PredIctIon of Al2o3 leAchIng recovery In the BAyer Process usIng stAtIstIcAl multIlIneAr regresIon AnAlysIs

This paper presents the results of defining the mathematical model which describes the dependence of leaching degree of Al2O3 in bauxite from the most influential input parameters in industrial conditions of conducting the leaching process in the Bayer technology of alumina production. Mathematical model is defined using the stepwise MLRA method, with R2 = 0.764 and significant statistical reliability – VIF<2 and p<0.05, on the one-year statistical sample. Validation of the acquired model was performed using the data from the following year, collected from the process conducted under industrial conditions, rendering the same statistical reliability, with R2 =


Introduction
Bayer's process of alumina extraction from bauxite ore is the dominant process for obtaining alumina for more than 100 years.Today, more than 90% of alumina production in the world is achieved by this process, despite the fact that alternative processes have been developed in the meantime.[1,2] Bayer process involves the leaching of bauxite with concentrated sodium aluminate solution at temperatures between 100°C and 250°C, depending on the mineralogical form of the aluminum in bauxite.Trihydrate bauxite type -gibbsite can be dissolved in caustic solution in the temperature range DOI: 10.2298/JMMB1002161D 100-180°C.Monohydrate bauxite forms (boehmite and diaspore) are dissolved in the temperature ranges: 130-180°C and 200-250°C, respectively [3].This process includes reactions with soluble silica compounds and titanium dioxide under certain conditions.[4] Process parameters influencing the leaching rate and the degree of Al 2 O 3 recovery are: mineralogical and chemical composition of the bauxite, granulation size distribution, caustic modules of the starting solution and its Na 2 O (caustic) content, temperature of the leaching process, stirring speed and duration of the process [5].The process of bauxite leaching, under industrial conditions of the Bayer technology for alumina production, is highly complex.The ability to predict the recovery of Al 2 O 3 during leaching, as the result of modeling the input parameters of the process, presents a large advantage for management of the process [4].
This paper presents the modeling of the leaching process of bauxite based on the results obtained under the industrial conditions, in order to predict the leaching degree of Al 2 O 3 (process output) depending on the parameters of the process (process input).Obtained model presents a great advantage based on its ability to predict the accurate output of the investigated process.

experimental data
The database used in this work was created on the basis of data collected from the industrial production in the alumina factory Birač in Zvornik (Bosnia and Herzegovina), whose capacity is 600,000 t of  Using the formula (1) to calculate the leaching degree of Al 2 O 3 by the use of "inert'' Fe 2 O 3 provides satisfactory results in industrial practice, with an accuracy of over 99%, from this reason it is used in this study.
For the modeling of the dependence of Al 2 O 3 leaching degree from bauxite -Y (process output) using established database, following technological parameters of the leaching process (process inputs) were used: X 1 -concentration of Na 2 O k in the leaching solution (g/l) X 2 -caustic ratio of the solution at the beginning of the leaching process X 3 -the moisture content of the bauxite (%) X 4 -the Al 2 O 3 content in the bauxite (%) X 5 -the SiO 2 content in the bauxite (%) X 6 -the Fe 2 O 3 content in the bauxite (%) X 7 -the TiO 2 content in the bauxite (%) X 8 -the CaO content in the bauxite (%) X 9 -the loss on calcination (%) and X 10 -caustic ratio of the solution at the end of the leaching process.
During the studied period, the database was generated from data measured during the industrial production in the factory Birač in Zvornik, during the stabile operation of the factory.At that time, as well as in the future period, processed bauxite mainly originated from the Vlasenica ore deposit (Bosnia and Herzegovina) which is of the boehmite type.Leaching temperature throughout the whole period was constant and equal to 245 o C, with the pressure in autoclaves "reactors" kept to 35 bar.Granulation of the bauxite in all cases after hydrocyclonic classification was 100% -74 μm, and the S : L ratio was 1 : 5. Rotational speed of the mechanical stirrers in the autoclaves was 31 rpm.Table 1 presents the experimental results in the form of descriptive statistics that were used for statistical modeling of process output-input dependence, i.e.Y = f (X 1 -X 10 ).A set of 300 data relates to the operation of the factory in 2008.
The values of standard deviations in Table 1 show that there is a normal Gaussian distribution with certain parameters.On this basis it can be concluded that this dataset is suitable for statistical analysis with multiple linear regression analysis (MLRA) technique.[5,6] If the satisfactory degree of fitting is not achieved then the methods of nonlinear regression analysis (NLRA) are to be used, including methods of artificial neural networks (ANNs).[7,8,9]

statistical analysis
Forced-entry method was used to rank the influences of all predictors X 1 -X 10 on the dependent variable Y, which in principle gives the possibility to define MLR dependence Y = f(X 1 -X 10 ), Table 2.The results indicate that the values of the variance inflation factor (VIF) are greater than two, indicating the existence of a larger share of collinearity among certain predictors.Also, the values of statistical significance (p>0.05)indicate that the Forced -entry method, in this case, does not provide statistically valid results.
Since the Forced-entry method did not give a satisfactory result based on the fact that it takes into consideration the impact of all predictors, it is useful to eliminate the impact of those predictors whose effect on output -Y is negligible.This requires the use of a stepwise regression analysis method, which solves the problem of collinearity by basing the order of the predictors entry on a mathematical criterion of ranking the individual predictors significance.[9] Acquired results of the stepwise method are shown in Table 3.Now the values of VIF are less than two, and the statistical significance is p<0.05,indicating a satisfactory statistical reliability of the results.
Redundant predictors were eliminated through the four step iteration, and the dependence of output -Y on the most important predictors: X 3 , X 5 , X 6 and X 10 is defined with the following equation: Since the value of R 2 = 0.560 is determined by the distribution of errors above 4σ, Fig. 2.a, in the further statistical analysis of this empirical data set, reduction of the statistical sample was carried out by removing the extremes exceeding the 4σ boundaries.After ten iterations, 55 elements were discarded and the initial set was reduced to 245 elements with the distribution error within 4σ boundaries, and undisturbed statistical reliability Fig. 2.b.With the reduced statistical sample of 245 elements, within the error a limit of 4σ, MLRA was performed using the stepwise method and the results are presented in Table 4.
Obtained results presented in Table 4 show statistical correctness in all cases because VIF<2 and p<0.05.It should be noted that the number of predictors in this case, compared to the previous (sample of 300 elements), increased from 4 to 6 with an increase in R 2 from 0.560 to 0.764.MLRA resulted with the following form of linear dependence of Y = f (X i ): Y = 92.847-0.587X 2 + 0.154X 3 -1.521X 5 -0.424X 6 -0.518.X 8 + 6. Validation of the mathematical model given by the equation (3) was carried out through its application on data obtained in regular production during the following year (2009), using the statistical dataset of 330 measurements.The results of defined model

Y
Fig.1shows the comparison between the Y-measured and the Y-calculated values, with the coefficient of determination R 2 = 0.560.Since the value of R 2 = 0.560 is determined by the distribution of errors above 4σ, Fig.2.a, in the further statistical analysis of this empirical data set, reduction of the statistical sample was carried out by

Figure 1 -
Figure 1 -Linear regression analysis of the Al 2 O 3 recovery based on equation (2) for the statistical sample of 300 elements

Fig. 3 .Fig. 3 .
Fig.3.shows the dependence between the measured and calculated values for Y, given by the equation (3), for reduced statistical sample of 245 elements.Model defined this way, given by the equation (3), has a satisfactory value of

Fig. 4 .
Fig.4.Validation of the model (3) on a sample of 330 measurements in regular industrial production during 2009.

Table 1 .
Descriptive statistics for the input (X 1 -X 10 ) and the output (Y) values of the bauxite leaching process in industrial process conditions for a set of 300 samples for 2008.

Table 2 .
MLRA values for a set of 300 samples, using the Forced-entry method

Table 3 .
MLRA values for a set of 300 samples, using the Stepwise method