Thermodynamic modeling of some properties of electrolyte solutions

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
1

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
3

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
4

Standard states
• Water is the solvent
RT ln( x ); 1
0
w w w w w
x w
1
• Ions, non-electrolytes and gases are
treated equally as solutes in water
* * *
RT ln( x ); 1
i i i i i
xi
0
i
5

Gibbs energy of transfer
Kamps, A.P-S., Ind. & Eng. Chem. Res., 44(2005)201-225
6

90
Relative permittivity
Relative permittivity of aqueous solutions
Relative permittivity .
80
70
60
50
40
30
20
NaCl, Hasted et al, 1948
Ethanol, Åkerlöf, 1932
10
0
0 10 20 30 40 50 60 70 80 90 100
Mass % solute
7

Conventional and ”Mixed solvent”
approach
RT ln x RT ln
* *
i i i i
ideal
excess
"Mixed solvent" approach:
Mixed solvent
RT ln x RT ln
Mixed solvent
i i i i
ideal
excess
In the ”Mixed solvent” approach, the standard state chemical
potential is a function of the solvent composition
8

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

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
10

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
11

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
12

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
13

Parameters
• H + , Na + , K + , NH 4+
, Ca 2+ , Mg 2+ , Mn 2+ , Fe 2+ ,
Co 2+ , Ni 2+ , Cu 2+ , Zn 2+ , Ba 2+ , Sr 2+
• F - , Cl - , Br - , NO 3-
, SO 4
2-
, HSO 4-
, OH - , CO 3
2-
,
HCO 3-
, S 2
O 8
2-
, SO 3
2-
, HSO 3- , HPO 4- , H 2 PO 4
-
• H 2 O, CO 2 , NH 3 , SO 2 , HNO 3 , H 3 PO 4 ,
C 12 H 22 O 11 , CH 3 OH, C 2 H 5 OH, n-C 3 H 7 OH, i-
C 3 H 7 OH, n-C 4 H 9 OH, i-C 4 H 9 OH, s-C 4 H 9 OH,
t-C 4 H 9 OH
14

Equilibrium calculations
• Speciation equilibrium
• +
• Solid-liquid equilibrium
• Vapor-liquid equilibrium
• Liquid-liquid equilibrium
15

Speciation equilibria
NH 3 (aq)+H 2 O NH 4+ (aq)+OH - (aq)
Equilibrium condition:
NH H O NH OH
3 2 4
- -
a a
- ln
RT a a
* * * 0 * *
NH OH NH H O NH OH
- -
4 3 2
4
*
NH
H O
3 2
-
16

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.
17

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
CO ( g ) CO ( aq)
2 2
P
RT ln y ˆ RT ln x
0, ig
* *
CO CO CO CO CO CO
P0
2 2 2 2 2 2
18

Liquid-liquid equilibrium
• Equilibrium between component i in
phase I and phase II
I
i
RT ln( x ) RT ln( x )
* * I * * II
i i i i i i
• Here the activity product of salts rather
than the activities of the individual ions
ions are compared
II
i
( x
) ( x
)
* I * II
i i i i
19

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
20

90
80
70
Extended UNIQUAC model
Experimental data
--
60
Temperature °C
50
40
30
20
10
0
-10
Ice
Mn(NO 3 ) 2·6H 2 O
Mn(NO 3 ) 2·4H 2 O
Mn(NO 3 ) 2·2H 2 O
Mn(NO 3 ) 2·H 2 O
-20
0 10 20 30 40 50 60 70 80 90 100
Mass percent Mn(NO 3 ) 2
21

120
100
Ca(OH) 2
Temperature °C
80
60
40
20
Calculated
Experimental
0
0 0.05 0.1 0.15 0.2
Mass percent Ca(OH) 2
22

K2CO3 salt fraction
Na2CO3
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Ice
Na 2 CO 3·10H 2 O
NaKCO 3·6H 2 O
Na 2 CO 3·7H 2 O
K 2 CO 3·½H 2 O
Extended UNIQUAC model
Experimental data
Na 2 CO 3·H 2 O
Na 2 CO 3·K 2 CO 3
Na 2 CO 3
-40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110
Temperature, °C
23

100
90
80
70
60
50
40
30
80
Na 2 CO 3 ∙H 2 O
20
90
Na 2 CO 3
100
10
0
T= 75.0°C
70
60
50
40
30
20
10
0
Na 2 CO 3 ∙K 2 CO 3
100
H 2 O(l)
24
90
80
70
60
Experimental
data, various
sources
Extended
UNIQUAC,
Equilibrium lines
and tie lines
50
40
30
20
K 2 CO 3 ∙½H 2 O
10
0 10 20 30 40 50 60 70 80 90 100
K 2 CO 3
0

90
80
Extended UNIQUAC model
Experimental data
__
5.9 molal NH 3
80°C
CO2 partial pressure, bar
70
60
50
40
30
0.6 m
1 m
6.8 molal NH 3
2 molal 4.1 m
NH 3
9 molal NH 3
12 molal NH 3
20
10
0
0 2 4 6 8 10
CO 2 mol kg -1
25

90
100
10
0
100
90
H 2 O
15°C
20
80
80
70
25°C
30
70
50°C
40
60
60
50
40
75°C
60
50
50
40
70
30
30
80
Thompson and Vener (1948)
20
Armstrong and Eyre (1910)
90
Gerardin (1865)
Schiff (1861)
10
KNO 3
Extended UNIQUAC model
100
Series2
0
Series3
0 10 20 30 Series4 40 50 60 70 80 90 100
26
C 2 H 5 OH
20
10
0

100.00
90.00
80.00
70.00
60.00
50.00
40.00
30.00
20.00
K 2 CO 3
100
10.00
0.00
Iino et al. (1971)
Do & Park (1974)
Extended UNIQUAC
Series2
Series3
30 °C Series4
Series7
20
90
80
70
60
50
40
30
10
0
100
H 2 O
0 10 20 30 40 50 60 70 80 90 100
27
iso-propanol
90
80
70
60
50
40
30
20
10
0

xy - diagram for iso-propanol - water, 1 bar
1
0.9
0.8
y iso-propanol
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Extended UNIQUAC model
Marzal et al. (1996)
Saturation with K2CO3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x iso-propanol
28

4000
J mol -1
3000
2000
1000
Integral heat of dilution to infinite dilution
Ext. UNIQUAC
Experimental, 12.5°C
Experimental, 25°C
Experimental, 40°C
Experimental, 60°C
Experimental, 80°C
0
-1000
-2000
-3000
0 2 4 6
Molality of KCl
29

50
J mol -1 K -1
Apparent molal heat capacity
0
-50
-100
-150
-200
Ext. UNIQUAC
Experimental, 10°C
Experimental, 25°C
Experimental, 40°C
Experimental, 100°C
0 0.5 1 1.5 2 2.5 3
Molality of NaOH
30

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
31

What is scale formation?
32

Pressure dependency
• Solubility product:
0
ln KP ln KP
( P P
0
0) ( P P0)
• Activity coefficients:
V
2RT
dis P dis
RT
2
V
ex
ex
* * iP ,
2
0
i
ln i, P
ln i, P
( P P
0
0) ( P P0
)
RT
2RT
33

Equilibrium expression
• The resulting equation for
equilibrium is:
ln K ( P P ) ( P P ) ln x
2
P 0 0 i i i,
P
i
0 0
• Alfa and beta have physical
meanings.
• We treat them as adjustable
parameters
34

BaSO 4 solubility at 500 bar
BaSO4 (m)
4.60E-05
4.10E-05
3.60E-05
3.10E-05
2.60E-05
2.10E-05
1.60E-05
1.10E-05
Extended UNIQUAC model
Blount (1977)
Lyashchenko and Churagulov (1981)
García A.V.,
Thomsen K.,
Stenby E.H.,
Geothermics
34(2005)61-
97
6.00E-06
1.00E-06
0 50 100 150 200 250 300
T ( o C)
35

SrSO 4 solubility isotherms
1.4E-03
1.2E-03
1.0E-03
Extended UNIQUAC model
Howell et al. (1992)
25 °C
SrSO4 (m)
8.0E-04
6.0E-04
4.0E-04
100 °C
2.0E-04
0.0E+00
200 °C
0 100 200 300 400 500 600 700
36
P (bar)

CaCO 3 solubility at 30 bar CO 2
0.030
CaCO3 (m)
0.025
0.020
0.015
0.010
Extended UNIQUAC model
Segnit et al. (1962)
Miller (1952)
García A.V.,
Thomsen K.,
Stenby E.H.,
Geothermics
35(2006)239-
284
0.005
0.000
0 50 100 150 200
T ( o C)
37

Inconsistent data
0.010
0.009
0.008
0.007
SrSO4 (m)
0.006
0.005
0.004
0.003
0.002
0.001
0.000
Howell et al. (1992)
Brower and Renault (1971)
Vetter et al. (1983)
Lucchesi and Whitney (1962)
Müller (1960)
0 1 2 3 4 5 6
NaCl (m)
38

Experimental setup
39

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!
40

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
41

Where do we find corrosion?
CO 2
H 2 O
Natural gas
GAS Line 16”
CO 2
H 2 O
NaOH
MEG
42

CO 2 Corrosion
• pH is lowered by dissolved CO 2 .
CO ( aq) H O( l) HCO ( aq) H ( aq)
2 2 3
Half cell reactions:
2H 2e H
2
Fe Fe 2e
The sum of reactions:
2
Fe( s) 2 CO ( aq) 2 H O( l) Fe ( aq) 2 HCO ( aq) H ( g)
2
2 2 3 2
43

CO 2 corrosion
• If pH is high enough, a protective
layer of FeCO 3 is formed
• Gas composition
– 1.6 mol % CO 2
– 0.1 mol % H 2 O
– Balance light alkanes
• Temperature 10 to 50°C
• Pressure 60 to 70 bar
44

• Inhibitors:
– NaOH
CO 2 corrosion
– Mono ethylene glycol (MEG)
• Liquid phase:
– 27 to 0.5 wt % NaOH in water
– 95 - 30 wt % MEG
• How much CO 2 can dissolve in this
solution?
45

CO 2 corrosion
• If sufficient data are available, the system
can be modelled
– No data for solubility of CO 2 in H 2 O – MEG
mixture published
– Few data for solubility of Na 2 CO 3 and NaHCO 3
in H 2 O – MEG mixture
• Highly non ideal solution
– High ionic strength (up to 10 molal)
– Mixed solvent solution
– Speciation equilibria
46

CO 2 –NaOH - H 2 O – MEG
measurements
• New measurements are required
• The solubility of Na 2 CO 3 and NaHCO 3 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
• Na 2 CO 3 ∙10H 2 O
• Na 2 CO 3 ∙7H 2 O
• Na 2 CO 3 ∙H 2 O
• Na 2 CO 3 ∙NaHCO 3 ∙2H 2 O
47

CO 2 – NaOH - H 2 O – MEG
measurements
10 < T°C < 50
0 < wt % MEG < 100
Saturated solutions
Equilibration
Automated accurate titration
48

100
90
80
70
60
50
40
30
20
CO 2 – NaOH - H 2 O – MEG
90
Na 2 CO 3
10
30
Na 2 CO 3 ∙10H 2 O
80
70
60
50
40
20
10
0
100
H 2 O+MEG
90
Saturated
80
100 Solid phase 0
0
Solubility isotherm
Tie-lines
Experiment
Na 2 CO 3 ∙NaHCO 3 ∙2H 2 O
70
Start
0 10 20 30 40 50 60 70 80 90 100
NaHCO 3
49
60
50
40
30
20
10
NaHCO 3 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

25
CO 2 – NaOH - H 2 O – MEG
g/100g solvent
20
15
10
5
model 25C
model 50C
model 80C
experimental 25C
experimental 50C
experimental 80C
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
0 20 40 60 80 100
wt% saltfree MEG
50

CO 2 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 FeCO 3
– 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
51

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
52

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
53

Myers, Sandler and Wood (MSW)
electrolyte EOS
• Myers et al., Ind. Eng. Chem. Res.
41(2002)3282-3297
• A R = A PR + A Born + A MSA
• Short range term :
• Long range terms:
PR EOS
• We use ion specific parameters.
• A PR :
• A Born , A MSA :
Explicit MSA, Born
ai , bi , kij
i
54

Modified MSW electrolyte EOS
• We replace the explicit MSA term with the
implicit MSA term
• A R = A PR + A Born + A imMSA
• Short range term :
• Long range terms:
• Ion specific parameters.
• A PR :
• A Born , A imMSA :
PR EOS
Implicit MSA, Born
ai , bi , kij
i
55

Electrolyte CPA EOS
• We replace the Peng-Robinson term with
the Soave-Redlich-Kwong + Wertheim
term
• A R = A SRK + A W + A Born + A imMSA
• Short range term :
• Long range terms:
• Ion specific parameters.
• A SRK :
• A Born , A imMSA :
56
SRK + Association
Implicit MSA, Born
ai , bi , kij
i

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
•
n · M / V
H O H O H O
2 2 2
( T , ) ( T , V, n )
r H O r H O
2 2
57

SRK + DH EOS
SRK DH SRK DH
i i i
ln ln ln
• 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
58
ai , bi , kij

Test system at 298.15 K
• H 2 O-(Na + , Ca 2+ , H + )-(Cl - , SO
2-
4 , OH - )
• 1300 experimental data points used
– Osmotic/activity coefficient
– Solid-liquid equilibrium data
– Apparent molar volume
M
,
S
nM n M n M n M
n
0
ex
ex
w w * S S w w
MS
S
nS
59

Results
1.35
1.30
1.25
NaCl
4.5 CaCl 2
4.0
3.5
Osmotic Coefficient
1.20
1.15
1.10
1.05
1.00
0.95
0.90
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
Osmotic Coefficient
3.0
2.5
2.0
1.5
1.0
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
0.85
0 1 2 3 4 5 6 7
Molality (mol/kg)
0.5
0 1 2 3 4 5 6 7 8 9 10
Molality (mol/kg)
Lin Y., Thomsen K., and de Hemptinne J-C., submitted to AIChE Journal
60

Results
2.0
Na 2
SO 4·10H 2
O
0.025
Ca(OH) 2
0.020
Na 2
SO 4
Molality (mol/kg)
1.5
1.0
0.5
Experimental Data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
Na 2
SO 4
NaCl
Ca(OH) 2
Molality (mol/kg)
0.015
0.010
0.005
Experimental Data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
CaSO 4·2H 2
O
0.0
0 1 2 3 4 5 6 7
NaCl Molality (mol/kg)
0.000
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
CaSO 4
Molality (mol/kg)
61

24
HCl
Results
1.15 HCl
Apparent Molar Volume
22
20
18
16
14
12
0 1 2 3 4 5 6 7
Molality (mol/kg)
NaOH
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
Density (g/cm 3 )
1.10
1.05
1.00
0.95
0.90
0.85
0.80
0.75
0 1 2 3 4 5 6 7 8
Molality (mol/kg)
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
SRK+DH
1.25 NaOH
5
1.20
1.15
Apparent Molar Volume
0
-5
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
Density (g/cm 3 )
1.10
1.05
1.00
0.95
0.90
0.85
Experimental data
MSW EOS
mMSW EOS
eCPA EOS
0.80
0 1 2 3 4 5 6
Molality (mol/kg)
62
0.75
0 1 2 3 4 5 6
Molality (mol/kg)

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:
•
2
a( T)
a a T 298.15 a T 298.15
0 1 2
•
( T)
T
298.15
•
0 1
63

Results
8
Temperature o C
120
100
80
60
40
20
0
Ice
Exp data
MSW EOS
mMSW EOS
CPA EOS
SRK+DH EOS
Na 2
SO 4
. 10H2 O
Na 2
SO 4
0 0.5 1 1.5 2 2.5 3 3.5 4
Molality (mol/kg)
CaCl 2
Molality (mol/kg) )
7
6
5
4
3
2
1
-40 o C
0 o C
Ice
Ca 2
Cl . 6H 2
O
-25 o C
-25 o C
-10 o C
NaCl . 2H 2
O
0 o C, y + 2.5
-20 o C
-10 o C, y + 1.5
-20 o C, y + 0.5
Exp data
MSW EOS
mMSW EOS
CPA EOS
0
0 1 2 3 4 5 6
NaCl Molality (mol/kg)
64

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
65

Results
Apparent molar volume cm 3 /mol
25
23
21
19
17
15
13
11
9
7
5
0 1 2 3 4 5 6
NaCl molality
MSW, 100 °C
mMSW, 100 °C
CPA, 100 °C
Experimental, 100 °C
Experimental, 25 °C
MSW, 25 °C
mMSW, 25 °C
CPA, 25 °C
66

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
67