Mathematical modeling of quantitative changes in hydrogen and oxide inclusions in aluminum alloy

. In this article, mathematical modeling of quantitative changes in hydrogen and oxide inclusions in aluminum alloys is justified, developed, and analytically implemented. Usually, the methods of linear algebra are mostly used; in particular, the solutions of systems of inhomogeneous algebraic equations are obtained based on the method of Gauss, Cramer, and inverse matrices using the Maple 13 software package. Quantitative changes in hydrogen and oxide inclusions in the alloy are determined by a change in the average dispersion of the loaded flux. The connectivity functions of the change of oxide aluminum in the alloy (cid:2) (% ) with an increase in temperature Т ( 0 С ) during the loading of the charge into the liquid bath are obtained. The connectivity functions to determine the change quantity of hydrogen (cid:3) (cm 3 /100 g) in the alloy depending on the time t (minute) holding of the heated charge in the period of research is obtained. Based on functional dependencies, graphs of changes in the mainly desired parameters and numerical indexes in tabular form for engineering and applied calculations are constructed. In particular, graphs of the change quantity of hydrogen and oxide inclusions in the alloy with an increase of average dispersion of the flux d, graphs of change quantity of hydrogen with an increase in temperature during loading of the charge into the liquid bath, changes of the quantity of oxide aluminum in the alloy (cid:2) (%) with an increase in temperature Т ( 0 С ) , patterns of change quantity of hydrogen in the alloy (cid:3) (cm 3 /100 g) and quantity of oxide (cid:4) (%) were plotted depending on the time of holding the heated charge in period research .


Introduction
One of the most important areas of engineering materials science is the production (getting, producing) of new effective, and promising alloys for foundry production.At the same time, for the further improvement of modern technologies for processing of materials and details for general mechanical engineering, a powerful lever arm for the development of theoretical foundations and innovative technologies is the correct justification and formulation of the problem of using analytical research, in particular, the development and analytical realization of mathematical models.
At present, the development of resource and energy-saving technologies in the smelting of aluminum alloys is of particular importance.Resource saving in melting aluminum alloys is of particular importance as heat exchange processes in which alloys are saturated with gas and non-metallic inclusions.Connecting with the development of the protective flux to reduce gas and other non-metallic inclusions, optimize the melting process and improve the quality of the obtained casting from aluminum alloys is one of the important tasks of our time.In this area, in many developed countries, such as the USA, Canada, Germany, France, Korea, Japan, Russia, Ukraine, and China, special attention is paid to reducing gas and other non-metallic inclusions in aluminum alloy casting.In general, numerous scientific works of foreign scientists and scientists from Uzbekistan are devoted to research, creation of new technologies or improvement of existing technologies for melting aluminum alloys, development of more effective compositions of protective fluxes used in the melting process of aluminum alloys [1]- [3].
Here is a brief overview of research conducted at many universities and institutes worldwide.In particular, the University of California (USA), Jinan University (China, scientists Min Zuo, Maximilian Sokoluk, Chezheng Cao), scientists from Canada (T.A. Utigard, R.R. Roy and K. Friesen), group of scientists from Great Britain and Italy Annalisa Pola, Marialaura Tocci, Plato Kapranos, scientists from Germany and Harbin University Jean Ducrocq, Szunyan Chan and R. Nabibullah (University of Pakistan) and others [4], [5].
Scientists of the CIS countries S.P. Zadrutsky, G.A. Rumyantsev, B.M. Nemenenok, I.A. Gorbel (National Technical University of the Republic of Belarus), S.V. Voronin, and P.S. Loboda (Samara National Research University named after Academician S.P. Korolev), V.A. Grachev (Penza State Technical University) and others conducted several research work to improve the mechanical properties of aluminum alloys obtained from gas melting aggregates [6], [8].
In the Republic of Uzbekistan, research works are underway to improve the technologies for melting aluminum alloys and protective flux composition for melting aluminum alloys, which helps to improve the quality of the melt.In addition, research was carried out to increase the melting process's efficiency and to use new protective materials and constructions to provide these technologies.For this, it is necessary to increase the priority of ongoing research works on the development of effective composition of protective flux, improving the efficiency of the use of flux during the melting process, which wildly used in aluminum alloy production.Scientists (E.Kh.Tulyaganov, N.Dj.Turakhodjaev and T.Kh.Tursunov, and others) investigated structural changes in casting from aluminum alloys depending on the mode of melt [9]- [11].
Based on an analytical review of world and domestic literature in the above areas was found that in many research of the authors' insufficient attention is paid to issues of the widespread use of mathematical modeling of the process under study.

Methods
In this article, the object of research is the mathematical modeling of quantitative changes in hydrogen and oxide inclusions in the composition of aluminum alloys.As known, the basis of the method of mathematical modeling is algorithmization.It should be emphasized that the word "algorithm" comes from the name of the Central Asian scientist of the 9th century, Muhammad ibn Musa Al-Khorezmi (who was born in the city of Khorezm, Uzbekistan) around 783 800 -about 850).Al-Khorezmi is a great mathematician, astronomer, geographer, and historian.Thanks to Al-Khorezmi, the terms "algorithm" and "algebra" appeared in mathematics.
Modeling of foundry production is carried out on the basis of analytical and numerical methods, particularly methods of finite difference and finite element.To complete a mathematical model, it seems necessary to be able to combine theory and practice to solve engineering problems; choose measuring facilities following the required accuracy and operating conditions; the ability to follow metrological norms and rules, to comply with the requirements of national and international standards; use the physical-mathematical equipment for solving problems; use the basic concepts, laws, and methods of thermodynamics, chemical kinetics, heat, and weight transfer, choose and apply appropriate methods for modeling physical, chemical and technological processes; use information and technologies in solving of problems.
The first step in our previous studies of obtaining promising alloys in foundry technology was to substantiate the problems and prospects for developing mathematical models of heat and mass transfer processes using linear algebra methods using the Lagrange interpolation polynomial.To develop and analytically implement the mathematical model as an object of study, the technologies developed and implemented in industrial production were chosen.In particular, the extraction of metals from liquid slag and increase of the operational properties of cast details from steel 45 [11]- [13].Based on the analytical implementation of mathematical models of the technological process, numerical values are determined, graphs of changes in the desired parameters are plotted, and recommendations are given for industrial production.
In works [14]- [18], technology was developed to determine the flux composition for melting aluminum alloy to reduce the content of gas inclusions in the resulting melts.In particular, the influence of the flux composition, the influence of the technology of loading the flux into the melt, and the mode of melt to gas content in the resulting aluminum alloy are determined.Now let's move on to the development and analytical implementation of a mathematical model for quantitative changes in hydrogen and oxide inclusions in an aluminum alloy.The mathematical model of this process was developed based on experimental data.
The task of the mathematical modeling, on another side, is the analytical determination of experimental data based on the composed mathematical function of the quantitative change in the content of hydrogen and oxide in the alloy and the mathematical method to determine their content without additional experimental research.Determining experimental data using a mathematical function is considered equivalent to the problem of uniquely determining the coefficients of a high degree polynomial [20], [21].It should be denoted, in further, all the tables with data from the experimental research and constructed graphs are given in the third section of the research results and discussion.
Firstly we determine the change in the quantity of hydrogen in the alloy with a change in the average dispersion of the loaded flux.Based on the data in the above table № 1, it is possible to write the followings: The system of algebraic equations is: Thus, the function which characterized the change in the amount of hydrogen in the alloy with a change in the average dispersion of the flux is as follows: Depending on this, it is possible to conclude that obtained results (3) help to unambiguously determine the amount of hydrogen in the alloy with a change in the average dispersion of the flux d.It seems necessary to determine the accuracy level to prove the results' reliability.
Let us now determine the quantitative change in the oxide additives in the alloy with a change in the average flux dispersion.Based on the data in Table 1, it is possible to write the followings: In this case, the system of algebraic equations will have the form: The solution to this system of inhomogeneous algebraic equation by the Gauss or Cramer's method using the Maple13 software package, we obtain the following desired roots of the equation: Based on this analytic solution, it is possible to conclude that it unambiguously determines the quantitative change in the oxide additives in the alloy with a change in the average flux dispersion d.
In further research, developing and implementing a mathematical model of experimental data is of interest.In this case, the task is to determine the decrease in the amount of hydrogen (cm 3 /100g) with an increase in temperature ( 0 С) during the loading of the charge into the liquid bath [22]- [26].
At the same time, it should be noted that in the mathematical modeling of experimental data, the integrity and uniqueness of experimental data are important; in other words, the need to establish the uniqueness of a function to determine the law of change of a parameter depending on another parameter during its natural change is significant.From this point of view, during the developing analytical expressions, the definition of a function with one variable is considered the main task: the development of an unambiguous connection function.The advantage of the developed mathematical function is that it includes not only experimental results but also creates the possibility of determining subsequent experimental data without expensive experiments [25]- [27].
Continuing the above research, we will define the following functions for this case analytically.
1. Determinations of the connectivity function of the decrease in the quantity of hydrogen (cm 3 /100 g) and the increase in temperature ( 0 С) during loading the charge into the liquid bath.
Based on the experimental data, which is given in In this case, the charge temperature is a natural variable.Table 2 gives five temperature values, so the degree of the desired polynomial will equal 4. Depending on this, the system of algebraic equations has the form: In this case, the following connectivity function was obtained to determine the decrease in the quantity of hydrogen (cm 3 /100 g) with an increase in temperature ( 0 С) during the loading of the charge as: Depending on the coefficient, the inhomogeneous system of the algebraic equation has the form: The followings roots of this system of algebraic equations are determined by the inverse matrix method: Depending on the obtained roots of the equations, we obtain the following connectivity function the change in aluminum oxide in the alloy E (%) with an increase in temperature Т ( 0 С) during the loading of the charge into the liquid bath: Depending on the numerical values, the nonhomogeneous system of the algebraic equation has the form: As in the previous case, using the method of an inverse matrix, the following roots of this system of algebraic equations are determined: Thus, the following connectivity function was obtained to determine the change in hydrogen content O (cm 3 /100 g) in the alloy depending on the holding time t (minute) of the heated charge during the research: 4. Now, let's move on to determining the regularity of changes in oxide content K (%)   depending on the holding time (minute) of the heated charge during the research.We present the experimental data following Table 3.
For this case, the form of the nonhomogeneous system of the algebraic equation has the form: As a result, the following function was obtained to unambiguously determine the regularity of changes in oxide content K (%) depending on the holding time (minute) of the heated charge during the research period: In general, the analytical studies make it possible to say that the degrees of connectivity functions in the form of polynomials ( 3), ( 6), ( 9), ( 13), ( 17), ( 21) can be increased based on the numerical values of experimental researches given in tables.It can be seen that in the connectivity functions, due to the coefficient value in front of the highest degree of the polynomial variable, it will be possible to obtain accurate data of 3 4 10 ..10 .This means that the error can be approximately zero, allowing a high degree of accuracy.In addition, based on the obtained connectivity function dependencies, using preliminary tabular data, it seems possible to obtain and other results without experimental research.
In concluding this section, it is possible to note that it seems useful to use integral equations for further research.In particular, for mathematical modeling of the heat transfer process during gas or electro-arc melting of aluminum alloys for each alloy layer, the law of conservation of energy can be written as the following integral equation: .
After some transformations, when passing to the limit case , Fqv y t is a function of the density of heat sources, which characterized the change in the energy influx in each internal layer of the alloy.
By integrating the differential equation ( 22) under exact initial and boundary conditions, it is possible to mathematically estimate the heat transfer process during the melting of aluminum alloys.

Results and Discussion
This section presents tables and graphs of changes in the main desired parameters based on the developed mathematical functional given in the previous section.In particular, table 1 gives the results of the experimental data, which determined the degree of dependence of hydrogen and oxide content in the alloy formed during the melting of the charge, depending on the average dispersion of the loaded flux.4. based on functional dependencies, graphs of changes in the main desired parameters and numerical values in tabular form for engineering and applied calculations are constructed.In particular, graphs of the change in the content of hydrogen and oxide inclusions in the alloy with an increase in the average dispersion of the flux d, graphs of the change in hydrogen amount with an increase in temperature during loading of the charge into the liquid bath, the change of aluminum oxide amount in the alloy E (%) with an increase in temperature Т ( 0 С), the regularity of change in hydrogen content in the alloy O (cm 3 /100 g) and percentage of oxide K (%) depending on the holding time (minute) of the heated charge during the research period are plotted.

Таble 1 .Fig. 1 .Таble 2 .Fig. 2 .Таble 3 .Fig. 3 .Fig. 4 . 2022 Fig. 5 .Fig. 6 .
Fig. 1.The graph of the change in the amount of hydrogen in the alloy with an increase in the average dispersion of the flux d.The following table of experimental data: Таble 2. Changes in the amount of hydrogen with increasing temperature during loading of the charge into the liquid bath The temperature of the charge during the loading its into the liquid bath, 0 С

3 .
Obtained the connectivity function of the change in aluminum oxide in the alloy E (%) with an increase in temperature Т ( 0 С) during the loading of the charge into the liquid bath; the connectivity function is obtained to determine the change in hydrogen content O (cm 3 /100 g) in the alloy depending on the holding time t (minute) of the heated charge during the research period.

Table 2
For this case, the roots of the system of algebraic equations are determined by the Jordan-Gauss method and get: