APPLICATION OF MATHEMATICAL MODELING FOR THE SYNTHESIS OF NEW ALLOYS

ABSTRACT

It has been determined that the creation of mathematical models of alloy synthesis processes is supported for the production of promising new alloys in the foundry industry.

АННОТАЦИЯ

Было определено, что создание математических моделей процессов синтеза сплавов поддерживается для получения перспективных новых сплавов в литейном производстве.

Keywords: mathematical modeling, synthesis and application, new alloy.

Ключевые слова: математическое моделирование, синтез и применение, новый сплав.

Scientists of Uzbekistan  and Tajikistan  have developed technologies for the high-temperature processing of industrial slags [1] and wastes [2], separation processing in electric furnaces [3] and a technology for extracting metals from liquid slag [1]. Let's give some examples [4, 5]. In addition, studies have been carried out on the anodic behavior and oxidation of the Zn22Al alloy doped with scandium, yttrium and erbium [6–8]. At the same time, reliable protection against the corrosive effects of the agents in which they operate is necessary. In the practice of protecting semi-finished steel products from corrosion, zinc-aluminum coatings such as “galfan” (Zn5Al, Zn55Al) [9–11] Zn22Al [12–14] and “galvalum” (Zn55Al-1.6Si) are currently used in various aggressive environments.

To develop a new alloy, it seems necessary to select an appropriate method of mathematical modeling while taking into account the technological process [15, 16]. Below are the tasks for the development and analytical implementation of the mathematical model.

In numerical programming problems with a wide range of applied problems, the extremum of a function (objective function) is determined which borders on a system of linear equations and inequalities. The mathematical model of the general linear programming problem has the form:

                        (1)

In other words, this task is reduced to the task of qualitative assessment of economic, technical technological processes and determination of optimal solutions. If problem (1) has solutions, then the objective function reaches an extremum.

We explain this problem with the following problem. Suppose a metallurgical plant produces three types of alloys from raw materials A and B. The percentages of raw materials in the alloys are presented in the table below. In total, the plant has 7 tons of raw materials of type A and 17 tons of raw materials of type B. A ton of the first alloy costs 1.5 million conventional units, the second - 1.6 million conventional units, the third - 1.2 million conventional units. In this case, the need for the third alloy should not exceed 12 tons, and the need for the first alloy should not be less than 5 tons. The question is, how many tons of each alloy should be produced to achieve maximum economic efficiency?

Table 1.

The percentages of raw materials in the alloys

To solve the problem, it is necessary to develop a mathematical model. In accordance with the general model (1), we introduce the target function:

                     (2)

The conditions set in the task require the implementation of the following inequalities:

                                        (3)

To solve problems (2) and (3), the MathCAD method will build the following sequence of operations:

                                     (4)

Then we have

                                              (5)

                                 (6)

Now we consider the issue of assessing the change in the percentage of oxygen in the alloy by changing the amount of aluminum in the electrode using mathematical modeling.

We determine the percentage of oxygen in the alloy corresponding to a change in the amount of aluminum in the electrode depending on the diameter of the crystallizer. According to general mathematical theories, this problem is solved using the regression model. From the point of view of linear algebra, the functional connection is determined by the solution of the system of linear algebraic equations. If we denote the interpolation in the form of a function, then it is necessary to solve the system of equations:

                                                     (7)

Thus, we can conclude that when the first alloy is generated, 6.15 tons of the second and 12 tons of the third alloy will achieve the greatest economic effect (31,892 million conditional units). From synthesized new alloys, machine parts are made.

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