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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 45, 2015 

A publication of 

 
The Italian Association 

of Chemical Engineering 

www.aidic.it/cet 
Guest Editors: Petar Sabev Varbanov, Jiří Jaromír Klemeš, Sharifah Rafidah Wan Alwi, Jun Yow Yong, Xia Liu  

Copyright © 2015, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-36-5; ISSN 2283-9216 DOI: 10.3303/CET1545099 

 

Please cite this article as: Suleman H., Nasir Q., Maulud A.S., Man Z., 2015, Comparative study of electrolyte 

thermodynamic models for carbon dioxide solubility in water at high pressure, Chemical Engineering Transactions, 45, 589-

594  DOI:10.3303/CET1545099 

589 

Comparative Study of Electrolyte Thermodynamic Models 

for Carbon Dioxide Solubility in Water at High Pressure 

Humbul Suleman, Qazi Nasir, Abdulhalim S. Maulud*, Zakaria Man 

Department of Chemical Engineering, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750, Tronoh, Perak, 

Malaysia 

halims@petronas.com.my 

The electrolyte thermodynamic models have been extensively studied for carbon dioxide – water system 

for the prediction of vapour liquid equilibrium at low pressures. However, no guidelines are available for 

selection of electrolytic models which are applicable at high pressure for prediction of thermodynamic 

properties. In this study, solubility prediction of limited Debye Huckel (DH), Pitzer Debye Huckel (PDH) and 

modified Three Characteristic Parameter Correlation (mTCPC) electrolyte models have been tested for a 

wide range of temperature (273 – 453 K) and pressure (0.1 – 7.2 MPa).The comparative study shows that 

introduction of electrolyte model improves the prediction accuracy when physical solubility of gas is low, 

either in high temperature or low pressure region. The mTCPC model gives improved prediction than non-

electrolyte model but requires additional parameters and complex calculations. New values for binary 

interaction parameters of UNIFAC for carbon dioxide – water system are also optimized. 

1. Introduction 

Absorption of carbon dioxide in water has found extensive applications in chemical process industries such 

as beverages, enhanced oil recovery, carboxylic acids, etc. Moreover, the recent developments in carbon 

capture have focused on solubility of carbon dioxide in water based solvents. The knowledge of vapour-

liquid equilibrium (VLE) of the CO2 – H2O system is a key factor in the design of such chemical processes 

(Kohl, 1997) and associated process development, like oil and gas sector (Nguyen et al., 2014). Since CO2 

slightly dissociates into ionic species upon dissolution in water, a lot of research has been channelled to 

discuss the effect of ionic equilibria at low pressures and temperatures (Edwards et al., 1978), but has 

limited information at high pressures. Chapoy et al. (2004) applied equation of state / excess Gibbs energy 

model (EoS/G
E
) to predict VLE for CO2 – H2O system for medium pressure range and low temperatures. 

Valtz et al. (2004) compared semi-empirical, EoS/G
E
 and statistical thermodynamic modelling techniques 

for limited pressure and temperature. Both studies neglected the ionic equilibria in liquid phase. The 

importance of ionic equilibria in modelling has been detailed by Li Yuen Fong et al. (2014). 

The purpose of this study is to analyze the effect of electrolytic correction on prediction of carbon dioxide 

solubility in water. The study encompasses a large data with range of pressures (0.1 – 7.2 MPa) and 

temperatures (273 – 453 K).  

2. Experimental Data 

The experimental data was taken from published literature and is given in Table 1. 

3. Determination of Liquid Phase Ionic Equilibria 

The dissociation of carbon dioxide in water is represented by following reactions (Edwards et al., 1978), 

where k1, k2 and k3 represents the equilibrium constants for reactions given in Eq(1) to Eq(3). 

  


   

1

1 2
2 2 3 3

[ ]
[ ]

[ ]

k k CO
CO H O HCO H HCO

H
 (1) 



 

 

590 

 


   

 
   

2
2 2 2 3 1 2 2

3 3 3 2

[ ] [ ]
[ ]

[ ] [ ]

k k HCO k k CO
HCO CO H CO

H H
 

(2) 

  


  

3

3
2

[ ]
[ ]

k k
H O OH H OH

H
 

(3) 

Table 1: Sources of Experimental Data for Carbon Dioxide – Water VLE Equilibrium 

Data Source No. of Data Points Temperature (K) Pressure (MPa) 

Bamberger et al.(2000) 12 323 – 353 4.05 – 7.08 
Campos et al. (2009) 17 303 – 323 0.10 – 0.54 
Dalmolin et al. (2006) 33 288 – 308 0.11 - 0.47 
Han et al.(2011) 16 313 – 333 0.37 – 2.00 
Houghton et al. (1957) 103 273 – 373 0.10 – 3.65 
Lucile et al. (2012) 29 298 – 373 0.54 – 5.14 
Muller et al.(1988) 48 373 – 453 0.60 – 7.21 
Stewart and Munjal (1970) 10 273 – 285 1.01 – 4.56 
Valtz et al. (2004) 32 298 – 318 0.51 -7.03 
Zawisza and Malesinska (1981) 27 323 – 453 0.15 – 4.62 

The electro-neutrality equation of the dissociation is given as follows 

   
  

2

3 3
[ ] [ ] [ ] 2[ ]H OH HCO CO  (4) 

Substituting values of ionic species and solving Eq(4) for hydrogen ion [H
+
] 

 
   

3

1 2 3 1 2 2
[ ] ( [ ] )[ ] 2 [ ] 0H k CO k H k k CO  (5) 

The concentration of carbon dioxide in liquid, [CO2] is determined by Henry’s law relationship. 


2 2 2

[ ]
CO CO

P H CO  (6) 

The values of equilibrium constants for formation of bicarbonate (k1), carbonate (k2), hydroxyl (k3) ions and 

Henry’s constant (HCO2) are taken from Edwards et al. (1978). The concentration of hydrogen ion is 

calculated from roots of the equation and value of [H
+
] was selected between range of 10

-3
 and 10

-7
 

(corresponding to the pH value of 3 and 7 for carbonic acid). Subsequent values of carbonate, bicarbonate 

and hydroxyl ions are found using corresponding equilibrium constants. 

4. Thermodynamic Model 

4.1 Base Model – (translated modified Peng-Robinson – LCVM – UNIFAC) 
Linear Combination of Vidal and Michelsen mixing rule (Boukouvalas et al., 1994) has been used to 

correlate excess Gibbs energy function with Translated Modified Peng Robinson equation of state 

(Magoulas and Tassios, 1990). The excess Gibbs energy function (G
E
) consists of two forces, namely 

short range and long rang forces and given by 

 
( ,  or )

E E E

UNIFAC ELECTROLYTE DH PDH mTCPC
G G G  (7)

 

The short range forces are determined by Universal Functional Activity Coefficient (UNIFAC) local 

composition model (Fredenslund et al., 1975). The long range forces are calculated by Limited Debye 

Huckel (Debye and Hückel, 1923), Pitzer Debye Huckel (Pitzer, 1973) or Modified Three Character 

Parameter Correlation electrolyte equations (Ge et al., 2007) in three different models, discussed further in 

following sections. 

4.2 Model 1 – Limited Debye Huckel Law with tmPR – LCVM – UNIFAC 
The limited Debye Huckel law is a reduced form of Debye Huckel equation that is applicable for weak 

solutions (0.01 molal). 

 


 
2

Limited Debye Huckel

1

i
E

i i

i

G x Az I  (8) 

where A is the Debye Huckel constant, z is the charge on specie and I is the ionic strength of solution. 

Subscript i represents ionic specie in liquid phase. 



 

 

591 

4.3 Model 2 – Pitzer Debye Huckel with tmPR – LCVM – UNIFAC 
The Pitzer Debye Huckel equation for electrolytes was developed by Pitzer. The model is applicable up to 

molalities of 6.0 with good prediction and given by Eq(9). Constants B and C are second and third virial 

coefficients and are fitted to experimental data. Constant b represents the ionic radius of the ions in the 

system. 



  
          


0.5
2 0.5 2 1.5

Pitzer Debye Huckel 0.5
1

2
ln(1 ) 2

1

i
E

i i i i i i

i

I
G x z A bI m v B m v C

bI b
 (9) 

where m and v are molality and stoichiometric coefficients of species. 

4.4 Model 3 – Modified Three Characteristic Parameter Correlation with tmPR – LCVM – UNIFAC 
The mTCPC model is applicable up to 6.0 molal. Constants b, S and n are fitting parameters, whereas T 

represents temperature.  



  
         


0.5 2
2 0.5

mTCPC 0.5
1

2
ln(1 )

1

ni
E

i i

i i

I S I
G x z A bI

bI b T v
 (10) 

4.5 Determination of Interaction and Fitting Parameters 
The binary energy interaction parameters (BIP) for UNIFAC and fitting parameters for mTCPC were 

optimized by following objective function. 



 
 
 
 


calc exp

1 exp

1 N

i

x x
OF

N x
 (11) 

where xcalc and xexp are the calculated and experimental solubility of carbon dioxide. The optimized values 

are given in Table 4 and compared to published literature for base model only. 

Table 2: Binary Interaction Parameters for UNIFAC 

Model 
 Binary Interaction Parameters * 

Electrolytic Interaction Parameters ** 
a12 a21 b12 b21 

Base Model 610.49 283.12 -2.93 2.72    
Literature 
(Voutsas et al., 
1996) 

601.10 271.80 -2.91 2.75    

Model 1 (LDH) 636.22 289.92 -2.64 2.89    
Model 2 (PDH) 636.22 289.92 -2.64 2.89 0.00 (b) 0.00 (B) 0.00 (C) 

Model 3 
(mTCPC) 

634.46 287.83 -2.67 2.99 1.18 (b) 0.98 (S) 0.64 (n) 

* Subscript 1 and 2 represents carbon dioxide and water. 
**The constants for electrolytic interaction parameters for electrolyte equation have been given in brackets. 

The fitting parameters for PDH were found insensitive (0.013 %) to solubility data of carbon dioxide in 

water. Hence, the original Debye Huckel equation was maintained for calculation due to low ionic strength, 

which veritably explains similar values of BIP in model 1 and 2. On the other hand, the values of fitting 

parameters (b, S and n) for mTCPC-LCVM model were maintained due to considerable effect on solubility 

prediction (16.75 %). Moreover, the solvation term in mTCPC-LCVM model explains the formation of ionic 

equilibria in liquid phase (Solbraa, 2002). 

5. Results and Discussion 

Table 3 compares the prediction of optimized UNIFAC BIP (in this study) with literature. Prediction of 

carbon dioxide solubility has been improved by newly optimized UNIFAC BIP values without an effect on 

vapor phase prediction, as compared to the prediction by BIP values found in literature. The root mean 

square error (RMSE) and mean absolute error (MAE) has reduced substantially in liquid phase prediction. 

Figure 1 compares the VLE prediction of LCVM (base) model at various temperatures and pressures with 

newly optimized BIP and BIP available in literature. The prediction of carbon dioxide solubility in water has 

been improved by newly optimized UNIFAC BIP in this study. 

 



 

 

592 

 
Table 3: Comparison of performance of BIP taken from literature and BIP optimized in this study 

Model Type 

Liquid Phase Vapour Phase 

No. of Data 

Points 

RMSE 

(x10
-4

) 

MAE 

(x10
-4

) 

No. of Data 

Points 

RMSE 

(x10
-2

) 

MAE 

(x10
-2

) 

Base Model 313 7.84 4.96 105 4.79 2.91 

Literature (Voutsas et 

al., 1996) 
313 11.25 5.73 105 4.92 2.94 

   

Figure 1: Comparison of VLE prediction of BIP found in literature (Voutsas et al., 1996) and optimized (in 

this study) against experimental data at various temperatures for liquid (a) and vapour (b) phase 

Table 4 compares RMSE and MAE for liquid phase prediction among electrolytic models. Figure 2 

represents the parity graphs for liquid phase prediction for base model, model 1, model 2 and model 3. 

Table 4: Comparison of Electrolytic Models for Carbon Dioxide – Water System 

Model Type No. of Points RMSE (x10
-4

) MAE in liquid phase (x10
-4

) 

Base Model 

313 

7.84 4.96 

Model 1 7.36 5.20 

Model 2 7.36 5.20 

Model 3 7.15 4.89 

The performance of all four models is comparable. The mTCPC – LCVM model has the smallest error but 

has trade-offs to introduce three new fitting parameters and complex calculations. Contrarily, base model 

efficiently predicts the discussed system without using complex set of equations, with slight loss of 

accuracy as compared to mTCPC – LCVM model. 

6. Conclusions 

At high pressures, introduction of primitive electrolytic models (DH and PDH) have a negative effect over 

the accuracy of prediction, while the mTCPC – LCVM model improves prediction accuracy, slightly, but 

involves complex calculations. Therefore, the selection between base model and mTCPC – LCVM model 

has certain trade-offs. In case, the ionic strength is increased (like carbon dioxide absorption in saline 

water or aqueous alkanolamine solutions), the mTCPC – LCVM model will positively outperform base 

model. 

 

100

1000

10000

0 0.01 0.02 0.03

P
re

ss
u

re
 (

K
P

a
)

Carbon Dioxide Solubility (mol. fr.)

273 K

323 K

373 K

448 K

Base (LCVM) Model

Voutsas et al.
1000

10000

0.5 0.6 0.7 0.8 0.9 1

P
re

ss
u

re
 (

K
P

a
)

Carbon Dioxide in Vapour Phase (mol. fr.)

323 K

373 K

453 K

Base (LCVM) Model

Voutsas et al.

(a) (b) 



 

 

593 

 

 

Figure 2: Comparison of prediction by base model (a), model 1 (b), model 2 (c) and model 3 (d) against 

experimental values 

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