Estimation of the Gibbs Free Energy for Calcium Aluminate Sulphate Hydrates Using Iterative Least Squares Regression Method

The iterative least squares method was used for the estimation of the Gibbs free energy △G of calcium aluminate sulphate hydrates the lime desulfurization product in alumina production. By means of the tree diagram theory, the optimal selection of exponential power Xi in the regression equation is realized. The recursive least square method was used to solve the regression equation, and a regression method for estimating the Gibbs free energy of calcium aluminate sulphate hydrates was obtained. The Gibbs free energies of two kinds of calcium aluminate sulphate hydrates at 343K were estimated by this method. The experimental results were consistent with the estimated Gibbs free energies. It shows that the estimated method is correct and the result is reliable. The Gibbs free energy of calcium aluminate sulphate hydrates at any temperature can be estimated by this means. The study will provide theoretical and experimental basis for the lime desulfurization and synthesis of calcium aluminate sulphate hydrates by process of alumina production.


Introduction
Calcium aluminate sulphate hydrates is an important compound in the production of alumina.
In the process of alumina production by sintering process, desilication is an essential process in the process of production. Improving the desilication depth of sodium aluminate solution plays a key role in improving the quality of alumina products and the decomposition rate of sodium aluminate solution. Calcium aluminate sulphate hydrates is the most effective desilication agent for calcium compounds. Under the optimum conditions, the desilication of sodium aluminate solution by sintering method was carried out, and the silicon content index of the refining solution was up to 12500. The synthetic conditions and properties of calcium aluminate sulphate hydrates are helpful to the study of desilication mechanism and the improvement of desilication efficiency, thus affecting the quality and yield of alumina produced by sintering process [1][2][3][4][5]. Therefore, it is important to study and optimize the synthesis thermodynamics of calcium aluminate sulphate hydrates for alumina production by sintering process.
The Bayer process of producing alumina, Sulfur pollution is very harmful Sulfur in bauxite mainly exists in the form of pyrite (in China). The pyrite reacts with sodium aluminate and caustic into the solution in the forms of sulfide, thiosulfate and sulfite, and finally transforms into sodium sulphate (Na 2 SO 4 ) [6][7][8].so that the alkali consumption is increased; the accumulation of sulfur in the production process reduces the decomposition rate of seed. Sulfide and thiosulfate aggravate the corrosion of steel equipment, resulting in higher concentrations of soluble iron in solution and polluting aluminum hydroxide and so on, therefore, it is important to remove sulfur compounds from the process to control Na 2 SO 4 content in a harmless range. It is of great significance to reduce the 2 1234567890''""
It was noted that the desulfurization efficiency is three times higher when calcium aluminate trisulfate hydrate is generated that than when calcium aluminate monosulfate hydrate is produced, leading to a reduction in the loss of Al 2 O 3 .
In order to study and optimize the synthesis of calcium aluminate sulphate hydrates it is necessary to know the Gibbs free energy generated by calcium aluminate sulphate hydrates However, the thermodynamic data such as Gibbs free energy of calcium aluminate sulphate hydrates are rarely reported in the literature, which makes it difficult to further study the related problems.
For this reason, we will study the thermodynamic conditions for the formation of calcium aluminate sulphate hydrates. A method for estimating the free energy of calcium aluminate sulphate hydrates Gibbs is proposed. The free energy of Gibbs and the free energy of reaction are calculated to predict the reaction results. The experimental results show that the calculated results are in agreement with the experimental results, The existence and characterization of calcium aluminate trisulfate hydrates 3Ca·Al 2 O 3 ·3CaSO 4 ·nH 2 O (Ca 6 Al 2 SO 18 ·nH 2 O) were determined, which provided the basis for the thermodynamic study of the synthesis and desulfurization of calcium aluminate sulphate hydrates.

Estimation of gibbs free energy using iterative least squares regression method
For the determination of Gibbs free energy of Complex oxysalt, the commonly used methods are experimental determination methods (such as calorimetry, hydrofluoric acid dissolution method, etc.) and empirical theory calculation method For the complex compounds such as calcium aluminate sulphate hydrates, the determination of Gibbs free energy is very difficult, complex and timeconsuming.
Karpov et al., [16] Nriagu, [17] Chen [18] and Tardyand et al. [19] Proposed empirical formulas for estimating the thermodynamic data of some minerals. It is generally believed that the "regression method", which is first applied to silicate minerals by karpov et al., and further improved by Chen, is a widely used and accurate method.
With the help of regression equation such as Chen estimated standard formation Gibbs free energy of 20 kinds of silicate minerals by using complex multiple regression extrapolations to the limit, and the value is compared with the experiment, the relative error is 1%. However so far, all literatures using iterative least squares regression method are those to estimate the standard Gibbs free energy of formation of silicate mineral. Due to the complex silicate is various oxide compound [20,21] [22], the Gibbs free energy of calcium aluminate sulfate hydrates is estimated by iterative least squares method.

IWMSE2018
IOP Publishing IOP Conf. Series: Materials Science and Engineering 381 (2018) 012099 doi:10.1088/1757-899X/381/1/012099 343K(70°C), but they failed to obtain an exact evidence of generation of calcium aluminate trisulfate hydrates (3CaO·Al 2 O 3 ·3CaSO 4 ·nH 2 O), and did not give the thermodynamic basis of both existing. So, this paper will use the above method to estimate the Gibbs free energies of the two calcium aluminate sulphate hydrates at the temperature, in order to determine whether they can generate and react in sodium aluminate solution. Because of lack of the corresponding thermodynamic data published as reference. The validity and reliability of the estimation will be experimentally tested here. G o value of each oxide under standard condition (298K, 25°C) can be found from the thermodynamic data manual. From this, G o of the calcium aluminate sulphate hydrates under standard condition can be calculated. For the estimation of G values of calcium aluminate sulphate hydrates under other temperatures, changing G o value of each oxide under standard condition into the value under the temperature. The thermodynamic data under certain temperature can be found from the thermodynamic data manual or got from calculation by the classical thermodynamic formula, Table 1 lists G o of thermodynamic data of each oxide under standard conditions [23 ] and the thermodynamic data of corresponding substance under 343K (70°C) [24].  The above method, for each ∑ΔG fi , selected an integer value x i , the x i is called the exponent of ∑ΔG fi . In this way, a form such as a list of the following functions is constituted: The y i =∑ΔG fi , noting y i as observation value of ∑ΔG fi , then fitting curve for the listed functions according to exponential equation, thus a best fitting curve can be obtained. The equation is: Where: a, b and c are for the undetermined coefficients; c stands for the ΔG f of the mineral.

The selection of the exponent xi
In equation (2), exponent x i corresponding ∑ΔG fi is selected artificially. How to select x i is the key of the regression method. In order to eliminate arbitrariness, according to the nature of exponential curve, the selection of x i should combine with tree diagram theory [25] . Figure 1 is that when the mineral combination form number p=4, the combination tree diagram of the corresponding exponent x i . In the figure from the straight branch from top 0 down along multi groups of x i value can be obtained. It can be known from figure 1 that the combination forms of a group of ∑ΔG fi corresponding x i are 0012, 0013, 0023, 0024, 0123, 0124, … Figure 1. When p=4, a tree diagram of possible combination form corresponding x.

The iterative least squares method to solve regression equation
In order to determine the values of a, b and c in the best fitting curve equation (2), firstly it was assumed that a0, b0 and c0 were the initial values, a1, b1 and c1 were the correction values of a0, b0 and c0 respectively, that is, a =a0 +a1, b =b0 +b1 and c =c0 +b1. Thus, The difference between the observation value and the best fitting curve value a exp(b x i )+c was R, there were Substituting equation (3)  can be calculated. According to this method, the values of the least squares fitting curve and its corresponding can be calculated for each x combination form. Then selecting the combination of minimum value in as a best combination, and its corresponding fitting curve is just the best fitting curve. The c value is the Gibbs free energy △G of the compound.
This paper adopts MATLAB programming to calculate [27] (Program omitting). The computational procedure is as Figure 2

Estimation of the gibbs free energies for two kinds of calcium aluminate sulphate hydrates
The values of the Gibbs free energies calculated by the tree diagram corresponding x i combination for the fitted two compounds, the results are in table 4.    (10) Using the above estimated Gibbs free energies, the directions of reactions (8) -(10) at 343K (70°C) were determined. When calculating the reaction equations (8) and (9), the thermodynamic data for each substance 343K (70°C) are shown in table 5.   (7), let n=31 calculate G 7 , and substitute G 7 into reaction (10), △G 10 was calculated .result also is -95.54 kJ/mol All the Gibbs free energies calculated for the desulfurization of a sodium aluminate solution were negative (△G 8 =-104.96 kJ/mol, △G 9,10 =-95.54 kJ/mol). These results show that the reactions to form the calcium aluminate monosulfate and trisulfate hydrates are favorable, i.e., two types of calcium aluminate sulfate hydrate should be produced in the sodium aluminate solution. It is impossible to compare and judge whether the thermodynamic calculation mentioned above is reliable because there are no related reports. We use experiments to verify it.

Experimental
A sodium aluminate solution was prepared in a high-pressure steel bomb using puriss sodium hydroxide and aluminum hydroxide. Puriss calcium hydroxide was placed in a crucible and calcined at 7 1234567890''""
The sodium aluminate solution was prepared with a molar ratio of 3:1 and then diluted with 4 times as much distilled water. The diluted sodium aluminate solution (500 mL) was placed in a 1 L stainless steel cup, and the Na 2 SO 4 reagent was then added to it. The experiments were performed in a constant temperature water bath (the temperature was automatically controlled with precision of ±1°C under , stirring by an agitator at a constant speed. A certain amount of calcium oxide was dissolved in water to make lime milk, which was slowly added to the sodium aluminate solution (24 g/L Na 2 SO 4 ) under stirring for 50 or 120 min at 70°C. The solution was decompressed and filtered after the reaction, and the filter cake quality was determined. The cakes were washed repeatedly with water and dried with cold wind. The reaction products were analyzed by chemical methods, X-ray diffraction and infrared spectroscopy.

Results and Discussion
As shown in the XRD spectrum of the products in Figure 3, both the calcium aluminate monosulfate and trisulfate Hydrates (Ca 4 Al 2 SO 10 ·12H 2 O and Ca 6 Al 2 (SO 4 ) 3 (OH) 12 respectively) were formed when the lime content was 17 g/L, the reaction temperature was 70°C, and the reaction time was 50 min. Notably, the content of these two compounds reached 96%.
The diffraction peaks of calcium aluminate monosulfate hydrate (3CaO·Al 2 O 3 ·CaSO 4 ·12H 2 O) were observed in the XRD pattern of the desulfurization product ( Figure 3) and had characteristic d spacings (characteristic values) of 0.890, 0.445, 0.399, 0.287, 0.242, and 0.223 nm. The diffraction peaks of calcium aluminate trisulfate hydrate [Ca 6 Al 2 (SO 4 ) 3 (OH) 12 ] were also observed and had d spacings of 0.980, 0.561, 0.554, 0.491, 0.388, 0.279, and 0.256 nm. The diffraction pattern was important for identifying the compound products, especially calcium aluminate trisulfate hydrate, which was not identified in previous work. These experimental results are consistent with the fact that the calculated △G o values were significantly less than 0, meaning that the formation of both calcium aluminate sulfate hydrates is favorable and they both should be produced in the sodium aluminate solution. Thus, the estimated Gibbs free energies of the calcium aluminate sulfate hydrates is consistent with the experimental results.
The absence of calcium aluminate trisulfate hydrate (Ca 6 Al 2 (SO 4 ) 3 (OH) 12 ) in previous experiments performed under the same conditions can be explained by the △G calculations and the experimental results presented here. Figure 4 shows the results of an experiment performed under the same conditions as the experiment in Figure 3, except the reaction time was 120 min. When this longer reaction time was employed, the reaction products were only calcium aluminate monosulfate hydrate and Ca 3 Al 2 (OH) 12 ; CaSO 4 and calcium aluminate trisulfate hydrate (Ca 6 Al 2 (SO 4 ) 3 (OH) 12  Subtracting reaction (9) from reaction (8) (7) and (8) The results demonstrating this reaction process are shown in Figures 3 and 4. The presence of calcium aluminate (Ca 3 Al 2 (OH) 12 ) was observed in Figure 4 because calcium aluminate monosulfate hydrate (3CaO·Al 2 O 3 ·CaSO 4 ·12H 2 O) is also a metastable material and decomposes into calcium aluminate and calcium sulfate after prolonged heating [8][9][10][11], shown as by reaction (12) 12 , respectively were formed.
3) Adding lime to a sodium aluminate solution to remove sulfur can lead to the formation of calcium aluminate trisulfate hydrate (3CaO·Al 2 O 3 ·3CaSO 4 ·6H 2 O). The diffraction peaks in the XRD pattern of calcium aluminate trisulfate hydrate were also observed and had d spacings of 0.980, 0.561, 0.554, 0.491, 0.388, 0.279, and 0.256 nm. This is an important characterization.

Acknowledgement
This research was financially supported by the National Science Foundation of China (Ggrant No. 21366009)