Mathematical modeling of a three-component system to investigate the formation and characteristics of glass

. The article investigates glass formation and its properties in a three-component system using mathematical modeling. The authors have applied the method of mathematical planning to increase the efficiency of research and to establish the dependences of glass properties on the composition. The authors propose a mathematical model of structure relaxation and two-dimensional stress field in grade glass products, considering the temperature dependencies of glass viscosity of a given chemical composition.


Introduction
Modern industries pay more and more attention to the use of automated data processing systems.Modern technological processes require considerable material and time expenditures.Many productions cannot be without the prior creation of process models.An example of such products can be glass melting.This area has a lot of competition and other problems of unsatisfactory design of production technology.
The choice of insufficient parameters of technological process, such as charge composition, temperature, and pressure of glass melting, caused tons of glass not satisfying the output parameters to be melted down.This leads to equipment downtime and huge losses.
Theoretical-experimental methods are an assumed structure or development of a theory with subsequent verification by experiment, refinement, and determination of necessary parameters.
Conducting experiments often requires considerable financial expenses and a long time.To prepare for experiments, it is necessary to use a mathematical approach aimed at minimizing the number of experiments without loss of quality and reliability of the results [1,2].
The solution to this problem is the mathematical planning of experiments and the application of specific practical tasks.

Methodology
This work optimizes the initial raw material components to determine the optimal composition of glass, with a minimum amount of research, and installation of physical and chemical properties of glasses depending on the composition of the initial components.To solve the problems, we use methods of control systems analysis, heat exchange theory, temperature stress theory, mathematical modeling, software development, algorithm theory, and numerical optimization.
We have developed a mathematical model of structure relaxation and two-dimensional stress field in grade glass products that considers temperature dependences of viscosity and elongation for a glass of certain chemical composition.

Results
We investigated the area of glass formation in the three-component system loess loam -ashslag-flotation waste of the fluorite concentrator [3,4,5].Table 1 presents the chemical composition of the initial raw materials.The complicated theoretical and practical question of research of glass technology (for effective solution of a concrete technical problem) is the development of composition with the set properties.
Such cases require consideration of the influence of each component on the physicochemical properties of the system under study.Such an approach, however, requires many melting of glass samples.
To reduce the number of experiments in the search for optimal compositions, it is advisable to apply the method of mathematical planning of the experiment [6,7].
Based on the experimental data, we calculated the coefficients of the regression equations to test their adequacy using the following ratios (% hereafter, mass content): 1. 40 loess loam ⨯ 40 ash-and-slag ⨯ 20 flotation waste of the fluorite concentrator.2. 30 loess loam ⨯ 40 ash-and-slag ⨯ 30 flotation waste of the fluorite concentrator.3. 30 loess loam ⨯ 30 ash-and-slag ⨯ 30 flotation waste of the fluorite concentrator.The adequacy of the regression equations was checked for each test point by Student's test.Then we compared the values got with the table value of the similar criterion at a significance level of 0.05.We considered the equation to be adequate if the calculated value of the Student's t-test was less than its tabulated value.
Table 2 contains coefficients of the regression equations of the glass properties.According to the regression values, we set the properties of glasses.

Discussion
The analysis of the data "Composition -properties" showed regular changes in the properties of glasses depending on the composition (Table 2).The glasses containing more flotation waste from the fluorite concentrator have lower chemical stability (from 0.76 to 1.61mg/dm) when treated with 6n.HCl) and a high coefficient of linear thermal expansion (55.42 -76.25 10 deg -1).

Conclusion
Increasing the proportion of ash and slag leads to a decrease in density and microhardness, while the chemical stability increases.
The results using mathematical modeling correlate with the experimental data.The application of the simplex lattice planning method allows us to calculate without error the most optimal compositions with predetermined properties and, as a result, to reduce the number of experimental studies.

Fig. 1 .
Fig. 1.Indexing of fourth-order experimental points for glasses of a three-component system.Glass melt had a dark brown, black color depending on the composition of the glass.The mathematical modeling regulated the properties depending on the content of components.The resulting models allow us to say about the properties of glasses limited by experiments in the composition of loess loam -ash-slag -flotation waste of the fluorite concentrator.We investigated the local area of the diagram as a triangle (Fig.1.).Х1 = 90, Х2 = 0, Х3= 10, Х1 = 30, Х2 = 30, Х3 = 40 и Х1 = 30, Х2 = 0, Х3 = 70.Regression equations of the second and incomplete third orders proved to be inadequate.We chose a fourth-order polynomial, considering the compositional nature of the simplex lattice plans.The regression equation for the study of the properties Ŷ = β1z1 + β2z2 + β3z3 + β12z1z2 + β13z1z3 + β23z2z3 + ν12z1z2(z1-z2) + ν13z1z3(z1-z3) + + ν23z2z3(z2-z3) + δ12z1z2(z1-z2) 2 + δ13z1z3(z1-z3) 2 + δ23z2z3(z2-z3) 2 + β1123z1 2 z2z3 + + β1223z1z2 2 z3, β1233z1z2z32  where Ŷ -calculated property; β, ν, δthe coefficients of the regression equation.Based on the experimental data, we calculated the coefficients of the regression equations to test their adequacy using the following ratios (% hereafter, mass content):1.40 loess loam ⨯ 40 ash-and-slag ⨯ 20 flotation waste of the fluorite concentrator.2. 30 loess loam ⨯ 40 ash-and-slag ⨯ 30 flotation waste of the fluorite concentrator.3. 30 loess loam ⨯ 30 ash-and-slag ⨯ 30 flotation waste of the fluorite concentrator.The adequacy of the regression equations was checked for each test point by Student's test.Then we compared the values got with the table value of the similar criterion at a significance level of 0.05.We considered the equation to be adequate if the calculated value of the Student's t-test was less than its tabulated value.Table2contains coefficients of the regression equations of the glass properties.According to the regression values, we set the properties of glasses.

Table 1 .
The chemical composition of the initial raw materials.

Table 2 .
Regular changes in the properties of glasses depend on the composition.