mixed solvent Mixed solvent

i i i i

ideal excess

RT x RT

RT x RT

In the ”Mixed solvent” approach, the standard state chemical potential is a function of the solvent composition

9

Model parameters

• Standard UNIQUAC parameters

–Volume parameter for each species

–Surface area parameter for each species

– Interaction energy parameter for each pair of species

• Temperature dependence of interaction energy parameter

• Number of parameters:

• eUNIQUAC ~ eNRTL << Pitzer

10

Databank

• Over 100,000 experimental data on electronic form – Activity/osmotic coefficient

– Enthalpy of mixing

– Heat capacity

– Degree of dissociation

– Gas solubility

– Density

– Salt solubility (Solid-liquid equilibrium)

– Liquid-liquid equilibrium

– Vapor-liquid equilibrium

11

Parameter estimation

• Critical review of data

• Non-linear least squares optimization

–Differences between experimental and calculated values are minimized

–The calculation of the difference depends on the type of data

–All data of same type weighted equally

12

Anchoring of parameters

• No binary solution of one ion in water

• Parameters of ions are relative to each other

• The hydrogen ion is used as anchor

–Parameters for the hydrogen ion are given fixed values

13

Thermal properties

• Excess enthalpy is calculated from the temperature derivatives of activity coefficients.

• By using thermal properties in the parameter estimation a better temperature dependency of activity coefficients is achieved

• Clear distinction between temperature dependency and concentration dependency

14

Parameters

• H+, Na+, K+, NH4+, Ca2+, Mg2+, Mn2+, Fe2+,

Co2+, Ni2+, Cu2+, Zn2+, Ba2+, Sr2+

• F-, Cl-, Br-, NO3-, SO4

2-, HSO4-, OH-, CO3

2-, HCO3

-, S2O82-, SO3

2-, HSO3-, HPO4

-, H2PO4-

• H2O, CO2, NH3, SO2, HNO3, H3PO4, C12H22O11, CH3OH, C2H5OH, n-C3H7OH, i-C3H7OH, n-C4H9OH, i-C4H9OH, s-C4H9OH, t-C4H9OH

15

Equilibrium calculations

• Speciation equilibrium

• +

• Solid-liquid equilibrium

• Vapor-liquid equilibrium

• Liquid-liquid equilibrium

16

Speciation equilibria

NH3(aq)+H2O NH4+(aq)+OH-(aq)

Equilibrium condition:

-3 2 4

- -3 24 4

3 2

* * * 0 * *

*

- -- ln

NH H O NH OH

NH H ONH OH NH OH

NH H O

a a

RT a a

17

Solid-liquid equilibrium

• At equilibrium, the chemical potential of the pure crystalline salt(hydrate) equals the sum of the chemical potentials of the salts components in solution

• It is required that other salts are not supersaturated.

18

Vapor-liquid equilibrium

• Equality of chemical potential in gas phase and in liquid phase (Gamma-phi method)

• Gas phase fugacities are calculated with the Soave-Redlich-Kwong equation of state

2 2

2 2 2 2 2 2

( ) ( )

0, * *

0

ˆln ln

CO g CO aq

ig

CO CO CO CO CO CO

PRT y RT x

P

19

Liquid-liquid equilibrium

• Here the activity product of salts rather than the activities of the individual ions ions are compared

I II

* * I * * II

* I * II

ln( ) ln( )

( ) ( )

i i

i i i i i i

i i i i

RT x RT x

x x

• Equilibrium between component i in phase I and phase II

20

Standard state properties

• The numerical values of standard state chemical potentials are needed before equilibrium calculations can be made

• Such values for most solutes and many salts have been published by NIST

• Those not found are fitted to experimental data

• Temperature dependence calculated with classical thermodynamic method

21

-20

-10

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100

Tem

pera

ture °

C

Extended UNIQUAC modelExperimental data--

Mass percent Mn(NO3)2

Ice

Mn

(NO

3) 2

·6H

2O

Mn

(NO

3) 2

·4H

2O

Mn

(NO

3) 2

·2H

2O

Mn

(NO

3) 2

·H2O

22

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2

Mass percent Ca(OH)2

Te

mp

era

ture

°C

Calculated

Experimental

Ca(OH)2

23

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110

salt

fra

cti

on

Extended UNIQUAC modelExperimental data

Temperature, °C

K2C

O3

Na

2C

O3

Na2CO3·K2CO3

NaKCO3·6H2O

Na2CO3

Na2CO3·H2O

Na2CO3·7H2O

Na2CO3·10H2O Ice

K2CO3·½H2O

24

100

90

80

70

60

50

40

30

20

10

0100

90

80

70

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Experimental

data, various

sources

Extended

UNIQUAC,

Equilibrium lines

and tie lines

K2CO3

Na2CO3

H2O(l)

T= 75.0°C

Na2CO3∙H2O

Na2CO3∙K2CO3

K2CO3∙½H2O

25

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10

CO2 mol kg-1

CO

2 p

arti

al

press

ure, b

ar

Extended UNIQUAC modelExperimental data__

80°C

12 molal NH3

9 molal NH3

2 molal

NH3

6.8 molal NH3

4.1 m1 m0.6 m

5.9 molal NH3

26

100

90

80

70

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Thompson and Vener (1948)Armstrong and Eyre (1910)Gerardin (1865)Schiff (1861)Extended UNIQUAC modelSeries2Series3Series4

C2H5OH

KNO3

H2O

15°C

25°C

50°C

75°C

27

100

90

80

70

60

50

40

30

20

10

0100

90

80

70

60

50

40

30

20

10

0

0.0

0

10.0

0

20.0

030.0

0

40.0

0

50.0

0

60.0

0

70.0

080.0

0

90.0

0100.0

0

0 10 20 30 40 50 60 70 80 90 100

Iino et al. (1971)Do & Park (1974)Extended UNIQUACSeries2Series3Series4Series7

iso-propanol

K2CO3

H2O

30 °C

28

xy - diagram for iso-propanol - water, 1 bar

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x iso-propanol

y iso

-pro

pa

no

l

Extended UNIQUAC modelMarzal et al. (1996)Saturation with K2CO3

29

Integral heat of dilution to infinite dilution

-3000

-2000

-1000

0

1000

2000

3000

4000

0 2 4 6Molality of KCl

J mol-1Ext. UNIQUAC

Experimental, 12.5°C

Experimental, 25°C

Experimental, 40°C

Experimental, 60°C

Experimental, 80°C

30

Apparent molal heat capacity

-200

-150

-100

-50

0

50

0 0.5 1 1.5 2 2.5 3Molality of NaOH

J mol-1 K-1

Ext. UNIQUAC

Experimental, 10°CExperimental, 25°C

Experimental, 40°CExperimental, 100°C

31

Pressure dependency

• No pressure dependency in activity coefficient model

• High pressure applications

–Scale formation in oil production equipment and reservoirs

–Scale formation in equipment used for producing geothermal energy

32

What is scale formation?

33

Pressure dependency

• Solubility product:

• Activity coefficients:

0

0

2

0 0ln ln ( ) ( )2

dis P disP P

VK K P P P P

RT RT

0

0

,* * 2

, , 0 0ln ln ( ) ( )2

ex exi P i

i P i P

VP P P P

RT RT

34

Equilibrium expression

• The resulting equation for equilibrium is:

• Alfa and beta have physical meanings.

• We treat them as adjustable parameters

0 0

2

0 0 ,ln ( ) ( ) lnP i i i P

i

K P P P P x

35

BaSO4 solubility at 500 bar

1.00E-06

6.00E-06

1.10E-05

1.60E-05

2.10E-05

2.60E-05

3.10E-05

3.60E-05

4.10E-05

4.60E-05

0 50 100 150 200 250 300

T (oC)

BaS

O4 (

m)

Extended UNIQUAC model

Blount (1977)

Lyashchenko and Churagulov (1981)García A.V., Thomsen K., Stenby E.H.,

Geothermics 34(2005)61-97

36

SrSO4 solubility isotherms

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0 100 200 300 400 500 600 700

P (bar)

SrS

O4 (

m)

Extended UNIQUAC model

Howell et al. (1992)25 °C

100 °C

200 °C

37

CaCO3 solubility at 30 bar CO2

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 50 100 150 200

T (oC)

CaC

O3 (

m)

Extended UNIQUAC model

Segnit et al. (1962)

Miller (1952)

García A.V., Thomsen K., Stenby E.H., Geothermics 35(2006)239-284

38

Inconsistent data

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0 1 2 3 4 5 6

NaCl (m)

SrS

O4 (m)

Howell et al. (1992)

Brower and Renault (1971)

Vetter et al. (1983)

Lucchesi and Whitney (1962)

Müller (1960)

39

Experimental setup

40

Measurements

• Analysis for Ba performed by ICP-MS (Inductively Coupled Plasma – Mass Spectrometry)

• Three different labs measured three different Ba contents in the same sample!

• We did not get any good data yet!

41

Corrosion in wet gas pipelines

• In another project, the Extended UNIQUAC model is being applied for describing and preventing corrosion

• Equilibrium calculations to be combined with electrochemical and transport aspects

• PhD student Philip Fosboel

42

Where do we find corrosion?

GAS Line 16”

CO2

H2O

Natural gas

CO2

H2O

NaOH

MEG

43

CO2 Corrosion

• pH is lowered by dissolved CO2.

2 2 3

2

2

2

2 2 3 2

( ) ( ) ( ) ( )

Half cell reactions:

2 2

2

The sum of reactions:

( ) 2 ( ) 2 ( ) ( ) 2 ( ) ( )

CO aq H O l HCO aq H aq

H e H

Fe Fe e

Fe s CO aq H O l Fe aq HCO aq H g

44

CO2 corrosion

• If pH is high enough, a protective layer of FeCO3 is formed

• Gas composition

–1.6 mol % CO2

–0.1 mol % H2O

–Balance light alkanes

• Temperature 10 to 50°C

• Pressure 60 to 70 bar

45

CO2 corrosion

• Inhibitors:

–NaOH

–Mono ethylene glycol (MEG)

• Liquid phase:

–27 to 0.5 wt % NaOH in water

–95 - 30 wt % MEG

• How much CO2 can dissolve in this solution?

46

CO2 corrosion

• If sufficient data are available, the system can be modelled

– No data for solubility of CO2 in H2O – MEG mixture published

– Few data for solubility of Na2CO3 and NaHCO3 in H2O – MEG mixture

• Highly non ideal solution

– High ionic strength (up to 10 molal)

– Mixed solvent solution

– Speciation equilibria

47

CO2 –NaOH - H2O – MEG measurements

• New measurements are required • The solubility of Na2CO3 and NaHCO3 is

being measured by titration – The total Na+ content can be determined – The carbonate/bicarbonate ratio is not

determined – Solvent composition changes during

precipitation of hydrates • Na2CO3∙10H2O • Na2CO3∙7H2O • Na2CO3∙H2O • Na2CO3∙NaHCO3∙2H2O

48

CO2 – NaOH - H2O – MEG measurements

Equilibration

Automated accurate titration

10 < T°C < 50

0 < wt % MEG < 100

Saturated solutions

49

CO2 – NaOH - H2O – MEG NaHCO3 solubility determined by titration of saturated solution

Two salt transition points marked by sudden density change of saturated solution

Solubility of salts can be determined if amount of precipitate is known. Raw data are used for parameter estimation

100

90

80

70

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

100 Solid phase

Saturated

Start

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Solubility isothermTie-linesExperiment

NaHCO3

Na2CO3

H2O+MEG

Na2CO3∙NaHCO3∙2H2O

Na2CO3∙10H2O

50

CO2 – NaOH - H2O – MEG

Preliminary results calculated with Extended UNIQUAC model based on model parameters determined from literature data. (Gärtner et al. J. Chem. Eng. Data, 49(2004)116-125)

0

5

10

15

20

25

0 20 40 60 80 100

wt% saltfree MEG

g/1

00g

solv

ent

model 25C

model 50C

model 80C

experimental 25C

experimental 50C

experimental 80C

0

5

10

15

20

25

0 20 40 60 80 100

wt% saltfree MEG

g/1

00g

solv

ent

model 25C

model 50C

model 80C

experimental 25C

experimental 50C

experimental 80C

51

CO2 corrosion

• When parameters in the model are determined, we can – Combine this thermodynamic model with a

diffusion model to determine corrosion rate

– Calculate speciation equilibria in the mixed solvent solution

– Calculate the saturation index of the protective coating of FeCO3

– Determine the optimal amount of NaOH to add to the solution to avoid corrosion

– Determine the optimal amount of MEG to add to avoid gas hydrate formation

52

Equation of state for electrolytes

• Current activity coefficient models are good but not perfect –No pressure dependency

–No density calculation

–Decreasing accuracy with increasing number of components

• Practical to use same equation of state for all components

• PhD student Yi Lin

53

Comparative study of four EOS

• Short range interactions: –Soave-Redlich-Kwong

–Peng-Robinson

–Wertheim association term for water

• Long range electrostatic interactions: –Mean spherical approximation (MSA)

• Implicit and explicit version

–Simpel Debye-Hückel term

–Born term

54

Myers, Sandler and Wood (MSW) electrolyte EOS

• Myers et al., Ind. Eng. Chem. Res. 41(2002)3282-3297

• AR = APR + ABorn + AMSA

• Short range term : PR EOS

• Long range terms: Explicit MSA, Born

• We use ion specific parameters.

• APR :

• ABorn , AMSA :

, , i i ija b k

i

55

Modified MSW electrolyte EOS

• We replace the explicit MSA term with the implicit MSA term

• AR = APR + ABorn + AimMSA

• Short range term : PR EOS

• Long range terms: Implicit MSA, Born

• Ion specific parameters.

• APR :

• ABorn , AimMSA :

, , i i ija b k

i

56

Electrolyte CPA EOS

• We replace the Peng-Robinson term with the Soave-Redlich-Kwong + Wertheim term

• AR = ASRK + AW + ABorn + AimMSA

• Short range term : SRK + Association

• Long range terms: Implicit MSA, Born

• Ion specific parameters.

• ASRK :

• ABorn , AimMSA :

, , i i ija b k

i

57

Approximation

• An approximation introduced by Myers, Sandler and Wood (Ind. Eng. Chem. Res. 41(2002)3282-3297) is implemented for the three EOS mentioned.

• The density of water is needed for calculating the relative permittivity of water from the relation of Uematsu and Franck

•

2 2 2

2 2

· /

( , ) ( , , )

H O H O H O

r H O r H O

n M V

T T V n

58

SRK + DH EOS

• Short range term : SRK

• Long range terms: Debye-Hückel

• Ion specific parameters.

• The Debye-Hückel parameter A is a function of temperature only

• Debye-Hückel term with no contribution to volume

, , i i ija b k

ln ln lnSRK DH SRK DH

i i i

59

Test system at 298.15 K

• H2O-(Na+, Ca2+, H+)-(Cl-, SO42-, OH-)

• 1300 experimental data points used

–Osmotic/activity coefficient

–Solid-liquid equilibrium data

–Apparent molar volume

0*

,

ex ex

w w S S w wS S

S S

nM n M n M n MM M

n n

60

Results

0 1 2 3 4 5 6 7

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35NaCl

Osm

otic C

oe

ffic

ien

t

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

0 1 2 3 4 5 6 7 8 9 10

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5 CaCl2

Osm

otic C

oe

ffic

ien

t

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

Lin Y., Thomsen K., and de Hemptinne J-C., submitted to AIChE Journal

61

Results

0 1 2 3 4 5 6 7

0.0

0.5

1.0

1.5

2.0

NaCl

Na2SO

4

Na2SO

4·10H

2O

Na

2S

O4 M

ola

lity (

mo

l/kg

)

NaCl Molality (mol/kg)

Experimental Data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

0.000

0.005

0.010

0.015

0.020

0.025

Ca(OH)2

CaSO4·2H

2O

Ca

(OH

) 2 M

ola

lity (

mo

l/kg

)

CaSO4 Molality (mol/kg)

Experimental Data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

62

Results

0 1 2 3 4 5 6 7

12

14

16

18

20

22

24

Ap

pa

ren

t M

ola

r V

olu

me

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

HCl

0 1 2 3 4 5 6 7 8

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15 HCl

De

nsity (

g/c

m3)

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

SRK+DH

0 1 2 3 4 5 6

-5

0

5

Ap

pa

ren

t M

ola

r V

olu

me

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

NaOH

0 1 2 3 4 5 6

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25 NaOH

De

nsity (

g/c

m3)

Molality (mol/kg)

Experimental data

MSW EOS

mMSW EOS

eCPA EOS

63

Temperature dependence

• Usually the ”a” parameter in cubic EOS is temperature dependent

• We chose to let the ion size parameter be temperature dependent too

• Seven different temperature dependence functions were tested

• The best was:

•

•

•

0 1 2

0 1

2( )

( )

298.15 298.15

298.15

a T a a a

T

T T

T

64

Results

0 0.5 1 1.5 2 2.5 3 3.5 4

0

20

40

60

80

100

120

Molality (mol/kg)

Tem

pera

ture

oC

Exp data

MSW EOS

mMSW EOS

CPA EOS

SRK+DH EOS

Ice

Na2SO

4 .10H

2O

Na2SO

4

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

-40 oC

-25 oC

-20 oC

-20 oC, y + 0.5

-10 oC

-10 oC, y + 1.5

0 oC

0 oC, y + 2.5

NaCl Molality (mol/kg)

CaC

l 2 M

ola

lity

(m

ol/

kg

) )

Exp data

MSW EOS

mMSW EOS

CPA EOS

NaCl .2H2O

Ice

-25 oC

Ca

2C

l .6H

2O

65

Temperature dependence

• The thermal properties (heat of mixing, heat capacity) of electrolyte solutions could not be well correlated by any of the EOS

• The same is the case for the apparent molar volume of these solutions

• Alternative to be tested:

– Use temperature dependent interaction parameters in the EOS as it is done in the activity coefficient model

66

Results

5

7

9

11

13

15

17

19

21

23

25

0 1 2 3 4 5 6

NaCl molality

Ap

pa

ren

t m

ola

r vo

lum

e c

m3/

mo

l

MSW, 100 °CmMSW, 100 °CCPA, 100 °CExperimental, 100 °CExperimental, 25 °CMSW, 25 °CmMSW, 25 °CCPA, 25 °C

67

Conclusions

• Activity coefficient models like the Extended UNIQUAC model is currently the only way to model properties of electrolyte solutions

• Electrolyte EOS based on cubic EOS need more developement before they can be used as engineering equations