Huan Zhou,Peng Wu,Wenxuan Li,Xingfan Wang,Kuo Zhou,Qing Hao

1 College of Chemical Engineering and Materials Science,Tianjin Key Laboratory of Brine Resources and Eco-utilization,Tianjin University of Science and Technology,Tianjin 300457,China

2 College of Marine and Environmental Sciences,Tianjin Key Laboratory of Brine Resources and Eco-utilization,Tianjin University of Science and Technology,Tianjin 300457,China

Keywords: Aqueous electrolytes Comprehensive thermodynamic model Aqueous Li+-Na+-K+-SO42-Phase diagram Thermodynamic properties

ABSTRACT It is still a challenging task to accurately and temperature-continuously express the thermodynamic properties and phase equilibrium behaviors of the salt-lake brine with multi-component,multitemperature and high concentration.The essential subsystem of sulfate type brine,aqueous Li+-Na+-K+-SO42- and its subsystems across a temperature range from 250 K to 643 K are investigated with the improved comprehensive thermodynamic model.Liquid parameters(ΔgIJ,ΔhIJ,and ΔCp，IJ)associated with the contributions of Gibbs energy,enthalpy,and heat capacity to the binary interaction parameters, i.e.the temperature coefficients of eNRTL parameters formulated with a Gibbs Helmholtz expression,are determined via multi-objective optimization method.The solid constantsof 11 solid species occurred in the quaternary system are rebuilt from multi-temperature solubilities.The modeling results show the accurate representation of (1) solution properties and binary phase diagram at temperature ranges from eutectic points to 643 K; (2) isothermal phase diagrams for Li2SO4-Na2SO4-H2O,Li2SO4-K2SO4-H2O and Na2SO4-K2SO4-H2O ternary systems.The predicted results of complete structure and polythermal phase diagram of ternary systems and the isothermal phase diagrams of quaternary system excellently match with the experimental data.

1.Introduction

Qinghai salt lakes includes chloride and sulfate two types,e.g.Chaerhan salt lake of chloride type composed by Na+,Mg2+,Ca2+,K+,Li+,Cl-,B2O3and H2O,and Taijinaier(East and west)and Yiliping salt lake of sulfate type composed by Na+,Mg2+,K+,Li+,,Cl-,B2O3and H2O.Understanding salt forming behaviors from those salt-lake brine under certain conditions is of major importance in various fields of science and technology,such as geochemistry of rock salt and potash salt deposits,potash and lithium extraction and effective use of salt lake resources[1].This requires not only a large number of reliable experimental data,but also the accurate and predictable thermodynamic models.

In the last 100 years,much efforts have been expended by physical chemists toward a fundamental theory for electrolyte solutions,e.g.,the typical fundamental models are established with the base of perturbation theory[2],or equation of state[3],or solvation concepts [4,5].Meanwhile,the semi-empirical models for various applications are developed and commonly used,such as Pitzer’s type models [6,7],extended UNIQUAC model [8],eNRTL model [9–11],OLI-MSE model [12,13],and modified MSA model[14,15].However,due to the complexity of electrolyte solution and actual brine system,the development of models with few parameters,accuracy and predictability has never stopped.

The electrolyte NRTL model provides a thermodynamic framework to represent thermodynamic properties of electrolyte solutions,which captures the electrolyte solution non-ideality over the entire concentration ranges with two binary interaction parameters for each pair of molecule-electrolyte and electrolyteelectrolyte to correlate the composition dependence of solution non-ideality [16].The temperature dependence of the binary parameters further correlates to the excess Gibbs energy,excess enthalpy,and heat capacity at 298.15 K,and gives the predictive ability of multi-temperature properties and phase equilibrium.Using ASPEN system,Chen and his group reported some results of aqueous systems of NaCl-Na2SO4[17],NaCl-KCl [18],Ca2+-Na+-K+-Cl-[19],Mg2+-Na+-K+-Cl-[20],etc.

In order to more accurately and temperature-continuously express the thermodynamic properties and phase equilibrium behaviors of salt-lake brine with multi-component,multitemperature and high concentration,based on Song and Chen’s framework [11],Zhou proposes (1) one expression of symmetric activity coefficients of Pitzee-Debye-Hückel term (PDH) for ionic species in multicomponent electrolyte systems[21],(2)the correction scheme of chemical contribution(the coming articles),and(3)the multi-objective method to obtain the thermodynamic constants of solid or ionic species and liquid parameters (this work).The satisfactory results have been achieved for the challenging system of MgCl2-CaCl2-H2O [21]and typical brine system of aqueous Mg2+-Na+-K+-Cl-[20].

The sulfate type salt-lake brine,when boron is temporarily neglected,involves 8 binary and 16 ternary systems,among which,the essential subsystem of aqueous Li+-Na+-K+-concerned in this study,composed of 3 binary and 3 ternary systems.Because the sulfate salt(i.e.Li2SO4,Na2SO4or K2SO4)solubility changes significantly with temperature rising from eutectic points to high temperature(i.e.643 K),seeing Figs.1–3,the modeling with finite parameters would be difficult.

In this study,the comprehensive thermodynamic framework and parameterization scheme are further presented; the liquid parameters and solid species constants are obtained by regressing available thermodynamic properties and solubility data from literatureviamulti-objective optimization method.With those optimized parameters,the modeling results of thermodynamic properties,binary and ternary phase diagrams at entire temperature ranges,and the predicted results of full structure and polythermal phase diagrams for ternary systems and isothermal phase diagram for quaternary systems have been fairly achieved.

Fig.1.Solubility diagram Li2SO4–H2O at temperature range from 250 K to 643 K.

Fig.2.Solubility diagram of Na2SO4-H2O system at temperature range from 270 K to 653 K.

Fig.3.Solubility diagram of K2SO4-H2O system at whole temperature range from 271 K to 630 K.

2.Thermodynamic Framework

2.1.Liquid parameters

The electrolyte solution non-ideality of excess Gibbs energy(Gex) is expressed as the sum of two contributions:

whereGexis the total excess Gibbs energy,Gex，PDHandGex，lcdenote the excess Gibbs energy of long range contribution and local interaction contribution.

The long range ion–ion interactions are represented with the extended Pitzer–Debye–Hückel (PDH) formula for multicomponent electrolytes:

wherenis the total mole number of the solution,Ris the gas constant,Tis the absolute temperature;Axis the Debye-Hückel parameter on mole fraction basis,which can be converted from the molality basis value ofA?by the relation of;Ixis the ionic strength in mole fraction;=Ix（xw→0） is the ionic strength at fused salt reference state; ρ=14˙9 is the closet approach parameter;NA=6.02214076×1023mol-1is Avogadro’s number,Qe=1.602176634×10-19C is the electron charge,kB=1.380649×10-23J·K-1is Boltzmann constant;ziis the charge number of speciesi;dw,Mw,ε are the density,molecular weight,and dielectric constant of water.The values ofb1-3are 78.54,31989.4,and 298.15.

The local interaction contributions that exist at the immediate neighborhood of any species are represented with the electrolyte NRTL local composition formulation:

whereiis the species index including molecular species m,cationic species c,and anionic species a;niandxiare the mole number and the mole fraction of speciesiin the system,respectively; ziis the charge number of ionic species and it is unity for molecular species;aijis the non-randomness factor(fixed at 0.2 in this study);τijor τjiare the ion-based asymmetric binary interaction energy parameter between liquid speciesiandj(solvent m,cation c and anion a),Note that the binary interaction parameter between the same species is taken as τii=0 or τjj=0.

The ion based binary interaction parameters τijare inferred from the salt-pairs based binary interaction parameters τIJ,via mixing rules [11,21]for each molecule-electrolyte (τca-w) and electrolyte-electrolyte(τca-c’a’).A Gibbs Helmholtz like equation suggested by Clarke and Glew[38]is proposed for the temperature dependence of the binary interaction parameters.

where ΔgIJ,ΔhIJ,and ΔCp，IJare the liquid phase parameters for one given system which represent,respectively,the contributions of Gibbs energy,enthalpy,and heat capacity to the binary interaction parameters at reference temperatureTref,i.e.298.15 K;τIJ,ΔgIJand ΔCp，IJare dimensionless; while ΔhIJis in the unit of temperature,i.e.K.

2.2.Liquid enthalpy

wherec1-5are the temperature coefficients offor ionic speciesi.which are obtained by regressing the heat capacities of binary aqueous solution at multi-temperature,and are further optimized by fitting the isobaric heat capacities of several binary systems at multitemperature simultaneously.?ln/?Tand ?2ln/?T2in Eq.15 at temperatureTare calculated from the first and second order difference,i.e.

Differential stephwas set to 0.01 K in this work.

2.3.Constants of solid and liquid species

Solid-liquid phase equilibrium (SLE) between liquid phase species (i) and a given solid (k) can be described as

whereAsp，kandKsp，kare activity product and solubility product constant respectively for a given solid speciesk,i.e.k=Mv+Cv-·nH2O;mi,γm，iare respectively the molarity and molarity based activity coefficient of ionic speciesiat temperatureTin the liquid phase;vi，kis the stoichiometric coefficients of speciesi(include hydrated water) in solid speciesk;

The relationship between mole fraction and molarity based activity coefficients of ionic species is as follows:

Fig.4.Information flow diagram of thermodynamic properties and SLE data regression.

3.Parameterization Scheme

3.1.Parameterize process

Modeling system includes the models of activity coefficients models,solution properties models and solid–liquid phase equilibrium models,and data bases of liquid parameters,solid constants and ionic constants.The original experimental data of thermodynamic properties,such as mean ionic activity coefficients,saturated vapor pressure,water activity,freezing point,isobaric molar heat capacity,and SLE data in the literature,are employed to determine solid constants and to fit liquid parameters.Fig.4 presents the information flows of parameterize process.

(1) Liquid parameters (ΔgIJ,ΔhIJ,and ΔCp，IJ) for interaction parameter τIJare estimated from thermodynamic propertiesviamulti-objective optimization of multi- properties calculation,and through the liquid properties path of ①and ③,or SLE path of ②and ⑤.

The Gibbs energy term ΔgIJis correlated to the excess Gibbs energy of aqueous single electrolyte systems at 298.15 K,which is estimated from the binary liquid properties of mean ionic activity coefficients or osmotic coefficients,saturated vapor pressure at 298.15 K.

With the Gibbs energy term ΔgIJbeing identified,the enthalpy term ΔhIJand heat capacity term ΔCp，IJare correlated to the excess enthalpy and excess heat capacity at 298.15 K,respectively.ΔhIJ,and ΔCp，IJare estimated simultaneouslyviafitting thermodynamic properties at multiple temperatures.

In order to fit those liquid parameters for the multiple properties at whole temperature range,firstly,we use the experimental data of activity coefficients or osmotic coefficients and saturated vapor pressure at 298.15 K to identify the excess Gibbs energy ΔgIJ,and then,we use the saturated vapor pressure,the heat capacity of binary solution,ionic mean activity coefficients and the freezing points of solution to identify ΔhIJ,and ΔCp，IJ viathe multi-objective optimization method.

3.2.Multi-objective problem

For the multi-objective problem,i.e.,the experimental data of activity coefficients,vapor pressure,heat capacity and freezing points are simultaneously employed to determine the liquid parameters,the fgoalattain function for multi-objective goal attainment in MATLAB is adopted.The multi-objective goal attainment problems can be specified by

Minimize δ

where δ is the slack variable(scalar)used as a dummy argument.F and goal are the vector of objective function and experimental data for different properties,respectively.fn（x） is property calculation function.W is relative importance factor of goals.

We express comparison of the model results and the literature data with average mean relative deviation(MRD).MRD represents a quantitative measure of the comparison with the model results for a specific data set,which are calculated using the following equation.

wherekis the number of data points,Estis the estimated value,andExpis the experimental value.

4.Results and Discussion

4.1.Solution properties and liquid parameters

? Li2SO4-H2O system

It is assumed that the dissolved Li2SO4in aqueous solution is ionized completely into Li+andBy contrast,Li+andwill be precipitated as solid Li2SO4·H2O(cr)or Li2SO4(cr)when reaching saturation at a given temperature.

Some readily accessible experimental thermodynamic data of Li2SO4+ H2O binary are listed in Table 4.The available data such as vapor pressure,mean ionic activity coefficient,osmotic coefficient,liquid enthalpy,heat capacity,freezing points and solubility provide a comprehensive database to identify the liquid parameters,i.e.temperature coefficients(ΔgIJ,ΔhIJ,ΔCp，IJ)of binary interaction parameters τIJ.These liquid parameters are identified simultaneously from fitting the experimental data covering a temperature range from 250.13 to 573.15 K,and the results are reported in Table 3,and the temperature dependence of binary interaction parameters τIJis shown in Fig.5.

Table 1Thermodynamic constants of liquid species for aqueous Li+-Na+-K+-SO42- system

Table 2Solid parameters in Li2SO4-Na2SO4-K2SO4-H2O system

Table 3Liquid parameters for aqueous Li+-Na+-K+- system

Table 3Liquid parameters for aqueous Li+-Na+-K+- system

①.

Vapor pressure data covering the temperature range from 298.43 to 505.15 K of Li2SO4saturated solution are reported by Applebey [23]and Campbell [24],and the smoothed data at temperature range from 276.15 K to 573.15 K and concentration from 2%to 22%are reported by Zaystev[44].Fig.6 shows the isothermal vapor pressure of Li2SO4unsaturated solution and multitemperature vapor pressure of Li2SO4saturated solution,which shows that the modeling results are quite consistent with the experimental data at temperature range from 298.15 to 573.15 K,but a little different from Zaytsev’s smooth data [44]at 323.15 K.

Freezing point is directly related to the water activity at temperature below 273 K.Campbell [24],Indelli [45]and Khu [46]reported freezing data of Li2SO4aqueous solution covering the temperature range from 250.13 to 273.11 K and the concentration from 0.1%to 27.9%.Fig.7 presents very satisfying matches between the model results and those experimental data.It can be inferred from Fig.7,the higher MRDs of freezing data in Table 3 is caused by the systematic error between different data sets.

Heat capacity for aqueous Li2SO4solution at temperature from 276.15 to 373.15 K and mass percentage from 2% to 24% are reported by Zaystev[44].Fig.8 shows the comparison of the modeling and literature heat capacity of Li2SO4aqueous solution,which denotes the quite consistent between both data.

Fig.5.Temperature dependence of eNRTL binary interaction parameters for Li2SO4,Na2SO4,K2SO4 aqueous binary system.

Fig.6.Comparisonofthe experimentaldata(symbols)forvaporpressureof Li2SO4-H2Osystemand calculatedresults(curves)ofthiswork.‘●’A.Indelli[45],‘Δ’A.N.Campbell [24],‘×’ Zaystev [44],‘- -’ Calculated result in this work,‘’ Calculated vapor pressure of solid saturated solution in this work.

The non-ideal properties such as ionic mean activity coefficients,osmotic coefficients and water activities at several temperatures of 298.15,313.15,323.15 K were respectively reported by Robinson [47],Rard [48],Gilchrist [49],Guendouzi [50].The comparisons of modeling results to those experimental data are reported in Table 4.The weighted average of MRDs calculated from all those properties and all data points is 1.15%.Fig.9 shows the comparisons of calculated mean activity coefficients to Zaytsev’s smoothed data [44]at temperature range from 273.15 K to 553.15 K,meanwhile the mean activity coefficient of Li2SO4saturated solution is also presented.Fig.9 denotes the modeling activity coefficients are quite consistent with Robinson’s data [47],but larger MRDs of 4.76% with Zaytsev’s data.It should be noted that many of Zaytsev’s data [44]are beyond Li2SO4solubility range.

Table 4Thermodynamic properties for liquid Li2SO4–H2O system

Fig.7.Comparison of the experimental data(symbols)for freezing points of Li2SO4-H2O binary system and the model results (curves).

Fig.8.Comparison of the literature data (symbols) and the model results (curves)of heat capacity of Li2SO4 aqueous solution.‘○’Zaystev[44],‘–’Calculated result in this work.

? Na2SO4-H2O system

Although the modeling results of aqueous Na2SO4systemviaeNRTL model have been reported by Yan and Chen [17].Due to the discussion in Section 1,we revisit Na2SO4+ H2O system in order to obtain more reasonable liquid parameters and solid constants based on the improved models.

Assuming that Na2SO4dissociates to Na+andcompletely in aqueous solution,Na+andwill precipitate as Na2SO4·10H2-O (cr) or Na2SO4(cr) upon saturation at a given temperature.The readily accessible solution properties of vapor pressure,freezing points,heat capacity,molality scale mean ionic activity coefficients,osmotic coefficients,and water activity are summarized in Table 5.

The obtained liquid parameters (ΔgIJ,ΔhIJ,ΔCp，IJ) for Na2SO4-H2O system are reported in Table 2,and the temperature dependence of binary interaction parameters τIJis shown in Fig.5.The MRDs between modeling results and experimental data are listed in Table 5.

Fig.9.Comparison of the literature data (symbols) model prediction results(curves) for Mean ionic activity coefficients of Li2SO4(aq) at temperature from 273.15 K to 493.15 K and concentration from zero to saturated.‘○’Zaystev[44],‘▲’Robinson[47],‘–’Calculated result in this work,‘--’Calculated Mean ionic activity coefficients of solid saturated solution in this work.

Apelblat[51],Leopold[52],Keevil[53],Kangro[54]and Bhatnagar [55]reported the vapor pressure data respectively,where in the isothermal vapor pressure data of Kangro [54]and Bhatnagar[55]are employed to regress liquid parameters,and the smoothed data of Zaystev [44]and non-isothermal vapor pressure data of Na2SO4saturated by Apelblat [51],Leopold [52]and Keevil [53]are used to compare model results.Fig.10 shows the comparisons of vapor pressure with temperature up to 640 K and concentration up to Na2SO4saturated,where the red curve is the predicted vapor pressure of Na2SO4saturated solution.Fig.10 denotes the calculated results of both isothermal pressure and multi-temperature pressure of Na2SO4saturated solution are very well consistent with experimental data.But Leopold [52]data is obvious deviates from Apelblat [51]and result in the larger MRD of 19.9% in Table 5.

Freezing points covering the temperature range from eutectic point of 271.5–273.11 K.Fig.11 denotes the very satisfying matches of calculated freezing points with that reported by Wuite[56],Deng [28]and Zaystev [44].

Heat capacity data reported by Likke [57],Rogers [58],Conti[59]and Saluja [60]are used to regress the liquid parameters.The total average of relative errors is 0.25%.Fig.12 plots the percentage of relative errors between calculated values and experimental data for heat capacitiesCpper kg aqueous Na2SO4solution up to 453.15 K and 2.63 mol·kg-1.The results show excellent agreement between model fitting results and experimental data.

The non-ideal properties of ionic mean activity coefficients γ±,osmotic coefficients ?and water activities αwwere respectively reported by Holmes [61],Stokes [62],Rard [63],Platford [29]and Baabor[64].MRDs for modeling results to those experimental data are reported in Table 4.The average relative errors are 1.39%,1.68%and 0.17%for γ±,?and αw,respectively.Fig.13 shows the comparisons of calculated mean activity coefficients to ionic mean activity coefficients data of Holmes [61]at temperature from 273.15 K to 498.15 K and concentration from 0.1 mol·kg-1to solid saturated.Meanwhile,the mean activity coefficient curves of Na2SO4saturated solution at entire temperature region are also presented.Fig.13 denotes the modeling activity coefficients are quite consistent with the experimental data.

? K2SO4-H2O system

The modeling results of aqueous K2SO4binary systemviaeNRTL model have been reported by Sanjoy and Chen[65].Here,we revisit K2SO4+ H2O system to obtain the liquid parameters and solid constants basing on own improved model systems.The results of liquid parameters (ΔgIJ,ΔhIJ,ΔCp，IJ) for K2SO4-H2O binary system are reported in Table 2,and the temperature dependence of binary interaction parameters τIJare shown in Fig.5.The solution properties and MRDs of calculations are summarized in Table 6.Limitedby the article length,the figures to describe the calculation results are omitted.

Table 5Thermodynamic properties for liquid Na2SO4-H2O system

Table 6Thermodynamic properties for liquid K2SO4-H2O system

Fig.10.Comparison of the experimental data for vapor pressure of Na2SO4 aqueous solution at temperature from 273.15 K to 640.15 K and those calculated from this work.‘×’ Apelblat [51],‘○’ Bhatnagar [55],‘’ Kangro [54],‘’ Leopold [52],‘●’Keevil [53],‘–’ modeling result of this work,‘–’ Calculated vapor pressure of solid saturated solution in this work.

Fig.11.Comparison of the experimental (symbols) data for freezing points of Na2SO4-H2O binary system and the model results (curve).

Fig.12.Relative error of heat capacity between experimental data of Na2SO4(aq)and those calculated with this model.

Fig.13.Comparison of the literature data (symbols) and model prediction results(curves) for mean ionic activity coefficients of Na2SO4 (aq) at temperature from 273.15 K to 493.15 K and concentration from zero to saturated.‘○’Zaystev[44],‘–’this work,‘--’Calculated Mean ionic activity coefficients of solid saturated solution in this work.

4.2.Binary phase diagrams and solid parameters

? Li2SO4-H2O system

Solubility data of Li2SO4-H2O binary system are summarized in Table 7.Three solid species Li2SO4(s),Li2SO4·H2O(s) and ice are found in the system at entire temperature range from 257.15 to 643.15 K.Monohydrate Li2SO4·H2O dominates the temperature range from 250 to 510 K,and nine authors reported its solubility data,but with high dispersion and uncertainty (see Fig.14).We choose the data of Friend [76],Linke [22],Sohel [25],Applebey[23]and Campbell [24]to regress Li2SO4·H2O(s) solid constants ofat 298.15 K,and coefficients of（T）.The results are listed in Table 4.The comparisons of model results and experimental data are shown in Fig.4.The empirical formula of Li2SO4-·H2O(s) solubility recommended by’ IUPAC-NIST solubility data[77]are also presented on Fig.14.

Fig.14 denotes the model can well express the solubility change of Li2SO4·H2O(s) with the temperature in the whole temperature range,and consistent with the empirical formula recommended by IUPAC-NIST.In particular,the model has good temperature ductility,which provides a guarantee for the prediction of the solubility in polythermal systems.

Anhydrous lithium sulfate Li2SO4(s) dominates the solubility region at temperature higher than 508 K,and its solid constantsat 298.15 K have to be extrapolated to fit the its solu bility data reported by Elenevskaya [27]and Marshall [26].The solid heat capacity coefficients of anhydrous lithium sulfate in regression are consistent with the data in tabulated data of physical property manual [40].The results are listed in Table 2.Fig.15 shows the model results which are in good agreement with experimental data in such a high temperature region up to 643 K.

Table 8Solubility data of Na2SO4-H2O system

Fig.14.Comparison of the experimental data for solubility of Li2SO4·H2O in water and the model results.

Fig.15.Comparison of the experimental data for solubility of Li2SO4 in water and the model results.

Table 7Solubility data of Li2SO4-H2O system

Fig.16.Comparison of the experimental data for solubility of Na2SO4·10H2O in water and the model results.

Fig.17.Comparison of the experimental data for solubility of Na2SO4 in water and the model results.

Fig.1 shows the Li2SO4-H2O phase diagram at entire temperature region.The MRDs of ice,Li2SO4·H2O(s) and Li2SO4(s) are 3.28%,0.99% and 3.77% respectively.The eutectic freezing point is determined to be 251.67 K (-21.48 °C) with composition as 27.44% (mass) Li2SO4(3.439 mol·kg-1),the invariant point of Li2-SO4·H2O and Li2SO4co-saturated is estimated to be 506.58 K(233.43 °C) with 23.04% (mass) of Li2SO4(2.72 mol·kg-1).

? Na2SO4-H2O system

Solubility data of Na2SO4-H2O binary system reported by ten authors are summarized in Table 8.Three solid species Na2SO4(s),Na2SO4·10H2O(s)and ice are found in the system at entire temperature region from 271.67 K to 653.15 K.Na2SO4·10H2O (s) dominates the temperature range from 271.95 to 305.53 K.We choose solubility data of Deng and Zhou [28],Platford R F [29],Halla[30],Ivanova [31],and Flatt [33]to regress Na2SO4·10H2O solid constants ofat 298.15 K.The results are listed in Table 2,and presented in Fig.16.

Solid species of anhydrous Na2SO4(s)cover a very large temperature region from 305.53 K to 653.15 K,and its solubility varying with temperature follows an unusual curve (see Figs.2 and 17).It is found that solid species of Na2SO4did not change,but its solubility decreases firstly and then increases at 305–510 K,and maintains a high level.When the temperature is at 510–653 K,the solubility decreases rapidly to a very low level.This phenomenon also occurs in the system of Li2SO4and K2SO4,and the reason should be the change of.Therefore,the expression ofheat capacity is re-optimized to adapt to the sulfate solubility change.The results show (1) the accurate solubility calculation of Na2SO4has been extended to 643.15 K; (2) thermodynamic constants of Na2SO4(at 298.15 K) via extrapolation are in good agreement with the manual data [40]; (3) the coefficients ofionic heat capacity can be applied to the sulfate systems of K2SO4-H2O,Li2SO4-H2O.The results are listed in Table 2,and presented in Fig.17.

Table 9Solubility data of K2SO4-H2O system

Table 10Solubility data of Li 2SO4-Na2SO4-K2SO4-H2O ternary and quaternary systems

Based on the calculated solubilities of Na2SO4,the transition point between Na2SO4·10H2O(s) and Na2SO4(s) at 0.1 MPa is estimated to be 305.55 K,which is consistent with the value of305.55 K reported by Linke[22]and the value of 305.65–306.15 K reported by Seidell [79].

Fig.18.Solubility in the system Li2SO4-Na2SO4-H2O at 298.15 K.

Fig.19.Solubility in the system Li2SO4-Na2SO4-H2O at 373.15 K.

Fig.20.Solubility in the system Li2SO4-K2SO4-H2O at 298.15 K.

Fig.21.Solubility in the system Li2SO4-K2SO4-H2O at 373.15 K.

Fig.22.Solubility in the system Na2SO4-K2SO4-H2O at 298.15 K.

Fig.23.Solubility in the system Na2SO4-K2SO4-H2O at 373.15 K.

Fig.24.Predictedpolythermal phasediagramofLi2SO4-Na2SO4-H2Osystem.‘O’Cavalca [84]andBodaleva [80],‘▼’Skarulis [85],‘Δ’Gou[86],‘j’Khu [46],‘?’Campbell [81],‘–’ Modeling result of this work.

? K2SO4-H2O system

Solubility data of K2SO4-H2O binary system reported by five authors are summarized in Table 9.Three solid species K2SO4(s),K2SO4·H2O(s)and ice are found in the system at entire temperature region from 271.25 K to 630.15 K.K2SO4·H2O occupies a very small temperature range.K2SO4solubility varying with temperature follows a very unusual curve (see Fig.3).Based on the new data ofionic heat capacity and optimized liquid parameters(ΔhIJ,and ΔCp，IJ),the solid constants of K2SO4(s) and K2SO4·H2O(s)are determined,and listed in Table 2.The calculated solubilities are presented in Fig.3,which show the excellent matches with the experimental data at entire temperature range.Furthermore,the minimum eutectic point and transition point between K2SO4·H2O(s) and K2SO4(s) at 0.1 MPa are estimated to be 271.55 K and 280.80 K which are consistent with the literature data.

Fig.25.Predicted complete structure 3d phase diagram of Li2SO4-Na2SO4-H2O system.‘o’sulubility data;‘’P1,P2,P3,P4,P5 are Transition points of three solid co-saturated in the ternary system.

Fig.26.Predicted polythermal phase diagram of Li2SO4-K2SO4-H2O system.‘▼’Druzhinin [90],‘j’ Yanko [91],‘×’ Campbell [81],‘●’ Spielrein [87]; ‘’ Invariable point of this work; ‘–’ Modeling result of this work.

Fig.27.Predicted complete structure 3 d phase diagram of Li2SO4-K2SO4-H2O system: ‘o’ sulubility data; ‘’ Invariable point of this work.

Fig.28.Predicted polythermal phase diagram of Na2SO4-K2SO4-H2O system.‘’Invariable point of this work; ‘–’ Modeling result of this work.

Fig.29.Predicted complete structure 3 d phase diagram of Na2SO4-K2SO4-H2O system: ‘O’ sulubility data; ‘’ Invariable point of this work.

4.3.Ternary phase diagrams and solid–liquid parameters

The solubility data for ternary and quaternary systems are summarized in Table 10.For Li2SO4-Na2SO4-H2O system,11 literature cover the temperature range from 249.75 K to 373.15 K,and 2 double of Li2SO4·Na2SO4and LiNa3(SO4)2·6H2O were found in the system.For Li2SO4-K2SO4-H2O system,8 authors have reported the solubility data covering the temperature range from 252.15 K to 653.15 K,and one double salt of Li2SO4·K2SO4was found.For Na2-SO4-K2SO4-H2O system,we use the classical solubility data [28]covering the temperature range from 270.45 K to 373.15 K,which contain 1 double of Na2SO4·3K2SO4.

The results of ternary double salts constants and liquid parameters are reported in Tables 3 and 2 respectively.The comparisons of the experimental data (Table 10) with the modeling results are selectively presented in the isothermal phase diagrams Figs.18–23,i.e.Li2SO4-Na2SO4-H2O system at 298.15 K and 373.15 K are shown in Figs.18 and 19; Li2SO4-K2SO4-H2O system at 298.15 K and 373.15 K are shown in Figs.20 and 21;Na2SO4-K2SO4-H2O system at 298.15 K and 273.15 K are shown in Figs.22 and 23.From these figures,we can conclude that the simulation results can express the experimental results and evaluate the quality of experimental data.

Table 11Invariant points in the ternary system

5.Phase Diagram Prediction

5.1.Prediction of complete ternary phase diagrams

As we know,it is difficult to determine the multi-temperature invariable points of the ternary systemviaexperimental method.Thus,it is necessary to predict the full structure or complete phasediagramsviamodeling,such as,to predict the univariate curve of two salts co-saturated and the invariable points of three salts cosaturated,and to deduct the polythermal and complete phase diagram of ternary or quaternary systems.

Figs.24–29 show the predicted results of polythermal and complete phase diagram of Li2SO4-Na2SO4-H2O system,Li2SO4-K2SO4-H2O system and Na2SO4-K2SO4-H2O system,respectively.The experimental data of the two salts co-saturated and isothermal data are compared on the diagram.From Figs.24–29,we can conclude that the predicted results are in perfect agreement with the experimental data,and the predicted structures of phase diagram are reasonable and believable.The multi-temperature invariable points of the ternary system are listed in Table 11.

5.2.Prediction of quaternary phase diagram

Fig.30.Predicted phase diagram of system Li2SO4-Na2SO4-K2SO4-H2O at 298.15 K.‘○’Lepeshkov[89],‘Δ’Campbell[94],‘–’Modeling result of this work,‘j’Predicted co-saturated points of this work.

Fig.31.Predicted phase diagram of system Li2SO4-Na2SO4-K2SO4-H2O at 273.15 K.‘○’ Zeng [95],‘–’ Modeling result of this work,‘j’ Predicted co-saturated points of this work.

Fig.32.Predicted phase diagram of system Li2SO4-Na2SO4-K2SO4-H2O at 323.15 K.‘○’ Lepeshkov [89],‘–’ Modeling result of this work,‘j’ Predicted co-saturated points of this work.

Fig.33.Predicted phase diagram of system Li2SO4-Na2SO4-K2SO4-H2O at 373.15 K.‘○’ Lepeshkov [89],‘–’ Modeling result of this work,‘j’ Predicted co-saturated points of this work.

Liquid parameters (ΔgIJ,ΔhIJ,and ΔCp，IJ) for interactions between salt pair and water (τca-w) or salt pair and salt pair(τca-c′a′)in aqueous Li+-Na+-K+-system,have been determined from binary and ternary systems,and the solid constants except one quaternary double salt (2Li2SO4·Na2SO4·K2SO4),have been determined from binary and ternary systems.Therefore,the quaternary phase diagram can be predicted.

Solubility data at 273.15 K,298.15 K,323.15 K and 373.15 K were determined by Lepeshkov [89],Campbell [94]and Zeng[95].We use the solubility data of quaternary double salt(2Li2SO4-·Na2SO4·K2SO4)to determine the double salt constants.The results are reported in Table 2.The comparisons of experimental data and predicted results of isothermal phase diagram at the four temperatures of 298.15 K,273.15 K,323.15 K and 373.15 K are presented in Figs.30–33 which denote the predicted polythermal results of the quaternary are well consistent with the experimental data,except the data [94]at 273.15 K.

6.Conclusions

A improved comprehensive thermodynamic model based on eNRTL is applied for the aqueous Li+-Na+-K+-quaternary and its subsystems cover temperature ranges from eutectic points to 643 K.Liquid parameters of 12 pairs and solid constants of 11 solid species are obtained by regressing available thermodynamic property data from literature and further optimized by fitting solid–liquid phase equilibrium data.

With the obtained parameters,(1) the model accurately represents thermodynamic properties at temperature ranges from 250 K to 633 K;(2)the phase diagrams of binary Li2SO4-H2O,Na2-SO4-H2O and K2SO4-H2O systems,and ternary Li2SO4-Na2SO4-H2O,Li2SO4-K2SO4-H2O and Na2SO4-K2SO4-H2O systems are accurately presented; (3) the complete phase diagrams of three ternary systems and polythermal phase diagram of quaternary system are excellent to be predicted.

Therefore,the results of this study show that the models and parameters have the strong ability to express the thermodynamics of sulfate type solution.The thermodynamic expression of real salt lake brine is expected.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully thank the financial support of the National Natural Science Foundation of China (U1707602,U1407204) and also thank the Yangtze Scholars and Innovative Research Team in University of Education of China,the Innovative Research Team of Tianjin Municipal Education Commission(TD12-5004).