# Simulation of Thermal Fields and Formation of Drops at Welding of Microsystems

ABSTRACT

The study is devoted to numerical modeling of the electrode melting process under the influence of an electric arc on it in the process of microwelding. The mathematical model of thermal conductivity in the electrode, taking into account the assumptions made, is applicable to all parts of the electrode. The values of the voltage and the strength of the welding current, which determine the value of the heat flux from the electric arc to the electrode, varied over a wide range. The researches allowed establishing the basic laws of heating and melting of the electrode.

## INTRODUCTION

When creating micromechanical products, various types of welding are often used, including electric arc welding, called micro-welding. Micro welding is characterized by small diameters of the electrodes used and low welding current forces. It should also be noted that micro-welding has a high degree of automation.

Micro-welding is a complex phenomenon and involves various physical and chemical processes. Understanding these processes allows you to create reliable microsystems with the required parameters. In this regard, mathematical modeling of the entire set of processes accompanying micro-welding is highly relevant and is used to study various welding technologies and solve current technological issues.

There are several main problems:calculation of thermal fields, the study of the heating of the electrode and the welded elements of the microsystem, the melting of the electrode, the drop formation processes, the formation of a micro weld, the cooling of the system, and the formation of residual stresses.

The change in temperature fields during welding was studied in Refs. [1, 2] by the finite element method.The joint evolution of temperature fields and the stress-strain state of the welding area were modeled in the three-dimensional setting in Ref. .Numerical simulation of the drop formation processes during electric welding in paper  was performed. The transient thermo-mechanical (coupled) analysis of temperature and residual stress distributions on welded plates was performed in Ref. . The aim of this work is the complex modeling of micro-welding processes within the framework of the concept of multi-level mathematical modeling of nanosystems .

This chapter presents the results of the first part of this task-simulation of thermal fields and drop formation processes at micro-welding on macro-level modeling.

## FORMULATION OF THE PROBLEM

The design of most welds involves a gap between the parts to be welded and, as a consequence, the volume of the gap must be filled with additional metal. The source of metal can be either a filler material or an electrode, such electrodes are classified as melting, in both cases, it is usually a wire.

The task will be considered on the example of a melting electrode. In the process of welding, an electric arc is established between the parts to be welded and the electrode, which heats both the part and the electrode.

In general, the process of melting the electrode during welding can be divided into two stages:

• 1. When the electrode is heated, the moment comes when the temperature in the electrode reaches the melting temperature (TJ, and the internal energy (H) stored in it becomes equal to the heat of fusion (AHJ. When two of these conditions are fulfilled, the process of melting (drop formation) begins in the contact zone of the electrode with the electric arc, while the temperature in the electrode continues to increase until the temperature in it reaches the boiling point (Jb); and
• 2. When the boiling point (Tb) in the electrode is reached and the internal energy accumulates in it equal to the heat of boiling (ДЯЬ), the material from which the electrode is made boils, which causes it to evaporate. At the moment of the beginning of the boiling of the electrode material, the dimensions of the melted portion of the electrode, diops, will no longer change. Denote the increment of the internal energy - AH.

When modeling the electrode melting process, we introduce the following assumptions:

• • There is no heat exchange between the protective gas and the electrode;
• • Thermal expansion of the electrode material is absent;
• • The heat flux generated by the electric arc is evenly distributed over the end of the electrode;
• • In the molten part of the electrode, the Reynolds number tends to zero (ReO);
• • The calculation is earned out until the boiling of the electrode material.

The assumption of a uniform distribution of the heat flux over the end of the electrode follows from the fact that the electrodes used in welding micromechanical products have a small diameter.

Taking into account the assumptions about the absence of heat exchange between the electrode and the protective gas and the uniform distribution of the heat flux over the end of the electrode, the process of thermal conductivity in the electrode itself can be described in a one-dimensional formulation.

The process of heating the electrode is described by the following mathematical model of thermal conductivity: where с, p, к, у are the specific heat capacity, density, theimal conductivity, electrical resistivity of the electrode material, respectively; T is the temperature of the electrode material; r is time; .r is the spatial coordinate; j is the density of the electric current flowing through the electrode; mm, Щ, are the specific mass rate of melting and boiling, respectively; /x is the distance from the end of the electrode to the coordinate where there is no influence from the heating of the electrode by the electric arc.

The assumption that the Reynolds number in the molten portion of the electrode tends to zero leads to the fact that the process of convective heat transfer in the molten portion as a whole does not affect the process of heat conduction in the electrode.

The boundary and initial conditions, taking into account the assumptions introduced, will be determined by the following relations: where q is the heat flux generated by the welding arc; r, - the moment of the beginning of the boiling of the electrode material. The processes of melting and boiling the material of the electrode were simulated in accordance with the following dependencies : where Дг is the time step; Ym, Yb, are the mass fiactions of the electrode material, melted, and boiled, respectively.

Low-carbon steel is considered as an electrode material. In this case, in accordance with [8, 9] and considering the fact that the parameters of low-carbon steels are close to iron , the thermophysical properties of the electrode material are taken equal to:

• • = 7770 kg/m3;£ = 44,4 J/(nv sec ■ K);
• • c = 557 J/(kg K); Tm = 1800 K;
• AHm = 247100 J/kgT Гь = 3145 K;
• • AHb = 6267123 J/kg; у = 1.510"4 * * 7Ohm m.

The current density is determined by the following relationship: where / is the current strength of the welding arc; S is the cross-sectional area of the electrode; d is the diameter of the electrode.

The heat flux uniformly distributed over the end of the electrode is defined as : where // - the efficiency of conversion of electric power of the welding arc into heat; X - concentration ratio of the welding arc; U is the arc voltage.

The type of electrode used is metal, in which case  the efficiency is assumed to be // = 0.7. The diameter of the electrode is assumed to be d = 2.5 ■ 10'4 m, in this case, 1= 6.4 • 107 nr2. We will accept the following ranges of welding arc parameters:

• Untin = 30 V - the minimum voltage of the welding arc;
• Umax= 60 V - maximum voltage of the welding arc;
• Imin = 0.225 A - minimum current of the welding arc;
• Imax = 0.625 A - the maximum current of the welding arc.

Table 4.1 shows the values of the heat flux of the welding arc, calculated from the relation (4.5) for the four combinations of voltage and the current strength of the welding arc.

TABLE 4.1 Heat Flow Welding Arc

 Heat Flow of Welding Arc     9,б2б-107 1,925-10s 2,674-10s 5,348-10s

The problem of heat conduction (4.1) with initial and boundary conditions (4.2) is solved using the method of control volumes [12, 13].

The discrete analog of the mathematical model (4.1) is implemented in an explicit scheme; as a result, the time step is limited from above: where Ax is the smallest size of the control volume in the entire computa- tional domain. The calculation is carried out with the following parameters of the calculated area: - the size of the control volume is: Ax = 10“3 m.

The control volume with the smallest size is “half,” that is: From Eq. (7) it follows that Ar =510-6 m.

Substituting the value of Axmm in (4.6), we determine that the time step should not exceed: Ax<1.218 • 10-6 sec. The time step is taken equal to At = 10"6 sec. The size of the computational domain determined by the value / is assumed to be equal, / = 15 • 103 m.

oo A oc

## RESULTS OF MODELING

A graphical representation of the simulation results is shown in Figures 4.1—4.4. FIGURE 4.1 Temperature profiles along the length of the electrode at different values of the heat flux welding arc: 1 - q = 9,626-107, 2-q= 1,925-10s, 3 - q = 2,674 10s, 4 - q = 5,348 10s, J/fnfsec)

Graphs in Figure 4.1 show that the temperature gradient in the electrode is proportional to the heat flux entering the end of the electrode, therefore, as the heat flux decreases, the depth by which the electrode warms up increases, and at q = 9,626- 107J/(m2-sec)it reaches the order value 7 mm.

The size of the electrode melt zone varies inversely with the heat flux from 0.84 nun at q = 9.626 - 107 J/(m2sec)to 0.18 mm at q = 5.348 • 10s J/(m2-sec).

This is shown in Figure 4.2. In addition, in the place of the electrode where the temperature reaches the value of 1800 K, the temperature gradient changes, which is associated with the expenditure of energy on the accumulation of heat of fusion in the material. FIGURE 4.2 Temperature profiles in the zone of molten electrode material at different values of the heat flux welding arc:l -q = 9,626-107, 2 -q = 1,925 -10s, 3 -q = 2,674 -10s, 4 -cq = 5,348- 10s, J/(m2-sec).

As can be seen from the graphs of Figure 4.3, when the temperature reaches 1800 К and 3145 К on the surface of the end of the electrode, the temperature stops rising. Graphically, this is displayed as horizontal sections, until the heat of melting and boiling is accumulated, respectively. It also follows from the graphs that, with a constant heat flux from the moment the melting point reaches the electrode material, the rate of temperature rises decreases. FIGURE 4.3 The temperature change on the surface of the end of the electrode at

different values of the heat flux welding arc:l - q = 9,626-107, 2-q= 1,925- 10s, 3 - q =

2,674-10s, 4-q = 5,348-10s, J/(m:sec).

The graphs shown in Figure 4.4, characterize the change in the heating tune of the electrode from the magnitude of the heat flux. FIGURE 4.4 Effects of heat flux on the heating time of the electrode: 1 - total heating tune before the boiling of the electrode material; 2 - tune of formation of the electrode material.

If the curve 1 in Figure 4.4 shows the total heating time on the surface of the electrode end from 293 К (initial temperature) until the boiling point of the electrode material begins, then curve 2 represents the tune of drop formation. This is the time from the moment of melting to the beginning of the boiling of the electrode material. From the graphs in Figure 4.4, it also follows that the dependence of the boiling time of the electrode material and the time of its drop formation on the heat flux generated by the welding arc is hyperbolic in nature.

The numerical values of time spent on boiling and dropping of the electrode material are given in Table 4.2    Wanning up time before the boiling of the electrode material, sec 0.15640 0.0418 0.0226 0.0062 Droplet tune of the electrode material, sec 0.122 0.033 0.018 0.005 Tlie proportion of the time of electrode drop in the total warm-up time,% 78.00 78.95 79.65 80.64

The values given in Table 4.2 show that when the heat flux entering the electrode from the welding arc changes in the studied range, the proportion of time spent on the electrode drop in the total heating time remains very constant and is about 79%.

## CONCLUSIONS

The conducted studies, considering the assumptions made, made it possible to establish the main parameters of the electrode melting process during electric arc welding of micromechanical products.So, the influence of the heat flux on the temperature gr adient in the electrode and the depth of the molten section are multidirectional.

A link was also established between the change in the heat flux and the tune parameters of the drop formation process.In general, it can be noted that the accounting of energy costs in the process of electric arc welding on the accumulation of heat of melting and boiling makes a significant contribution to the overall melting pattern of the electrode. The studies performed will allow investigating in more detail the various processes during micro-welding at different structural levels.

## ACKNOWLEDGMENTS

The works was earned out with financial support from the Research Program of the Ural Branch of the Russian Academy of Sciences (project 18-10-1-29) and budget financing on the topic “Experimental studies and multi-level mathematical modeling using the methods of quantum chemistry, molecular dynamics, mesodynamics, and continuum mechanics of the processes of formation of surface nanostmetured elements and meta- materials based on them” (project 0427-2019-0029).

## KEYWORDS

• boiling • drop formation • electrode • micro-welding • numerical simulation}} 

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•  thermal conductivity