Application of the Pitzer Model for Describing the Evaporation ...

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Application of the Pitzer Model for Describing the

Evaporation of Seawater

Alexander Kellera, Jakob Burgerb, Hans Hassea, Maximilian Kohns1,a

a University of Kaiserslautern, Laboratory of Engineering Thermodynamics,

Erwin-Schrödinger-Str. 44, D-67663 Kaiserslautern, Germany b Technical University of Munich, Campus Straubing for Biotechnology and Sustainability,

Chair of Chemical Process Engineering, Schulgasse 16, D-94315 Straubing, Germany

Abstract

The present work deals with the fractional crystallization of salts from

seawater. New experimental data for the crystallization sequence and the liquid

phase ion molalities for aqueous model solutions that contain the major seawater

components are presented. For describing the studied electrolyte solutions, the

Pitzer model was used with parameters adopted from the literature. This fluid

property model was combined with an equilibrium model of the fractional

crystallization. The comparison of the simulation results with experimental data

from both the literature (for seawater) and this work (for model solutions) shows

generally good agreement for seawater-like solutions, but also reveals limitations

of the model.

Keywords: Pitzer equations, Evaporation, Crystallization, Chemical

equilibrium, Seawater

1 Corresponding author; [email protected]; phone: +49-631 /

205-4028; fax: +49-631 / 205-3835

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1. Introduction

The crystallization of seawater occurs in several industrial processes, e.g. in

solar evaporation plants for sea salt recovery or in desalination plants for drinking

water purification. In most cases, the focus is on NaCl as a food grade salt [1].

However, besides NaCl, many other salts crystallize during the evaporation of

seawater. Using fractional crystallization, these salts can be obtained separately

[2]. For a simulation of such processes, thermodynamic models are needed that

describe the multicomponent aqueous electrolyte solutions. In this work, it is

explored how well a Pitzer model from the literature for the description of solid-

liquid equilibria (SLE) in seawater, which is extended here by explicitly

considering several dissociation equilibria, performs for describing the

evaporation and fractional crystallization of highly concentrated salt solutions.

Experimental data for the crystallization sequence of solutions from different

natural seawater deposits have been reported by several authors. Babel and

Schreiber [3] provide an overview of the topic. Millero et al. [4] collected

seawater composition measurements from several authors to define a reference

composition of seawater. Recently, Abdel Wahed et al. [5] measured the major

seawater ions in solar evaporation ponds in Egypt. Of particular interest for the

present article is the work of McCaffrey et al. [6], who measured ion molalities

during evaporation processes in a sea salt recovery plant on the Bahamas.

There are several models for describing activities and SLE in electrolyte

solutions that were especially trained for sea salt solutions. Bromley [7] developed

a one-parameter extended Debye-Hückel [8] model for thermodynamic properties

of strong electrolytes in aqueous systems. Bromley et al. [9] used that model to

calculate boiling point elevations and osmotic coefficients for seawater at

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different salinities. The most popular model for describing crystallization in salt

solutions is the Pitzer model [10–13], which employs several adjustable

parameters. Different parameterizations of the Pitzer model have been developed

for describing seawater [14,15]. Harvie et al. [16] provide a parameter set for the

Pitzer equations to describe equilibrium states of natural seawater with the major

components Na+, K+, Mg2+, Ca2+, H+, Cl-, SO42-, OH-, HCO3

- CO32, CO2 and H2O

at 298.15 K. Spencer et al. [17] extended that model by including temperature-

dependent parameters for the subsystem Na+, K+, Mg2+, Ca2+, Cl- and SO42-.

Marion and Farren [18,19] further modified the temperature-dependent parameters

of Spencer et al. [17] to describe the influence of SO42- and carbonate components

more accurately.

The models mentioned above have been used to describe a variety of processes

and phase equilibria of salt solutions. For example, the Debye-Hückel and Pitzer

models are implemented in the computer program PHREEQC [20]. It was used by

Kasedde [21] to calculate the crystallization sequence during evaporation of water

from Lake Katwe in Uganda. Charykova et al. [22,23] determined Pitzer model

parameters for arsenates, selenites and sulfates for the modeling of the weathering

zone of ore deposits. Altmaier et al. [24] used the model of Harvie et al. [16]

together with experimental solubility data to calculate equilibrium constants for

the dissolution of different magnesium hydroxide salts. Hamrouni and Dhahbi

[25] used a Pitzer model to predict calcium precipitation in desalination plants at

different temperatures. Of direct relevance for the present work is the

crystallization sequence upon water removal, e.g. by evaporation. In this field,

Harvie et al. [2] simulated sediment sequences of evaporated salt lakes and

compared the simulation results to layers of sediments observed by different

authors [26]. They distinguished two cases: equilibrium crystallization, where

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crystallization is considered as an equilibrium process, and fractional

crystallization, where crystallization is considered as an irreversible process. They

found that the sediment layer sequences observed in the potash deposit Zechstein

II are best described with the equilibrium crystallization model.

The crystallization sequence, the ion molalities during the course of the

evaporation process, and especially the points at which certain salts start to

precipitate from the solution are important for the design of fractional

crystallization processes. The main scope of this work is to assess how well a

thermodynamic model for seawater-like systems predicts the ion molalities during

the course of the evaporation process by comparison to experimental data. This

model employs the Pitzer equations as parameterized by Harvie et al. [16] and is

extended here by explicitly accounting for the CO2 solubility in seawater, the

dissociation equilibria of carbonic and sulfuric acid, and the self-ionization of

water. First, the model is applied to the seawater evaporation process studied by

McCaffrey et al. [6], and their experimental data are compared to the model

predictions. Then, own lab scale evaporation experiments with aqueous solutions

containing several subsets of the major seawater components are carried out at

298.15 K, and the ion molalities during the course of the evaporation process are

compared to the model predictions.

2. Model

In the present work, aqueous solutions containing the species Na+, K+, Mg2+,

Ca2+, CO2(aq), HCO3

-, CO32-, HSO4

-, SO42-, Cl- as well as H+ and OH- are

considered. In the aqueous solution, the chemical equilibria of carbonic acid,

sulfuric acid and the self-ionization of water are taken into account. The aqueous

solution is in thermodynamic equilibrium with a gas phase and eventually one or

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more solid phases. Water can evaporate from the solution into the gas phase, and,

conversely, the CO2 solubility in the aqueous solution is considered. The several

solids that can precipitate from the solution are assumed to form individual phases

of the pure salts. The salts that might be in equilibrium with the aqueous solution

are listed in Table 1.

[Table 1 about here.]

Technical details of the implementation of the dynamic evaporation process are

given in the Supporting Information.

The conditions for the thermodynamic equilibrium are discussed in the

following. First, the normalizations of the chemical potentials are introduced

briefly. Since we only consider processes at 298.15 K and 101.3 kPa in the

present work, in all following equations we omit the temperature and pressure

dependence of the chemical potentials for ease of notation. The chemical

potentials of all solutes i are normalized to infinite dilution according to Henry’s

law on the molality scale

0 0ln( / ) ln

i i i iRT b b RT , (1)

where 0

i is the reference chemical potential of solute i in the hypothetical

ideal solution at unit molality, ib is the molality of solute i, the unit molality

0b = 1 mol kg-1 is introduced to render the second term dimensionless, and i

is

the activity coefficient of solute i on the molality scale. The chemical potential of

the solvent water is normalized according to Raoult

0

W W WlnRT a , (2)

where 0

W is the chemical potential of pure water and W

a is the activity of

water.

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In the present work, the activity coefficients of all species are described with an

extended Pitzer model which is presented in the Supporting Information. The

parameterization of the Pitzer model is taken from Harvie et al. [16].

The chemical equilibrium of a reaction R is characterized by its equilibrium

constant KR defined as

n

m

n n

n

R

m m

m

b

Kb

. (3)

Here, m are all reactants of the reaction, and n are all products of the reaction.

bi and γi are the molality and activity coefficient on the molality scale of

component i in the solution, and νi is the stoichiometric coefficient of component i

in the reaction. The reactions considered in the present work and their equilibrium

constants, which are taken from the literature [27–29], are shown in Table 2.


The solubility of CO2 in the aqueous solution is calculated according to

Henry’s law on the molality scale

2 2 2 2

m (aq) (aq) (g)

CO CO CO CO H b p , (4)

Here, 2

m

COH is the Henry’s law constant of CO2 in pure water on the molality

scale, 2

(aq)

COb is the molality of CO2 in the aqueous solution,

2

(aq)

CO is the activity

coefficient on the molality scale of CO2 in the aqueous solution, and 2

(g)

COp is the

partial pressure of CO2 in the gas phase. The value 2

m

COH = 2.98 MPa kg mol-1 at

298.15 K is taken from the correlation of Rumpf and Maurer [30]. The gas phase

is assumed to be much larger than the aqueous solution, thus serving as an infinite

reservoir of CO2. The partial pressure of CO2 in the gas phase is therefore

constant, and the value 2

(g)

COp = 3.5·10-5 MPa [31] is used. We assume that the

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solution, which is always in contact with the atmosphere, is always saturated with

CO2.

The aqueous solution and the gas phase are always present. The dynamic

evaporation process leads to the precipitation of one or more salts. The condition

for the precipitation of a salt S can be formulated as the inequality

SP, AP,S SK K . (5)

Here, SP,SK is the solubility product and AP,S

K is the so-called activity product.

These properties are defined as

0

SP,exp n n

S

n

KRT

(6)

and

W,

AP, W

( ) ( ) Sn

S n nn

K b a

. (7)

Here, n are all ionic species the salt S comprises, νn is the stoichiometric

coefficient of ion n in the salt S, and 0

n is that ion’s standard chemical potential.

Furthermore, bn and n are the molality and activity coefficient of ion n in the

aqueous solution, aW is the water activity in the solution, and W,S is the number of

moles of crystal water in one mole of S.

For a given salt, the solubility product is a constant, whereas the activity

product is a function of the composition of the aqueous solution. Hence, Eq. (5) is

to be understood as follows: If the solubility product and the activity product are

actually equal, the aqueous solution is in equilibrium with the salt S. If the greater-

than sign holds in Eq. (5), this is not the case. In the present work, metastability is

not taken into account, i.e. if the solubility product is reached in the aqueous

solution, the salt immediately precipitates.

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The standard chemical potentials 0

n needed for calculating the solubility

products are taken from the literature [16]. The salts considered in the present

work, together with their solubility products, are compiled in Table 1.

The Supporting Information contains figures that illustrate how the activity

coefficients of the ions in solution, the water activity and the activity products of

several salts change during the course of a typical evaporation process.

3. Experiments

Overview

For the purpose of the present work, McCaffrey et al. [6] provide excellent

experimental data for the evaporation of real seawater. However, in order to

enable studying specific subsystems of real seawater and the evaporation of

seawater with greater detail, dedicated lab scale experiments were carried out in

the present work. Testing the validity of the model for these subsystems gives a

feeling for the robustness of the results for multicomponent solutions.

In total, six different salt solutions were prepared and evaporated. The initial

compositions of the solutions as determined from the masses of water and salts

are listed in Table 3.


Two sets of three experiments were carried out: In the first set of experiments

(M1-M3), test solutions were investigated, each of which contained three or four

different ionic species. In the second set of experiments (S1-S3), the studied

solutions had a composition close to that of seawater at three different stages of

evaporation. Six different ionic species (Na+, K+, Mg2+, Ca2+, Cl-, and SO42-) were

present in the solution at the beginning of experiment S1. The initial composition

of the solution for experiment S1 was adopted from the literature [4] to resemble

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seawater. Experiment S2 started with the composition that was found at the end of

experiment S1, and experiment S3 started with the composition found at the end

of experiment S2. As practically all Ca2+ precipitates from the solution during the

course of experiment S1 (see Section 4), the solutions in experiments S2 and S3

only contained five different ionic species. Splitting up the process into these three

consecutive experiments was necessary as the entire process could otherwise not

have been studied in the equipment that was available for the studies of the

present work. To prevent direct precipitation of salts at the start of the

experiments S2 and S3, both solutions were slightly more diluted with water than

the analysis of the samples taken at the end of the previous experiment would

suggest. Due to slow evaporation resulting from high salinities at the end of

experiment S3, the series of experiments was stopped at that point.

Chemicals

The suppliers and purities of the salts used for the preparation of the solutions

for experiments M1-M3 and S1-S3 as well as for the calibration solutions are

listed in Table 4.


Prior to the experiments, all salts were dried in an oven at 373.15 K for 3 or

more hours, except for MgCl2 ∙ 6 H2O which was dried for 2 weeks in a desiccator

at 298.15 K and atmospheric pressure. The calibration samples for the analysis on

the ion chromatograph were prepared in a water-free atmosphere in a glove box.

Milli-Q water with a specific electrical resistance greater than 18.2 MΩcm was

used for sample preparation and dilution.

Experimental setup

The evaporation experiments were performed in a cylindrical crystallizing dish

with a diameter of 190 mm. At the beginning of each experiment, the crystallizing

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dish was filled with about 250 g of salt solution. The experiments were carried out

at atmospheric pressure, which typically equaled 101.3 kPa with fluctuations of

about 3 kPa during a single experiment. Water was evaporated from the salt

solution by blowing preheated dried air (temperature 308.15 K, relative humidity

less than 0.01) over the liquid surface with a fan. With this setup, a water

evaporation rate of up to 25 g/h could be obtained. The solution was held at a

constant temperature of 298.15 K by a temperature control and a heating plate.

The temperature was measured with a Pt100 temperature sensor. The uncertainty

of the temperature measurement was 0.1 K.

The evaporation experiments were divided into N stages. In each stage j, the

crystallizing dish with the salt solution was first weighed to determine the mass of

the solution before the evaporation. Then, water was evaporated from the solution

by the fan for a time interval of about 20 minutes and the remaining solution was

weighed again to determine the mass of water mW,evap,j that had evaporated during

stage j. After taking a sample from the liquid phase with a syringe, the

crystallizing dish with the remaining solution was weighed again to determine the

mass msample,j of that sample. A typical sample had a mass of about 0.5 g. All

samples were filtered with a 45 µm syringe filter to hold back crystals.

Throughout this work, the progress of the evaporation process was measured

by the evaporation degree. For a single stage j, the evaporation degree j

is

defined as the ratio between the initial mass of water mW,j and the mass of

remaining water when the fan was stopped at the end of the stage, i.e.

W,

W, W,evap,

j

j

j j

m

m m

. (8)

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The evaporation degree k

for the entire process up to stage k is then defined

as the product of the evaporation degrees of all single stages

1

k

k j

j

, (9)

Since both evaporation and taking samples change the mass of water, the

variable mW,j in Eqs. (8) and (9) is calculated as

sample,

W, W,0 W,evap,

1 ,

= ions

1

jl

j l

l i l i

i

mm m m

b M

(10)

where W,0

m denotes the mass of water in the beginning of the experiment, ,i l

b

denotes the molality of ion i in the sample of stage l determined from the analysis

explained in the next chapter and Mi denotes the molar mass of ion i. From this

procedure together with the accuracy of the balance, which is 0.01 g, the relative

uncertainty of jis estimated to be below 0.5 %.

Analysis

The solutions in the experiments M1-M3 were analyzed for cations only. The

molality of the only anion, Cl-, was calculated from the electroneutrality

condition. The artificial seawater solutions in the experiments S1-S3 were

analyzed for Na+, K+, Ca2+, Mg2+, Cl- and SO42-. The analysis was performed

using a Metrohm IC 930 Flex ion chromatograph equipped with a conductivity

detector. More details about the IC equipment are given in the Supporting

Information. Prior to the analysis, the samples were gravimetrically diluted with

water. The accuracy of the ion chromatography analysis was estimated by

measuring several test solutions of known composition. It was found that the

relative uncertainty of the ion molalities is better than 2 % for Na+ and K+, better

than 6 % for Ca2+ and Mg2+, and better than 7 % for SO42- and Cl-.

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4. Results and Discussion

A comparison of experimental data, both from the literature and the present

work, with model predictions is carried out in this section. In all figures, we

compare the molalities of the ions in solution during the dynamic evaporation

process. Above the plots, arrows indicate the regions in which the salts precipitate

according to the model. The numerical results of the experimental data from the

present work are presented in the Supporting Information.

In addition to the ion molalities and precipitated salts, we also discuss the ionic

strength I of the solution, which is defined as

2

0

1

1 N

i i

i

I b zb

, (11)

where b0 = 1 mol/kg is the unit molality, bi is the molality of species i, and zi is

the charge number of species i. Hence, the ionic strength I is a dimensionless

quantity.

The ionic strength is often employed as a measure to provide a rough estimate

for the range of applicability of thermodynamic models for electrolyte solutions.

The higher the ionic strength, the greater the difficulty in describing the

thermodynamic properties. The uncertainty of the experimental data of the ionic

strength is calculated in a straightforward manner from the uncertainties of the

experimental ion molalities.

Before discussing the results, we note that in none of the studied systems, a

precipitation of carbonate or bicarbonate salts was observed. This is due to the

very small amounts of CO2, HCO3- and CO3

2- in solution. Hence, the precipitation

of such salts and the molalities of these three species are not discussed further. We

note here that none of the model results change to a noticeable extent when the

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possibility of CO2 uptake and its consequences are removed entirely from the

model.

In Figure 1 and Figure 2, the experimental data for the ionic strength and the

ion molalities in the evaporation process in a desalination plant obtained by

McCaffrey et al. [6] are shown together with the model predictions as a function

of the evaporation degree. Figure 1 shows the full range of the evaporation degree

(1 < < 45), whereas Figure 2 provides a zoom into the range 1 < < 5.

[Figure 1 about here.]


From the beginning of the experiment until = 2.25, the molality of Ca2+ rises. At

= 2.25, a kink in the molality of Ca2+ is found, which indicates that a Ca-

containing salt precipitates. A similar kink is found for the molality of SO42-,

which suggests that the precipitate is CaSO4 ∙ 2 H2O. The model prediction agrees

with the experimental finding. Both the experiment and the model show that the

CaSO4 ∙ 2 H2O precipitation stops at about = 10. In the experiments, at this point

the Ca2+ molality presumably falls below the experimental limit of detection,

because a value of zero is reported by McCaffrey et al. [6] for > 10. At = 10,

also an onset of the precipitation of NaCl is clearly observed both in the

experiment and in the simulation, which is indicated by the kinks in the molalities

for Na+ and Cl-. From that point on, NaCl is the dominating precipitating species,

and only minor amounts of other salts comprising multiple ions are found in the

simulations. From = 10 until = 36 small amounts of glauberite (Na2Ca(SO4)2)

precipitate. From = 36 until the end of the experiment, the simulation predicts

the precipitation of small amounts of polyhalite (K2Ca2Mg(SO4)4 ∙ 2 H2O). As

indicated above, due to the minor amounts of both salts compared to the NaCl

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precipitation, this prediction can neither be supported nor disproved by the

experimental data.

Overall, the agreement between the experimental data and the model prediction is

good. However, over the course of the evaporation process small discrepancies

accumulate, and some deviations become apparent towards the end of the process.

At the end of the experiment, the ionic strength of the solution reaches a value of

about I = 10. It appears that the Pitzer model is capable of describing solutions up

to this ionic strength quite reliably. Nevertheless, a closer investigation based on

dedicated laboratory experiments is desirable that enables to identify limitations

of the employed model. For this purpose, the experiments M1-M3 and S1-S3 were

carried out in the present work. In the following, the results of these experiments

are presented and compared to corresponding predictions from the model.

The results from experiment M1 (solutes: NaCl + KCl) are presented in Figure

3, where the ionic strength and the molality profiles of the ions Na+, K+ and Cl- are

depicted as a function of the evaporation degree.


In both experiment and simulation, from = 1 to = 1.2 the concentrations of

all ions increase. From = 1.2 to = 1.9 the molality of K+ decreases as KCl

precipitates. At = 1.9 the invariant point is reached and the molalities of the ions

remain constant at bNa+ = 5.29 mol/kg and bK+ = 2.18 mol/kg, as NaCl and KCl

precipitate simultaneously. The experimental ion molalities at the invariant point

are in good agreement with literature data for the invariant point in the studied

system [26]. The molalities predicted by the model are generally in good

agreement with the experimental data, showing only a slight underestimation.

Consequently, also the invariant point is predicted to be at somewhat lower

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molalities of Na+ and K+ compared to the experiments. The ionic strength

increases until the invariant point where it reaches its maximum at about I = 7. In

general, the model based on the Pitzer equations is able to predict the dynamic

evaporation in this system comprising three different ions in good agreement with

the experiment.

To further investigate the accuracy of the model, the solution studied in

experiment M2 additionally contains a divalent cation, namely Ca2+. The

evaporation of an aqueous solution containing NaCl + KCl + CaCl2 is examined

experimentally at 298.15 K and the results are compared to simulations. The ionic

strength and molality profiles of the ions Na+, K+, Ca2+ and Cl- for the dynamic

evaporation of an aqueous solution containing NaCl + KCl + CaCl2 are depicted

in Figure 4.


From = 1 to = 1.2 the molalities of all ions increase. Again, the first salt to

precipitate is found to be KCl from = 1.2 onwards. At = 1.6, an onset of the

precipitation of NaCl is observed. As a result, the molality of K+ decreases more

slowly due to the lower availability of the shared anion Cl-. The experiment was

stopped at = 3.2 as a large amount of salt crystals had formed in the solution. At

this point the molality of Ca2+ was still increasing, i.e. the invariant point was not

reached yet. Over the entire dynamic evaporation process, the experimental data

and the model predictions are in good agreement. The quality of the model

predictions is comparable to the one obtained for the simpler system studied in

experiment M1. The main difference is that the simulation overestimates the Na+

and Ca2+ molalities after the beginning of the NaCl precipitation. The ionic

strength at the end of the experiment is I = 10, which is similar to the one at the

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end of the seawater evaporation experiments by McCaffrey et al. [6]. In general,

the model gives reliable predictions in this system with four ions, in which the

maximum ionic strength is I = 10.

However, this changes if a different divalent cation is studied. The solution

studied in experiment M3 also comprises four different ions, but Ca2+ is replaced

by Mg2+. The dynamic evaporation of an aqueous solution containing NaCl + KCl

+ MgCl2 is depicted in Figure 5.


Again, the first precipitating salt is KCl, which is followed by NaCl. However,

at this point already some discrepancies arise between simulation and experiment.

The model predicts the onset of the NaCl precipitation at about = 1.5. In the

experiments, it is found at = 1.8. As a consequence, with increasing the

deviations in the ion molalities accumulate. At about = 3.7, the model predicts

the formation of carnallite (KMgCl3 ∙ 6 H2O) and a corresponding stop of the

precipitation of KCl. However, this behavior is not confirmed in the experiments.

There, the molality of Mg2+ steadily increases until the end of the experiment,

which was stopped at = 5 due to a large amount of crystals in the solution, i.e.

again the invariant point was not reached yet. At the end of the experiment, the

ionic strength reaches I = 15. This highly concentrated solution certainly poses

significant challenges to any thermodynamic model. However, the problems with

the model arise already at lower ionic strengths of about I = 10, where the

precipitation of NaCl is predicted at a too early stage in the process. When

compared to the rather good agreement obtained for the Ca2+-containing system, it

becomes apparent that the Pitzer model, at least in the parameterization employed

here, has difficulties in describing Mg2+-containing solutions. Interestingly, this

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does not pertain the molality of Mg2+ itself, which is still in quite good agreement

with the experimental data for a large part of the dynamic evaporation process. In

contrast, the interactions of Mg2+ with other species are likely overestimated by

the Pitzer model, resulting in a shift in activity coefficients and an according shift

in the crystallization sequences. Additionally, Mg2+ readily takes part in the

formation of many possible mixed salts comprising multiple ions, which opens up

additional possible pitfalls for the model, such as the wrong prediction of

carnallite precipitation in the solution studied here.

To study multicomponent solutions more closely, a second series of

experiments (S1-S3) was carried out in which aqueous solutions containing

several subsets of the major seawater components were studied. The initial

solution was a pre-concentrated seawater solution (S1 in Table 3). The ionic

strength and molality profiles of the ions are depicted in Figure 6.


Right from the beginning of the evaporation process, the model predicts that

CaSO4 ∙ 2 H2O precipitates from the solution. It is hard to discern from the

experimental molalities of Ca2+ if this is actually the case because the amount of

Ca2+ in the solution is rather low. However, from = 1.5 onwards the

experimental molality of SO42- increases more slowly than that of the other ions,

while the molality of Ca2+ is found to stall. Thus CaSO4 ∙ 2 H2O probably does

precipitate, but the precipitating amount is overestimated by the model. Until

about = 4.5, good agreement is found between model and experiment for the

molalities of the other ions. Then, the model predicts that at first two salts

comprising multiple ions start precipitating, namely glauberite (Na2Ca(SO4)2) and

bloedite (Na2Mg(SO4)2 · 4 H2O). For glauberite, only small amounts are found as

most of the Ca2+ has already precipitated before. Shortly afterwards, from about

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= 5 onwards also a precipitation of NaCl is predicted. When compared to the

experimental trends for the ion molalities, at first good agreement can be observed

for Na+, Cl- and Mg2+. The agreement for Ca2+ is less satisfactory, mainly because

most of the Ca2+ ions in the simulation have already precipitated previously as

CaSO4 ∙ 2 H2O. As a consequence, also the experimental and simulated molalities

of SO42- do not agree. With increasing , deviations arise also for the molalities of

Na+, Cl- and Mg2+. This suggests that the different types of salts found in the

simulations might be reasonable, but their amounts are not estimated correctly.

Over the entire process, mainly as a result of the electroneutrality balance that has

to be preserved in the simulations, the molality of K+ is slightly underestimated by

the model. However, from both the simulation results and the experimental data it

is evident that K+ is the only ion that does not take part in a precipitation process

over the course of experiment S1.

The precipitation of a seawater-like solution was continued in experiment S2.

The ionic strength and molality profiles of the ions over the course of experiment

S2 are depicted in Figure 7.


The solution studied in experiment S2 was initially slightly diluted with water (as

compared to the end of the previous experiment S1) to hinder the formation of

crystals at the start of the experiment. Despite this dilution, the simulation

suggests that bloedite precipitates directly. This behaviour is not confirmed in the

experiment: The molalities of all ions in solution increase until = 1.1, where

NaCl starts to precipitate. At = 2.9, the simulation predicts the precipitation of

leonite (K2Mg(SO4)2 · 4 H2O), while the bloedite precipitation stops. Leonite is

predicted to precipitate until = 3.3, from where on kainite (KMgClSO4 · 3 H2O)

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precipitates. In general, deviations between experimental data and the model

predictions accumulate over the entire course of the process due to the initial

incorrect prediction that bloedite precipitates. Still, the onset of the NaCl

precipitation is captured correctly by the model. In addition to NaCl, the

experimental data suggest that a K+-, Mg2+- and SO42--containing salt forms from

= 3 onwards, but the uncertainties in the data do not allow for an unambiguous

determination of its exact stoichiometry.

For the third stage of the seawater evaporation, experiment S3, the ionic

strength and molalities of the ions are depicted in Figure 8.


Comparing the model results and the experimental data reveals similar

problems as for experiments S1 and S2, which is mostly due to the wrong

prediction of SO42--precipitation. From the beginning on the model predicts the

precipitation of three salts: NaCl, carnallite and MgSO4. MgSO4 can form as

different hydrates, whose activity products are all close to the solubility product.

The simulation predicts the precipitation of all these differently hydrated MgSO4-

salts, hence it is labelled MgSO4 · x H2O in the figure. The experimental data

suggest that this is at least somewhat reasonable, as kinks appear in the molalities

of the ions Mg2+ and SO42- at about = 1.1. The experimental molality of Na+ is

decreasing from the beginning of the experiment on. Towards the end of the

studied process the model again gives a false prediction by suggesting the

formation of kainite. However, from = 1.7 on, the predicted molalities remain

constant, as the quinary invariant point of the Na+, K+, Mg2+, Cl-, SO42- system is

reached. At this point, the predicted molalities are bNa+ = 0.07 mol/kg,

bK+ = 0.05 mol/kg, bMg2+ = 5.77 mol/kg, bCl- = 11.26 mol/kg, bSO42- = 0.04 mol/kg,

20

which is in good agreement with the experiment and literature data for the

invariant point in the studied system [32].

Considering the large deviations associated with the formation of

SO42--containing salts such as bloedite or kainite when modeling the experiments

S2 and S3, it is instructive to rerun the simulation and simply remove the

possibility that these salts form. However, when doing so, no improvement of the

representation of the seawater evaporation process is observed. The reason is that

in the calculation, the activity products of many complex salts comprising

multiple ions are close to their solubility products. This is especially true at high

ionic strengths. If one of the complex salts is removed, simply another one takes

its part and introduces discrepancies between model and experiment.

Nevertheless, at the invariant point of the quinary system experiment and model

agree, which may be due to the fact that some of the model parameters were fit to

these points.

5. Conclusions

In the present work, the ionic strength and molality profiles of the major

seawater ions Na+, K+, Ca2+, Mg2+, Cl- and SO42- during evaporation experiments

were compared to simulations using a Pitzer model from the literature [16], which

was extended here by explicitly considering several dissociation equilibria.

Experimental results for the evaporation of real seawater in a desalination plant

were taken from the literature. They were complemented with own experiments

for three test systems as well as a seawater-like solution.

The comparisons between experiment and model show that for simple systems

at moderate ionic strengths (I < 10), the simulation reproduces the experimental

results well. Also the process from the seawater desalination plant can be

21

described satisfactorily. By contrast, some important deviations are observed at

high ionic strengths (I > 10), especially for solutions containing Mg2+ and SO42-.

These deviations are further complicated by the possibility of the formation of

many complex salts comprising several different ions, which also impact the

molalities of all other ions that precipitate together with Mg2+ and

SO42-. However, considering that solutions at such high ionic strength are

challenging to describe, the overall results obtained with the Pitzer model are

quite satisfactory. Nevertheless, for describing real fractional crystallization

processes also quantitatively, a reparameterization of the Mg2+- and SO42--

parameters should be considered and a closer examination of the solubility

products of the complex salts should be carried out.

6. Acknowledgements

We thank the KSB Stiftung for funding this work [KSB Projekt 1299].

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Figure 1: Ionic strength and molalities of Na+, K+, Ca2+, Mg2+, Cl-, and SO4

2-

plotted against the evaporation degree during the evaporation of real seawater.

Symbols are experimental plant data by McCaffrey et al. [6], lines are model

predictions. The experimental uncertainties are always within symbol size.

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Figure 2: Same as Figure 1, but enlarged for the range of evaporation degrees

1 < < 5.

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Figure 3: Ionic strength and molalities of Na+, K+, and Cl- in the solution

studied in experiment M1 (cf. Table 3) plotted against the evaporation degree .

Symbols are experimental data from this work, lines are model predictions. The

experimental uncertainties are always within symbol size.

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Figure 4: Ionic strength and molalities of Na+, K+, Ca2+, and Cl- in the solution




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Figure 5: Ionic strength and molalities of Na+, K+, Mg2+, and Cl- in the solution




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Figure 6: Ionic strength and molalities of Na+, K+, Ca2+, Mg2+, Cl-, and SO42- in

the solution studied in Experiment S1 (cf. Table 3) plotted against the evaporation

degree . Symbols are experimental data from this work, lines are model

predictions. Error bars indicate experimental uncertainties. Where not shown, the


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Figure 7: Ionic strength and molalities of Na+, K+, Mg2+, Cl-, and SO42- in the

solution studied in Experiment S2 (cf. Table 3) plotted against the evaporation




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Figure 8: Ionic strength and molalities of Na+, K+, Mg2+, Cl-, and SO42- in the

solution studied in Experiment S3 (cf. Table 3) plotted against the evaporation




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Table 1: Solubility products log10(KSP) of salts calculated with Eq. (7), using

the standard chemical potentials by Harvie et al. [16].

Salt log10(KSP) Salt log10(KSP)

NaCl 3.646 K2Ca(SO4)2 · 6 H2O 461.167

KCl 2.057 Mg2CaCl6 · 12 H2O 40.028

MgCl2 16.529 K2Ca2Mg(SO4)4 · 2 H2O -31.647

MgCl2 · 6 H2O 10.259 CaCO3 -19.356

CaCl2 17.705 CaMg(CO3)2 -39.334

CaCl2 · 4 H2O 13.164 CaNa2(CO3)2 · 5 H2O -21.693

CaCl2 · 6 H2O 9.541 MgCO3 -18.038

CaSO4 -10.044 Na2CO3 · 10 H2O -1.899

CaSO4 · 2 H2O -10.547 MgCO3 · 3 H2O -11.898

MgSO4 · H2O -0.282 K2CO3 · 3/2 H2O 6.983

MgSO4 · 4 H2O -2.042 KNaCO3 · 6 H2O -0.267

MgSO4 · 5 H2O -2.958 Na2CO3 · H2O 1.109

MgSO4 · 6 H2O -3.765 Na2Ca(CO3)2 ·2 H2O -21.283

MgSO4 · 7 H2O -4.331 Na6CO3(SO4)2 -1.778

Na2SO4 · 10 H2O -2.827 KHSO4 1.329

K2SO4 -4.090 K8H6(SO4)7 -24.882

Na2Mg(SO4)2 · 4 H2O -5.404 NaHCO3 -0.928

NaK3(SO4)2 -8.756 Na3H(CO3)2 · 2 H2O -26.212

Na2K6(SO4)4 -17.512 KHCO3 0.648

Na2Ca(SO4)2 -12.076 NaHCO3 -0.928

Na4Ca(SO4)3 · 2 H2O -13.061 K8H4(CO3)6 · 3 H2O -79.089

KMgCl3 · 6 H2O 9.971 K2NaH(CO3)2 · 2 H2O -20.960

KMgClSO4 · 3 H2O -0.443 K3H(SO4)2 -8.157

K2Mg(SO4)2 · 4 H2O -9.162 Na3H(SO4)2 -1.875

K2Mg(SO4)2 · 6 H2O -9.965 Na2CO3 · 7 H2O -1.059

Table 2: Chemical equilibrium constants log10(KR) taken from the literature.

Reaction R

phase

log10(KR) source

CO2(aq) + H2O ↔ HCO3

- + H+ -6.111 [27]

HCO3- ↔ CO3

2- + H+ -9.452 [27]

H2O ↔ H+ + OH- -14.001 [28]

HSO4- ↔ H+ + SO4

2- -1.993 [29]

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Table 3: Initial molalities of the salt solutions used in the evaporation

experiments.

Name Salts Molality / mol/kg

Na+ K+ Ca2+ Mg2+ Cl- SO42-

M1 NaCl + KCl 2.73 2.14 4.88

M2 NaCl + KCl + CaCl2 1.77 1.39 0.95 5.06

M3 NaCl + KCl + MgCl2 1.83 1.41 1.17 5.57

S1 Seawater 1 1.04 0.02 0.02 0.16 1.17 0.14

S2 Seawater 2 3.67 0.23 1.25 5.35 0.52

S3 Seawater 3 0.34 0.20 4.21 7.73 0.61

Table 4: Suppliers and purities of the salts used in the experiments.

Salt Supplier Purity

g/g

NaCl Sigma-Aldrich > 0.998

KCl Merck > 0.995

CaCl2 Roth > 0.980

Na2SO4 Sigma-Aldrich > 0.990

MgCl2 ∙ 6 H2O Fluka > 0.980

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