Estimating solubilities of ternary water-salt systems using simulated annealing algorithm based generalized regression neural network

Fluid Phase Equilibria 505 (2020) 112357

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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d

Estimating solubilities of ternary water-salt systems using simulated annealing algorithm based generalized regression neural network Xianze Meng a, Yunpeng Fu a, Junsheng Yuan b, c, * a

Key Laboratory of Marine Chemistry Theory and Technology, Ministry of Education, College of Chemistry and Chemical Engineering, Ocean University of China, Qingdao, 266100, China b School of Chemical Engineering, Hebei University of Technology, Tianjin, 300130, China c School of Chemical Engineering and Materials, Quanzhou Normal University, Quanzhou, 362000, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 May 2019 Received in revised form 23 September 2019 Accepted 7 October 2019 Available online 10 October 2019

Water-salt system phase diagrams are the theoretical guidance for the effective separation of high salinity wastewater. However, the published solubilities are numerous but unsystematized. In this paper, a mathematical model for estimating the stable equilibrium solubilities of ternary water-salt systems with a large temperature range (273.15e423.15 K) was established by the generalized regression neural network (GRNN) coupled with simulated annealing (SA) algorithm. The principle of constructing the database was to select saturation points in the validation set. The Griewank benchmark function was utilized to verify the reliability of the SA-GRNN model. The equilibrium solubilities of various ternary water-salt systems were predicted by SA-GRNN and compared with experimental data. The results show that the optimized spread parameters are from 0.0001 to 0.0570. Besides, the root-mean-square error (RMSE) and coefficient of determination (R2) for each ternary system are from 0.0001 to 0.0022 and 0.9958 to 0.9999, respectively, reflecting the good generalization and robustness of SA-GRNN model. NaCleKCleH2O system was selected to compare the reliability of SA-GRNN with the classical electrolyte modeldPitzer theory, since NaCleKCleH2O system holds the lowest prediction accuracy by SA-GRNN among the studied ternary in this work. It is found that the RD values of SA-GRNN forecast data (0.00 e0.14%) are much lower than those of the Pitzer model calculated data (0.05e5.11%). The results show that SA-GRNN has better accuracy and stability. The proposed model can successfully predict the solu2  bilities of ternary water-salt subsystems in Naþ, Kþ, Ca2þ, Mg2þ//Cl, SO2 4 , CO3 , HCO3 e H2O system with high accuracy. As an example of application, the unpublished solubility data for an intricate ternary water-salt system of Naþ, Mg2þ//SO2 4 e H2O containing diversified crystalline hydrate and saturation points at 358.15 K were predicted. The prediction reflects the applicability of the model to estimating the solubilities of complex ternary systems. © 2019 Published by Elsevier B.V.

Keywords: Solubility Ternary water-salt system Simulated annealing Generalized regression neural networks Spread parameter

1. Introduction Experimental and theoretical studies on phase equilibrium of water-salt systems are helpful to guide the separation of salt resource and utilization of high salinity wastewater in industrial process such as salt chemical industry [1,2], environmental protection [3,4], coal chemical industry [5] and even material industry [6]. Since the 1900s, researchers have compiled experimental data handbooks on seawater system and its subsystems composed of

* Corresponding author. School of Chemical Engineering, Hebei University of Technology, Tianjin, 300130, China. E-mail address: [email protected] (J. Yuan). https://doi.org/10.1016/j.fluid.2019.112357 0378-3812/© 2019 Published by Elsevier B.V.

Naþ, Kþ, Mg2þ, Ca2þ//Cl, SO2 4 - H2O [7,8]. Compared with the experimental methods, simulation methods are time-saving and labor-saving. Since the 1920s, many researchers have tried to predict the solubilities of water-salt systems with various calculation models from the perspective of thermodynamic phase equilibrium. Debye and Hückel [9] proposed a theory as the theoretical explanation of mean activity coefficients for a single strong electrolyte dilute solution (concentration less than 0.1 mol/kg). In 1973, Pitzer proposed a semi-empirical model based on Debye-Hückel theory, establishing a mathematical relationship among activity coefficient, permeability coefficient and excess Gibbs free energy. In the 1980s, Harvie, Moller and Weare put forward the HMW model [10], expanding the application of Pizter theory. Lu and Maurer [11] raised an activity

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coefficient model for the water-salt system on the basis of ion hydration theory. In 1995, perturbation theory [12] was established based on statistical mechanics and molecular mechanics theory. However, the complex process of computation and various parameters make the perturbation theory difficult to be popularized. The improvement of computer technology and the attention of international academia are promoting the research and application of artificial neural networks (ANNs). ANNs are not only mathematical models for parallel computing that imitating biological neural networks, but also rapidly developing machine learning approaches. They have been extensively used in solving engineering problems, such as linear regression, pattern recognition, and signal processing. As shown in Fig. 1, the artificial individual neurons imitate the structure and mechanism of biological neuron to deal with information.

2. Data acquisition Aiming at constructing an accurate and reliable model to estimate the ternary water-salt system solubilities at different temperatures, as much phase equilibrium experimental data as possible were gathered for training neural networks. The phase equilibrium data about ternary water-salt subsystems in Naþ, Kþ, Ca2þ, Mg2þ// 2  Cl, SO2 4 , CO3 , HCO3 - H2O system with the temperature ranging from 273.15 to 423.15 K were collected from published literatures [7,8]. The aggregate of experimental data set containing electrolyte composition and temperature range are given in Table 1.

3. Estimating model 3.1. GRNN model

u¼

n X

wi xi :

(1)

y ¼ f ðu þ bÞ:

(2)

i¼1

where xi are input signals; wi are weight values; u is the output value of linear combiner; b is the bias value for u; f(~) is the activation function; y is the output signal for the individual artificial neuron. At first, the neural network initializes a set of network weights and regards the supervisory signal as the expected output. Then, the errors between the actual signal and supervisory signal reflect the network weight updates. This adjustment will continue unless the error exceeds the expected accuracy. Thus, the training of the neural network is transformed into the algebraic problem of finding the minimum value of a function. In this work, we regard the phase diagram for ternary water-salt 2  subsystems in Naþ, Kþ, Ca2þ, Mg2þ//Cl, SO2 4 , CO3 , HCO3 - H2O system as a data set of electrolyte solubilities in aqueous solution under a certain temperature condition. Many types of ANNs models are sensitive to factors such as transfer function, the number of hidden layers, and activation function. Accuracy of the generalized regression neural network (GRNN) model proposed by Specht [13] only depends on spread parameter, thus avoiding the effects of subjective factors on the prediction results. However, manually adjusting the spread parameter is sometimes difficult to balance the prediction accuracy and data smoothness. To solve this problem, spread parameter is considered as the internal energy and optimized with simulated annealing (SA) algorithm [14]. The optimal SA-GRNN network model is determined at the minimum energy state so that the solubility data can be accurately predicted.

As shown in Fig. 2, GRNN consists of four layers: an input layer, a pattern layer (hidden layer), a summation layer and an output layer [13]. The pattern unit (p1, p2, …, pn) is consist with radial basis function (RBF) neurons which can be mapped directly to the hidden space without weight connection. The neuron number in each input unit is equal to the number of input samples. The summation layer includes two units, the numerator (Sn) unit, and the denominator (Sd) unit. Sn equals to the sum of RBF neurons vector values, while Sd equals to the arithmetic sum of the RBF neurons. The estimated value (Y) equals to Sn divided by Sd. Besides, when the mapping relationship has determined, the prediction results are theoretically constant. The algebraic basis of GRNN model is nonlinear regression analysis. Supposing f(X, y) denotes the joint probability density function of vector random variable X and scalar random variable y. The estimated value Y can be calculated by following Eq. (3):

 ∞ ð

∞ ð

Y ¼ E½yjX ¼

yf ðX; yÞdy ∞

(3)

∞

The estimated density function b f ðX; yÞ using nonparametric estimator proposed by Parzen is shown as follows [15]:

b f ðX; yÞ ¼

n X

   T   X  Xi 2s2 , exp  X  X i

i¼1

h  . i exp  y  yi 2s2

, nð2pÞðpþ1Þ=2 sðpþ1Þ

(4)

where Xi, yi represent sample values of vector random variable X and scalar random variable y respectively; p corresponds to the dimension of X; n is the number of observed samples and s is the width coefficient, that is, the spread parameter. Substituting Eq. (4) into Eq. (3), the estimated value can be represented as follows:

Y¼

n X i¼1 ∞ ð

   T   X  Xi 2s2 exp  X  X i . i h  2s2 dy y exp  y  yi

∞

 X  Xi

,

n X

  T exp  X  X i

i¼1



2s2

 ∞ ð ∞

Fig. 1. Structure of artificial neuron model.

f ðX; yÞdy:

. i h  2s2 dy exp  y  yi

(5)

X. Meng et al. / Fluid Phase Equilibria 505 (2020) 112357

3

Table 1 Descriptions of chemical samples used in this research [7,8]. No.

System

Electrolytes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

NaeK e Cl NaeCl e SO4 NaeCa e Cl NaeMg e Cl KeCa e Cl KeMg e Cl MgeCa e Cl NaeMg e SO4 KeCl e SO4 KeMg e SO4 MgeCl e SO4 NaeHCO3 e CO3 NaeSO4 e HCO3 NaeSO4 e CO3 NaeCl e HCO3 NaeCl e CO3

NaCl NaCl NaCl NaCl KCl KCl MgCl2 Na2SO4 KCl K2SO4 MgCl2 NaHCO3 Na2SO4 Na2SO4 NaCl NaCl

KCl Na2SO4 CaCl2 MgCl2 CaCl2 MgCl2 CaCl2 MgSO4 K2SO4 MgSO4 MgSO4 Na2CO3 NaHCO3 Na2CO3 NaHCO3 Na2CO3

Temperature range/K

Numbers of data points

273.15e423.15 273.15e423.15 273.15e398.15 273.15e423.15 273.15e368.15 273.15e423.15 273.15e383.15 273.15e373.15 273.15e373.15 273.15e373.15 273.15e373.15 273.15e348.15 273.15e318.15 273.15e373.15 273.15e323.15 273.15e373.15

134 124 119 106 109 159 94 121 173 115 118 89 68 107 80 99

Fig. 3. Operation flowchart of SA-GRNN algorithm. Fig. 2. GRNN model structure.

3.2. SA-GRNN model As mentioned above, the performance of GRNN network is only based on the selection of spread parameter, while intelligent optimization algorithm is the key method to obtain the optimal spread parameter. In 1983, Kirkpatrick [16] first used SA algorithm to resolve combinatorial optimization problem. The SA algorithm is easy to implement. Moreover, its search strategy avoids falling into the local optimal solution in operation, making the result more reliable than traditional optimization algorithm. In this work, the root-mean-square error (RMSE) between the estimated value of test sample and experimental value is considered as the fitness value of SA algorithm:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,ffi u n  2 uX t xie  xip yerror ¼ n

(6)

i¼1

where n is the sample number of validation set, xe corresponds to the experimental value, while xp is the predicted value for each sample in a validation set. The computational flow of SA algorithm is indicated by Fig. 3. Step 1: Data normalization. Converting all the temperatures and solubilities to 0e1 to balance the effects of the magnitude differences on estimating results. Step 2: Train GRNN with the samples from training set. For a

ternary water-salt system, the target electrolyte concentration is linked with the system temperature (Tws) and the equilibrium electrolyte concentration. Accordingly, the concentration Csalt1 can be estimated from the input layer with the other two parameters (Tws and C salt2):

Csalt1 ¼ f ðTws ; Csalt2 Þ

(7)

Step 3: Obtain the simulated value of validation set by trained GRNN. Step 4: Set initial temperature (Tsa), iteration times, initial solution (Xspread) and target accuracy for the SA algorithm. To obtain the optimal solution, high temperature and sufficient iteration times are necessary. The initial temperature is 1000 K, the iteration times is 500, and the target precision is 1E-9. Step 5: Establishing a new solution. Step 6: Calculating the increment of evaluation function.





Dy ¼ yerror spread0  yerror ðspreadÞ

(8)

Step 7: According to the Metropolis criterion, if Dy < 0, the spread will be updated with spread0 , otherwise, the current solution will be accepted with the probability of expð  Dy =Tsa Þ. If the increment is lower than the target accuracy, the current solution will be considered as the optimal solution. Step 8: Tsa decreases and gradually approaches zero. Repeating steps 5 to 7.

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4. Results and discussion 4.1. Performance test of model Benchmark function is a key method [17] to verify the feasibility of numerical optimization algorithms. Griewank benchmark function [18] with complex peak-valley relationship is a classical test function for objective optimization. The analytical expression and function graphic are shown in Eq. (9) and Fig. 4(a), respectively.

f ðxÞ ¼

d X i¼1

x2i

.

4000 

d Y

 . pffi  cos xi i þ1

(9)

i¼1

As shown in Fig. 4(b), one thousand groups of data are randomly selected as input samples, additionally, the corresponding target values are regarded as the output samples within the scope of definition: x2½ 5; 5 and y2½  5; 5. The selected data points are evenly distributed in the definition domain, which commendably represents the entire data. Sixty percent of the samples are regarded as the training set, the other twenty percent is collected to the validation set, while the remaining twenty percent (200 groups) are considered as the testing set. After network training with Griewank benchmark function, the optimal spread parameters of SA-GRNN and GRNN are 0.2637 and 0.0000 on a global scale, with RMSE values of 0.0036 and 0.0100, respectively. The prediction results for the two neural network models are presented in Fig. 5. Based on these results, in contrast to GRNN model, SA-GRNN model exhibits better generalization ability and higher accuracy. Moreover, comparing the spread parameters obtained from the two models, the coupled SA algorithm obtains the best value in global scope; while the GRNN modeling lacks optimization ability, showing poor accuracy. In summary, the modeling results on the basis of Griewank benchmark function show the effectiveness and robustness of SA-GRNN. Therefore, SA-GRNN algorithm has been proved to be a reliable and effective method for predicting twodimensional data sets and suitable for the prediction of the system involved in this paper.

Fig. 5. Prediction results for training set samples. (“True” are values computed by Griewank benchmark function, “SA-GRNN” are values estimated by SA-GRNN model, “GRNN” are values estimated by GRNN.)

of one electrolyte, the other two factors are the input parameters that affect the concentration value. Thus, the solution process can be regarded as a two-dimensional geometric optimization problem. The spread parameter of each system is the only parameter that needs to be optimized for neural network modeling. The SA-GRNN model proposed in this paper is applied to forecast stable equilibrium solubility. By comparing with the experimental values, the RMSE of predicted values can be considered as the objective function of the optimization algorithm. Besides, RMSE is also known as the performance function, which is utilized to evaluate the fitting degree of the network after training. Coefficient of determination (R2), as shown in Eq. (10), is commonly used in the regression model to evaluate the coincidence degree between predicted and actual values.

4.2. Simulation of stable equilibrium solubility Three columns of data are listed in stable equilibrium solubility data of ternary water-salt systems: the ambient temperature and the concentration of two electrolytes. In terms of the concentration

R2 ¼ 1  RSS=TSS ¼ 1 

X ðyi  fi Þ2 i

, X

ðyi  yÞ

2

(10)

i

where RSS is the residual sum of squares; TSS is the total sum of

Fig. 4. (a) Graphic of Griewank benchmark function; (b) Distribution of selected data points.

X. Meng et al. / Fluid Phase Equilibria 505 (2020) 112357

squares; y corresponds to the actual value; y represents the mean of the actual value; f is the predicted value. Fig. 6 shows the comparison of estimating data with experimental data for total points. Although the predicted values of some data points are well consistent with the experimental values, many data still have larger deviations. The spread parameter changes for each system are influenced by the number and interval of data points. These changes only represent the smoothness of the fitting surface, whereas, they are irrelevant with the prediction performance of the network. Importantly, the performance function of the network is RMSE, which indicates the degree of deviation between the predicted value and the actual value within the validation set. As shown in Table 2, the spread parameters are between 0.0001 and 0.0863, which are well optimized. The RMSE and R2 of the random data set modeling method of the ternary systems are scattered over the range of [0.0005, 0.0062] and [0.9354, 0.9912], respectively. Therefore, optimizing the component of data sets rather than just following the stochastic principle may have a beneficial impact on prediction performance. To better optimize the distribution of data sets, all the endpoints and two-phase saturation points should be incorporated into the validation set. If the number of saturated samples accounts for less than 20% of the total points, the goal could be reached by randomly adding single-phase saturation points into the validation set. After retraining the SA-

5

Table 2 Optimized spread parameters. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

random data set network parameters

optimized data set network parameters

spread

RMSE

R2

spread

RMSE

R2

0.0015 0.0029 0.0012 0.0005 0.0051 0.0002 0.0031 0.0010 0.0863 0.0003 0.0001 0.0008 0.0064 0.0017 0.0005 0.0032

0.0008 0.0013 0.0023 0.0027 0.0024 0.0005 0.0041 0.0027 0.0050 0.0044 0.0037 0.0062 0.0045 0.0037 0.0051 0.0056

0.9912 0.9870 0.9843 0.9740 0.9899 0.9938 0.9611 0.9670 0.9563 0.9552 0.9645 0.9640 0.9354 0.9598 0.9694 0.9417

0.0570 0.0006 0.0003 0.0044 0.0001 0.0024 0.0364 0.0003 0.0033 0.0002 0.0005 0.0005 0.0019 0.0103 0.0001 0.0006

0.0022 0.0001 0.0003 0.0001 0.0004 0.0001 0.0004 0.0003 0.0001 0.0012 0.0002 0.0002 0.0001 0.0006 0.0007 0.0004

0.9958 0.9993 0.9996 0.9998 0.9994 0.9997 0.9990 0.9995 0.9997 0.9969 0.9995 0.9996 0.9998 0.9985 0.9997 0.9991

GRNN, the results of regression analysis for each ternary watersalt system are illustrated in Fig. 7 and Table 2, respectively. As shown in Table 3, the spread parameters are between 0.0001 and 0.0570, which are well optimized. The optimized RMSE between

Fig. 6. Regression analyses for systems with random distribution data set: KCleNaCleH2O (a); Na2SO4eNaCleH2O (b); CaCl2eNaCleH2O (c); MgCl2eNaCleH2O (d); CaCl2eKCleH2O (e); MgCl2eKCleH2O (f); CaCl2eMgCl2eH2O (g); MgSO4eNa2SO4eH2O (h); K2SO4eKCleH2O (i); MgSO4eK2SO4eH2O (j); MgSO4eMgCl2eH2O (k); Na2CO4eNaHCO3eH2O (l); NaHCO3eNa2SO4eH2O (m); Na2CO3eNa2SO4eH2O (n); NaHCO3eNaCleH2O (o); Na2CO3eNaCleH2O (p). AX is the former electrolyte and AY/BX is the latter electrolyte for each system.

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Fig. 7. Regression analysis for ternary water-salt systems with optimized spread parameters: KCleNaCleH2O (a); Na2SO4eNaCleH2O (b); CaCl2eNaCleH2O (c); MgCl2eNaCleH2O (d); CaCl2eKCleH2O (e); MgCl2eKCleH2O (f); CaCl2eMgCl2eH2O (g); MgSO4eNa2SO4eH2O (h); K2SO4eKCleH2O (i); MgSO4eK2SO4eH2O (j); MgSO4eMgCl2eH2O (k); Na2CO4eNaHCO3eH2O (l); NaHCO3eNa2SO4eH2O (m); Na2CO3eNa2SO4eH2O (n); NaHCO3eNaCleH2O (o); Na2CO3eNaCleH2O (p). AX is the former electrolyte and AY/BX is the latter electrolyte for each system.

Table 3 Estimated solubility of ternary water-salt system Naþ, Mg2þ//SO2 4 - H2O. System

Spread parameter

T/K

Saturated concentrations of electrolytes (wt. %) Na2SO4

MgSO4

Naþ, Mg2þ//SO42 - H2O

0.0653

358.15

30.1 28.5 26.9 25.6 24.1 22.2 19.0 16.0 13.6 10.4 7.2 5.6 4.0 2.8 1.2 0.0

0.0 2.7 5.4 7.2 10.4 12.2 15.2 18.1 20.6 24.9 29.3 31.4 33.3 34.5 35.6 36.0

a

Equilibrium solid phasea

Na2SO4 Na2SO4 Na2SO4 Na2SO4 þ Van Van Van Van þ Low Low Low Low Low Low Low þ M1 M1 M1 M1

Van, 3Na2SO4$MgSO4; Low, Na2SO4$MgSO4$2.5H2O; M1, MgSO4$H2O.

0.0001 and 0.0022 has proved the superior network performance of the retraining network under the newly allocated data sets. The optimized R2 for each system is generally above 0.9900, revealing

the good approximation of experimental values with estimating values by the regression analysis. The optimized R2 of Naþ, Kþ//Cl H2O system is 0.9958, which is the lowest in all the ternary systems.

X. Meng et al. / Fluid Phase Equilibria 505 (2020) 112357

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With fixed ambient temperature, whether estimating the KCl concentration using the NaCl concentration or vice versa, the estimated values of most data points fit well with the experimental values. Similar situations occurred in other systems. Furthermore, the overall deviation is significantly reduced, and the fitting degree is greatly improved. 4.3. Comparison with the Pitzer model The reliable Pitzer theory has been widely used since it was proposed in 1970s. It is a semi-empirical thermodynamic model, which links excess Gibbs free energy with electrolyte activity coefficient and solution osmotic coefficient. The derivation process of Pitzer theory is described in our previous studies [4,19] and can obtain the solubility of equilibrium systems. Referring to the KCleNaCleH2O system with minimum coefficients of determination (R2 ¼ 0.9958), the equilibrium solubilities at 298.15 K, 323.15 K and 378.15 K were calculated using the Pitzer model and SA-GRNN model and compared with the literature values. Relative deviation (RD) between experimental and calculation value is listed in Table S1 (Supporting Information) and Fig. 8. The results of Pitzer theory in blue have a maximum RD of 5.11%, while the results of SAGRNN model in orange are more accurate with maximum RD of 1.49%. Fig. 8 shows that SA-GRNN model has advantages in accuracy and stability of estimating solubilities. 4.4. New solubilities prediction We attempt to predict solubilities of various temperatures that have not been published in previous studies with the newly constructed model. The modeling steps follow the approach described above. The solubilities in literatures help to establish ANNs and the internal logic between the electrolyte solubility and temperature. The solubilities of related electrolytes are collected as the validation sets to optimize the spread parameters, while the solubilities of all ternary subsystems are regarded as the training sets to construct neural networks. A simple solubility estimation method is proposed, which will help to guide the research of salt separation and zero discharge processes for high salinity wastewater. This method avoids traditional complicated experiment and classical thermodynamic models with many physical and empirical parameters, thus greatly reducing time and labor. As an example of application, the estimated solubility for an intricate ternary water-salt system of Naþ, Mg2þ//SO2 4 - H2O containing diversified crystalline hydrate and saturation points at 358.15 K is listed in Table 3. Also, liquidus

Fig. 9. Liquidus for ternary water-salt system Naþ, Mg2þ//SO2 - H2O at 348.15 K, 4 358.15 K and 373.15 K

for the system at 348.15 K and 373.15 K along with estimated liquidus at 358.15K are shown in Fig. 9. The estimated liquidus lies in the middle of the experimental liquidus, meanwhile, the good smoothness of the estimated liquidus reflects the credibility of the prediction results. It is noteworthy that the regression performance of SA-GRNN model is excellent in the range of published temperature for the seawater ternary subsystems studied in this paper, but its temperature extrapolation ability is insufficient. However, the model is still of great value in the prediction of unknown phase equilibrium solubility because the temperatures and the data points that published are sparsely discrete, which is a common phenomenon exist in ternary water-salt systems. For example, the solubility of the ternary system Naþ, Mg2þ//Cl- H2O at 253.15 K, 263.15 K, 273.15 K, 298.15 K, 323.15 K, 348.15 K, 373.15 K, 398.15 K and 423.15 K have been published in previous literatures. But large gaps of 25 K still exist between different temperatures for the above system. Besides, only six published data points exist in the system of Naþ, Kþ//Cl H2O at 283.15 K, the same as the system at 313.15 K. The phenomenon of sparse temperature and data points are common problems for water-salt systems. The optimum temperature and maximum salting-out in the separation process require stepless phase equilibrium data. The SA-GRNN model can well solve the problem of sparse temperature and data points in the temperature range mentioned in Table 1, while maintaining the accuracy of predicted results. 5. Conclusion

Fig. 8. Solubilities relative deviation using Pitzer theory and SA-GRNN model for ternary water-salt system Naþ、Kþ//Cl-H2O at 298.15 K, 323.15 K and 373.15 K.

A model for predicting equilibrium solubilities of ternary systems with a large temperature range (273.15e423.15 K) was established by SA-GRNN algorithm. The reliability of the model was validated by Griewank benchmark function. Then, the stable equilibrium solubilities of some ternary water-salt subsystems were predicted and compared with the experimental data. In addition, spread parameters for modeling were obtained by the optimized algorithm. The RMSE and R2 involved in each ternary system were in the range of [0.0001, 0.0022] and [0.9958, 0.9999] respectively, reflecting the good accuracy and robustness of the

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X. Meng et al. / Fluid Phase Equilibria 505 (2020) 112357

constructed SA-GRNN model. Moreover, compared with Pitzer theory, SA-GRNN model has good accuracy and stability. In summary, the proposed model was able to successfully predict the solubilities for ternary water-salt subsystems in Naþ, Kþ, Ca2þ, 2  Mg2þ//Cl, SO2 4 , CO3 , HCO3 - H2O system with extreme accuracy. Finally, the unpublished solubility data for an intricate ternary water-salt system of Naþ, Mg2þ//SO2 4 - H2O containing diversified crystalline hydrate and saturation points at 358.15 K was predicted. The ANN model will be conducive to the research of salt separation and zero discharge processes for high salinity wastewater. Funding sources This work was supported by the National Key Technologies Research and Development Program [2016YFB0600504]; Chinese Postdoctoral Science Foundation [2017M611142]; Natural Science Foundation of Hebei Province [B2017202246]; Science and Technology Planning Project of Fujian Province [2017Y4014] and Program for Changjiang Scholars and Innovative Research Team in University [IRT14R14]. We thank Syed Faizan Ali Shah for his linguistic assistance during the revision of this manuscript. Declaration of competing interest The authors declare no competing financial interest. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2019.112357. References [1] S. Tursunbadalov, L. Soliev, Phase equilibria in multicomponent water-salt systems, J. Chem. Eng. Data 61 (7) (2016) 2209e2220, https://doi.org/ 10.1021/acs.jced.5b00875. [2] S. Tursunbadalov, L. Soliev, Investigation of phase equilibria in quinary watersalt systems, J. Chem. Eng. Data 63 (3) (2018) 598e612, https://doi.org/

10.1021/acs.jced.7b00799. [3] M. Pannach, S. Bette, D. Freyer, Solubility equilibria in the system Mg(OH)2 MgCl2 - H2O from 298 to 393 K, J. Chem. Eng. Data 62 (4) (2017) 1384e1396, https://doi.org/10.1021/acs.jced.6b00928. [4] X. Meng, X. Wu, J. Yuan, Solid-liquid equilibrium of the quaternary system Ca2þ, Mg2þ//SO32e, SO2e 4 -H2O at 298.15 K, J. Chem. Eng. Data 63 (12) (2018) 4339e4349, https://doi.org/10.1021/acs.jced.8b00385. [5] H. Lu, J. Wang, J. Yu, Y. Wu, T. Wang, Y. Bao, D. Ma, H. Hao, Phase equilibria for the pseudo-ternary system (NaClþNa2SO4þH2O) of coal gasification waste water at T ¼ (268.15 to 373.15) K, Chin, J. Chem. Eng. 25 (7) (2017) 955e962, https://doi.org/10.1016/j.cjche.2016.08.016. [6] J. Maous, M. Kaddami, Experimental solid-liquid equilibria of the ternary system Co(NO3)2þMn(NO3)2þH2O at 20  C, 0  C and -15  C, Fluid Phase Equilib. 489 (2019) 16e22, https://doi.org/10.1016/j.fluid.2019.01.006. [7] H. Stephen, T. Stephen, Solubilities of Inorganic and Organic Compounds, vol. 3, Pergamon Press, Oxford, 1963. [8] T. Deng, H. Zhou, X. Chen, Salt-water System Phase Diagrams and Applications, Chemical Industry Press, Beijing, 2013, pp. 293e308. [9] P. Debye, E. Hückel, The theory of electrolytes. I. Lowering of freezing point and related phenomena, Phys. Z. 24 (1923) 185e206. [10] C. Harvie, N. Moller, J. Weare, The prediction of mineral solubilities in natural waters: the Na-K-Mg-Ca-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O system to high ionic strengths at 25  C, Geochem. Cosmochim. Acta 48 (1984) 723e751, https://doi.org/10.1016/0016-7037(84)90098-X. [11] X. Lu, G. Maurer, Model for describing activity coefficients in mixed electrolyte aqueous solutions, AIChE J. 39 (9) (1993) 1527e1538, https://doi.org/10.1002/ aic.690390912. [12] J. Wu, J. Lu, Y. Li, Application of the new perturbation method to predict densities of single electrolyte aqueous solutions, Fluid Phase Equilib. 107 (1) (1995) 45e59, https://doi.org/10.1016/0378-3812(94)02670-V. [13] D.F. Specht, A general regression neural network, IEEE Trans. Neural Netw. 2 (6) (1991) 568e576, https://doi.org/10.1109/72.97934. [14] X.S. Yang, Nature-Inspired Optimization Algorithms, Elsevier, London, 2014, pp. 67e75. [15] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Stat. 33 (3) (1962) 1065e1076, https://doi.org/10.1214/aoms/ 1177704472. [16] S. Kirkpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671e680, https://doi.org/10.1016/B978-0-08-0515816.50059-3. [17] G. Kong, L. Zhang, Generalized regression neural network modeling method based on genetic algorithm, Comput. Eng. Des. 38 (2) (2017) 488e493, https:// doi.org/10.16208/j.issn1000-7024.2017.02.039. [18] A. Griewank, Generalized descent for global optimization, J. Optim. Theory Appl. 34 (1) (1981) 11e39, https://doi.org/10.1007/BF00933356. [19] D. Li, J. Yuan, F. Li, S. Li, Solid-liquid stable equilibrium of the quaternary system NaCl-KCl-CaCl2-H2O at 303.15 K, J. Chem. Eng. Data 60 (6) (2015) 1886e1891, https://doi.org/10.1021/acs.jced.5b00109.