Desalination 189 (2006) 278–286

0011-9164/06/$– See front matter © 2006 Elsevier B.V. All rights reserved

*Corresponding author.

Modelling of a decarbonation reactor for geothermal waters:application to geothermal waters of Chott El Fejjej

P. Cézaca*, A.S. Manzolab, M. Ben Amorb

aGroupe Procédés, Laboratoire de Thermique, Energétique et Procédés (LaTEP), UPPA –ENSGTI,rue Jules Ferry, BP 7511, 64075 Pau Cedex, France

Tel. +33 (5) 59 40 78 30; Fax +33 (5) 59 40 78 01; email: pierre.cezac@univ-pau.frbLaboratoire des Procédés Chimiques, Institut National de Recherche Scientifique et Technique,

BP 95, 2050 Hammam-Lif, Tunis, Tunisie

Received 8 April 2005; accepted 20 July 2005

Abstract

The geothermal waters exploitation infrastructures in Chott El Fejjej (Tunisia) are constantly being damaged bya scaling phenomenon. Experimental work has been carried out on a new kind of decarbonation reactor: the opendouble partition reactor (RODP), which may be used in the construction of a new plant at the site of Chott El Fejjej.The objective of this paper is to propose a model, which may contribute to the design and optimisation of theprocess of decarbonation. Our model is based on the thermodynamic laws, which describe the physical and chemicalphenomena occurring in an electrolytic three-phase continuous reactor. However, due to electrostatic forces, thesekinds of systems cannot be successfully described by the concept of an ideal solution. Indeed, the calculation ofactivity coefficients is required. So, by using a group contribution method to calculate the activity coefficient, themodel remains user-friendly. The non-linear system obtained is solved by using Newton–Raphson’s method.Confrontation with experimental results shows that the model correctly describes the ODP reactor.

Keywords: Modelling; Geothermal waters, Electrolytes; Thermodynamic; Activity; Precipitation

1. Introduction

To fight against desertification in southernTunisia, the government utilises fossil watersdrawn from wells with depths varying from 2,000to 3,000 m. Extensive scaling takes place withinthe pipes, following the release of CO2. These

deposits pose a real problem for the exploitationof such waters. INRST of Tunis puts forward anew process of decarbonation: the open doublepartition reactor [1,2]. Through seeding of calciumcarbonate recovered on site during air bubbling,the ODPR constitutes a highly interesting firstexample of simple and clean technology. In fact,there is no chemicals addition. Indeed, the air

doi:10.1016/j.desal.2005.07.013

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bubbling causes the crystallisation of CaCO3 onthe seed through heterogeneous nucleation. Thismainly experimental work should lead to the con-struction of a large industrial reactor at the site ofChott El Fejjej in a short time. For the concept orfor the process optimization, modelling is the mostaccurate tool. One distinguishes two principaltypes of models: models of representation, com-monly called “black box” models, and models ofknowledge, which are the best means of capitaliz-ing on the knowledge acquired from the system.On the one hand, the former are based on the abili-ty to match input/output. They are based on theregression of experimental data, without any theo-retical knowledge of the physical phenomena inplay. However, by their own nature, these sorts ofmodels are not predictive, since they are derivedfrom specific experimental conditions. Their prin-cipal advantage lies in the speed of their solution.They are particularly adapted to “real time” appli-cations like control. On the other hand, the latterrely on the description of physical and chemicalphenomena present in the reactor. Yet, they dohave their drawbacks, especially since they areoften confronted with the complexity of the pro-cesses and connections between chemical kinetics,thermodynamics, hydrodynamics and masstransfer.

The model that we propose is a hybrid model,which positions itself between these two concepts.Indeed, the core of our model is the thermo-dynamic description of the electrolytic multiphasesystem, which constitutes the reactor. In this res-pect, it is a knowledge model. However, our de-scription of the phenomena is subject to onesignificant assumption. We assume that thevarious phases at the outputs are at thermo-dynamic equilibrium. Given this assumption, thecapacities of extrapolation of our model aredefinitely higher than those of the “black box”model. A key problem in any thermodynamicdescription of concentrated complex aqueouselectrolytic systems is how to accurately expressthe activity coefficient of electrolytes in saline

solutions. For saline water, Pitzer’s model [3] isstill widely used. So, for this model and similarones, the key issue is how to determine the re-quired parameters. Therefore, we use a group con-tribution method, which is an effective method topredict thermodynamic properties with onlylimited parameters. The model described in thispaper to represent the CaCO3 scale formation is athermodynamically based model. It is based onthe mathematical laws ruling the physical andchemical equilibria occurring during scale format-ion. It can predict the phase distribution and phasecomposition with a minimum of empirical param-eters.

2. Model description

We use three types of equations:• Mass balance equations.• Constraint equations: equilibria.• Constitutive equations: thermodynamics.

Inflow quantities, temperature and pressure areassumed to be given. Solid salts are supposed asbeing pure. The tank is assumed as being iso-thermic, perfectly agitated and in steady state. Thevarious phases at outputs are supposed to be atthermodynamic equilibrium. Hydration numbersused in the model of activity coefficients are inde-pendent of the salt concentrations. Given theseassumptions, the unknown factors at the outputof the reactor are the following:• The molar flow rate of water.• The molality of each species in the aqueous

phase.• The molar flow rate of each solid phase.• The molar flow rate of the component in the

gaseous phase.

3. Mathematical formulation

The asymmetrical convention is used. Themole fraction scale is used for the water. We usemolality (moles of the species per kg of water) asthe variable of the composition of the solutes.

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3.1. Chemical and physical equilibrium

We consider two types of equilibrium: phaseequilibrium and chemical (or ionic dissociation)equilibrium. For chemical equilibria and liquid–solid equilibria, we use the chemical potential asthe equilibrium criterion. Whereas fugacity willbe the criterion for the liquid–vapour equilibrium.

3.2. Liquid–solid and chemical equilibria

The mathematic equations related to solid–liquid and chemical equilibria are respectively:

( )1

0ts

i

n

i SLi

a mK Tν

=

− =∏ (1)

and

( )1

0ts

i

n

ii

a mK Tνψ

=

− =∏ (2)

where nts is the number of true species in theoverall system. Chemical potential of pure speciesi in its standard state is only dependent on tempera-ture. Likewise, the equilibrium constant mK is afunction of temperature only.

3.3. Liquid–gas equilibrium

The equilibrium condition at T and P for ispecies between the liquid and gas phases is:

( ) ( ), , , ,l gi if T P x f T P y= (3)

For i species in the gaseous mixtures, fugacityis written as:

( ) ( ), , , ,g gi i i i if T P y y T P y P= ⋅φ ⋅ (4)

and for i species in the liquid solution as:

( ) ( ), , , ,l Refi i i i i if T P x x T P x f= ⋅ γ (5)

For water, Eq. (3) is written as follows:

( ) ( )( )

,pure ,pure, , ,

, ,

x lw w w w

gw w w

x T P x f T P

y T P y P

⋅ γ

= ⋅ φ(6)

with:

( )

( )( ) ( )( )

,pure

,pure

Pointing factor

,

, dsat

w

lw

Pl sat sat lw w w w

P T

f T P

T P T P T v P= φ ⋅ ⋅ ∫(7)

For other species in aqueous phase, it is writtenas:

( ) ( )( )

, , , ,

, ,

m mi i i i

gi i i

m T P m H T P

y T P y P

∞⋅ γ

= ⋅ φ(8)

where Him(T,P) is the Henry constant in the molali-

ty scale.

3.4. Partial mass balance

Electrolyte species can be decomposed intochemical elements associated with a given oxida-tion number, called “basic elements”. This break-ing up generates a matrix B = [bi

j] in which bij

equals the number of basic elements j for the truespecies i. The main interest of this decompositionis that dissociation reactions are conservative withrespect to the basic elements, so that partial massbalances can be simplified to:

( )in out 0ji i ib n n⎡ ⎤⋅ − =⎣ ⎦∑ (9)

We chose to eliminate one partial mass balanceand include the electroneutrality equation.

3.5. Numerical procedure

The set of equations constitutes a non-linearsystem in which all the thermodynamic propertiesare expressed as functions of composition, pres-sure and temperature (see Table 1). The overall

P. Cézac et al. / Desalination 189 (2006) 278–286 281

system is solved, in steady state, by Newton–Raphson’s method. The Jacobean matrix is esti-mated by numerical perturbations. The programis performed on Fortran77.

4. Thermodynamic models

4.1. Activity coefficients

A group contribution method is used to esti-mate activity coefficients. Achard [4] developedthe coefficient activity model used here. It com-bines a term of Pitzer Debye–Hückel (long-rangeinteraction) of the type used by Chen et al. [5]with a modified UNIFAC equation [6] (short-range interaction). It is based on the solvation con-cept. This approach appears to be particularlyuseful in making reasonable estimates for thosestrongly non ideal solutions for which data issparse or totally absent. However, any group con-tribution method is necessarily approximate, be-cause the contribution of a given group in onecomponent is not necessarily the same as that inanother component. Moreover, this modelassumes the dissociation of all electrolytes intoconstituent ions. This hypothesis has been success-fully used by several authors [7–10]. The expres-sion for the excess Gibbs energy is constructed as

Table 1Mathematical expression of the general model

Type of equation Equation Chemical equilibria

( )1

0ts

i

n

ii

a mK TνΨ

=

− =∏

Physical equilibria Liquid–solid Liquid–gas Water: Species i:

( )1

0ts

i

n

i SLi

a mK Tν

=

− =∏

( ) ( ) ( ), , ,. , , . , . , , .x l pure l pure gw w w w w w wx T P x f T P y T P y Pγ = φ

( ) ( ) ( ), ,. , , . , . , , .m l m gi i i i i i im T P m H T P y T P y P∞γ = φ

Partial mass balances ( )in out. 0ji i ib n n− =⎡ ⎤⎣ ⎦∑

Electroneutrality . 0i iz m =∑

the sum of the two terms

exexexSRLR GGG

RT RT RT= + (10)

where GLRex represents the contribution of long-

range electrostatic interactions, GSRex stands for the

short-range interaction contributions.

4.2. Modified UNIFAC Larsen with solvation(ULS)

The UNIFAC model is composed of two terms:• A combinatorial term, which describes the

influence of the size and form of a functionalgroup on the non-ideality of the solution. Theonly parameters required are the structuralparameters of the molecules. They are com-puted for each molecule by dividing them intofunctional groups. Each subgroup k is charac-terised by two parameters, one relating to thevolume and one relating to the surface calcu-lated from structural parameters at crystallinestate Rk,c and Qk,c. The number of molecules ofwater fixed around the ion is called the hydra-tion number. In this model, it is assumed thatthe solvation phenomena are constant in temp-

282 P. Cézac et al. / Desalination 189 (2006) 278–286

erature and ionic strength. The combinatorialterm is given as:

ln ln 1c i ii

i ix x⎛ ⎞Ω Ωγ = + −⎜ ⎟⎝ ⎠

(11)

where

2/ 3

2 / 3i i

ij j

j

x rx r⋅Ω =

⋅∑(12)

with Ωi — volume fraction of i, xi — molefraction of i, ri — volume parameter of i.

• A residual term, which corresponds to inter-molecular forces. It is given as:

( )( ) ( )ln ln lnR i ii k k k

kγ = ν ⋅ Γ − Γ∑ (13)

νk(i) is the number of groups of kind k in the

molecule i; Γk is the residual activity coefficientof group k in the solution and Γk

(i) the residualactivity coefficient of group k in the referencesolution i. They depend on the interactionenergy parameters. The interaction energyparameters between two cations or two anionsare fixed equal at 2500 K. Interaction energyparameters between cations and anions arefixed equal at 0 K. Interaction energy betweenwater and water is fixed equal at –700 K. Thus,the energy parameters required are the inter-action energies between ions and water and aspecific energy parameter per salt.

Finally, the short-range activity coefficient γiSR

is given as:

ln ln lnSR C Ri i iγ = γ + γ (14)

4.3. Pitzer Debye Hückel (PDH) contribution(long-range interactions)

The activity coefficient expression of this con-tribution is based on an asymmetrical convention

and is given as:

12

1 312 2 2 22

12

1000ln

2 2ln 11

PDHi

i i x xx

x

AMs

z z I III

φ⎛ ⎞γ = − ⋅⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞⎛ ⎞ ⋅ −⎢ ⎥⎜ ⎟⋅ ⋅ + ρ ⋅ +⎜ ⎟⎢ ⎥⎜ ⎟ρ ⎝ ⎠ ⎜ ⎟⎢ ⎥+ ρ ⋅⎝ ⎠⎣ ⎦

(15)

where:

312 2221

3 1000A s

s

N d eAD k TΦ

⎛ ⎞π ⋅ ⋅⎛ ⎞= ⋅ ⋅⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠(16)

with Ms — molar weight of the solvent (g. mol–1),NA — Avogadro number (mol–1); ds—solventdensity (g.cm–3); e — electron charge (4.802654×10–10 eu); k — Boltzman constant (1.38048×10–16

erg.K–1); Ds — solvent dielectric constant; Ix —ionic strength of solution in mole fraction scale;ρ — minimum distance parameter between twoopposite charge ions arbitrarily fixed at 17.1 byAchard [4]; zi — number of charges of ion i.

The global activity coefficient is then com-puted with the expression:

ln ln lnSR LRi i iγ = γ + γ (17)

The model is presented in detail in a previouspaper [10]. The parameters necessary for proposedexamples in this paper are given in Tables 2–5.

4.4. Chemical and physical equilibrium constants

Required thermodynamic data for the chemicalequilibria were calculated from the softwareSupcrt92 [11]. The physical equilibrium constantfor the calcium carbonate is calculated by thePlummer and Busenberg [12] correlation.

4.5. Liquid–gas equilibria: the Henry constant

The solubility of gases in aqueous phases is

P. Cézac et al. / Desalination 189 (2006) 278–286 283

Table 5Energy parameters for the electrolyte CA

an important parameter in chemical engineering.Dissolved electrolytes lower the solubility of agas compared to that in pure water. This pheno-menon is called the “salting out” effect. It is oftentaken into account, by empirical models, for com-plex aqueous phases like those proposed bySchumpe [13] or Hermann et al. [14]. However,for complex media, an empirical approach re-quires many empirical parameters. Moreover, thecontinuity with a thermodynamic rigorousdescription does not appear clearly. This is particu-larly true for the description of liquid fugacity.We use a classical model proposed by Perry andGreen [15]. Thus, the Henry constant is calculatedthrough Eq. (18)

Table 2Structural parameters Rk,c and Qk,c

Component Rk,c Qk,c Na+ K+ Mg2+ Ca2+ CO3

2– HCO3- Cl–- OH– SO4

2– H+

0.1517 0.391 0.0478 0.1613 0.9379 0.6313 0.986 0.3912 2.8557 0.4661

0.2847 0.535 0.1319 0.2967 0.9591 0.7366 0.9917 0.5354 2.0149 0.6018

Table 3nhi parameters at infinite dilution

Component nhi Na+ K+ Mg2+ Ca2+ CO3

2– HCO3

– Cl– OH– SO4

2– H+

2.606 2.957 3.928 3.077 0 0 0 0 0 2.959

Table 4Interaction energy parameters between ions and water

Ion Ui,w (K) Na+ K+ Mg2+ Ca2+ CO3

2– Cl– OH– SO4

2– H+

401.50 531.54

–589.56 –649.87 –700.0

–1053.97 –1302.46 –1156.93

143.76

Electrolyte CA UCA (K) CO2 CaCO3 MgSO4 K2SO4 Na2SO4 CaCl2

0 0

–1296.99 –162.19 –300.98

0

( )3 2

ln

R Rxi i i

R

i i

T TH A BT T

TC DT

⎛ ⎞ ⎛ ⎞= ⋅ + ⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞

+ ⋅ +⎜ ⎟⎝ ⎠

(18)

Hix is expressed in atm, T in Kelvin and TR =

273.15 K. Parameters Ai, Bi, Ci and Di are listedin Table 6.

Hix depends of the variable of composition,

which is a molar fraction in the expression pro-posed. Thus, we must convert it for molality since

( ) ( ), ,1000

m xsi i

MH T P H T P= ⋅ (19)

where Ms is the molar weight of the solventexpressed in grams.

284 P. Cézac et al. / Desalination 189 (2006) 278–286

Table 6Henry constant parameters for CO2 from [15]

5. Application of the model to the modelling ofthe decarbonation of the geothermal waters ofChott El Fejjej

Manzola [2] has described in detail the opendouble partition reactor. It consists of a 53-L tankin which seed grain, which grows by crystallisa-tion, is fluidised by the upward flow of water andair bubbling. Air is injected in the lower part ofthis tank. A distributor assures the uniform dis-persion of air in the aqueous phase. The releaseof dissolved CO2 constitutes the principle motorof scaling [1,2]. The double partition allows thecollection of decarbonated water (by overflowfrom the first tank). The decarbonated water con-tains grains of CaCO3. Fig. 1 is a simplified repre-sentation of the first reactor. There were threeinputs and two outputs, E1, E2, E3 and S1, S2,respectively (Fig. 1). Input E1 was made up ofgeothermal water. The mass flow rate of waterwas fixed at 13.25 kg.min–1 or 736.11 mol.min–1.Input E2 (seeding) was made up of pure CaCO3solid with a molar flow rate equal to 26.47×10–2

mol.min–1. Input E3 was made up of atmosphericair. The flow rate was kept constant at 140 L.min–1.The gas phase is supposed to be perfect and satu-rated with water. We suppose that the nitrogenand oxygen are incondensable. Moreover, thetemperature is constant at 55°C. Given these con-ditions, the molar flow rate of CO2 gas is keptconstant at 1.819×10–3 mol.min–1.

Table 7 is the summary of the apparent com-position of the geothermal waters of Chott ElFejjej that we used at input E1.

CO2 A –12.027 B 21.674 C –18.038 D 14.979

Fig. 1. Simplified representation of the reactor of decar-bonation.

The objective of the experimental studyundertaken by Manzola [2] was not the validationof the model. Thus, the model provides moreinformation than the experimental data we have.Only the bicarbonate concentration and the pHwere recorded in experiments at the input andoutput of the reactor. The comparison betweencalculated values and experimental values at theinput E1 for calcium and bicarbonate species issummarised in Table 8.

The predicted values correspond very well tothe measurements. Indeed, the deviation betweenthe calculated results and experimental data is lessthan 1% for HCO3

– molality. The pHs are equal.The reactor is supplied directly with geother-

mal water. The physicochemical analysis of thiswater was carried out by Manzola [2]. Table 9

Table 7Apparent species and their composition in the geother-mal waters

Initial apparent species Initial concentration (mmol.kg–1)

CO2 982.340 CaCO3 0.979 MgSO4 2.093 K2SO4 0.602 Na2SO4 7.65 CaCl2 9.988

P. Cézac et al. / Desalination 189 (2006) 278–286 285

Table 8Comparison between calculated concentrations and ex-perimental analysis on site for calcium and bicarbonatespecies at the input E1

Calculated concentration (mmol.kgw–1)

Experimental analysis (mmol.kgw–1)

Bicarbonates 1.956 1.960 pH 6.60 6.60

compares the computed values and the experi-mental values for the majority of the species.

The deviation between the calculated and theexperimental values are very weak, less than 2%.This result indicates that our thermodynamicmodel can correctly represent the behaviour ofthis type of water. The characterization of theaqueous phase without taking into account solidliquid equilibrium also makes it possible to checkthat the water of CF9 is truly in a metastable ther-modynamic state. Indeed, CaCO3 should precipi-tate if this water is at the thermodynamic equilib-rium. The release of dissolved CO2 causes the cal-cium carbonate precipitation.

Table 10 presents the calculated concentrationsand the experimental values at the output S1.

The calculated pH is close to the experimentalpH with a variation of 0.2 points: respectively 7.68

Table 9Physicochemical analysis of the geothermal water at theinput E1

True species Calculated concentration (mmol.kgw–1)

Analysis (mmol.kgw–1)

Cations: Ca2+ Mg2+ Na+

10.97 2.09

15.3

10.98 2.09

15.09 Anions: SO4

2– Cl–

10.35 19.97

10.49 19.97

Table 10Comparison of calculated values and experimental analy-sis on site at Output S1. Temperature: 55°C

pH Calculated Experimental 7.68 7.49 HCO3

– (mmol.kgw–1)

Calculated concentration

Experimental analysis

1.53 1.64

and 7.49. The pH is dependent on the ion H+,which is also implied in many reactions. More-over, its coefficient of activity is in turn dependenton all the other components. Consequently, thisrelatively weak variation leads to the accurateprediction of the behaviour of the model. A 6.7%deviation, between the HCO3

– concentration ob-tained by simulation, and the one obtained duringthe experiment is observed. This result seems toshow that the thermodynamic pathway is not toofar away from the real mechanisms, even if theyare only a global representation of the phenomenaimplied. A model must be adapted to each use.Our objective was to propose a model for the de-sign and the optimization of the process. Theestablished model is adapted to this field. Its rigor-ous and robust thermodynamic core positionsitself as a solid base for an evolution towards amodel which takes into account the descriptionof more phenomena like the mass transfer.

6. Conclusion

A modelling approach of a continuously stirredtank decarbonation reactor is presented. Themodel proposed is a hybrid model, which placesitself midway between the black box concept andthe knowledge model. The core of the model isthe thermodynamic characterization of the in-volved phases (solid, gas and aqueous). The cal-culation of the coefficient of activity, through agroup contribution method, makes it possible todescribe the non ideality of the aqueous systemwith great flexibility. All the systems that present

286 P. Cézac et al. / Desalination 189 (2006) 278–286

the same decomposition in functional groups canbe described without any other supplementaryparameters than those we propose. Moreover, thecomparison between computed values and experi-mental values shows that this global approachgives good results as well as has an undeniableadvantage over the “black box” model: its pre-dictive character. Indeed, through mathematicalequations, this type of model manages to take intoaccount experimental conditions (pH, composi-tion, etc.). Thus, when these conditions change,the descriptive formulation of the model makes itpossible to obtain accurate and realistic results.This is contrary to empirical models, which de-pend on empirical parameters obtained directlyfrom experimentation. The prospects for theevolution of this model are numerous. Thermo-dynamic description is necessary, but insufficientfor a rigorous description of the implied pheno-mena. Thus, the suggested model is only the firststage. A second stage could be the integration ofthe description of the mechanisms implied incrystallization: nucleation and growth.

Symbolsa — Activityb — Number of basic elementsf — Fugacity, PaG — Gibbs energy, J.mol–1

H — Henry constant, Pam — Molality, mol.kg–1

mK — Equilibrium constant modelP — Pressure, PaT — Temperature, Kv — Molar volume, m3.mol–1

x — Molar fraction (liquid phase)y — Molar fraction (gas phase)z — Charge of the ion

Greekν — Stoichiometric coefficientϕ — Fugacity coefficient

Subscriptsi — Component iw — WaterSL — Solid liquid equilibriumΨ — Chemical equilibrium

Superscriptsl — Liquid phaseg — Gas phaseRef — Reference states — Solid phasex — Molar fraction scalem — Molality scalesat — Saturation state∞ — Infinite dilution

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