# A mathematical approach of the temperature field in case of electron beam welding.

1. INTRODUCTION

The Electron Beam Welding (EBW) is a welding process which uses the heat coming from a concentrated beam composed of high-velocity electrons impinging upon the surfaces to be joined.

This paper presents a mathematical and numerical model for the 2D temperature distribution which emerges during the stationary case of EBW, based on the equation of heat conduction. The convection and radiations losses are not taken into account in this paper because of the short heating time, and the small area we have considered (consequence of the fixed position of the EB). The numerical scheme which solves the mathematical model--derived by the finite difference method--is transposed into a MathCAD application that offers us a numerical solution for the thermal field. The temperature distribution and its peak value--closely connected to the depth to which the welding is performed--allow us to state that the developed mathematical model is an effective tool in predicting the temperature distribution and the depth to which the welding is performed, as well as in selecting process parameters in such a way as to enable the welding. Temperature distributions and shapes of melted zones at EBW are described using other methods too: using the line and point source theory (Eagar, 2002), and statistical analysis (Koleva & Mladenov, 2000).

2. ELECTRON BEAM WELDING

Among the most important applications of EB technology one could mention: melting, cutting, welding, microscopic scanning, etc. (http://www.twi.co.uk). A wide range of physical processes appear when matter is submitted to an electron bombardment, the main effect being heat generation. The EBW procedure belongs to the category of fusion welding which is characterized by a high intensity, and the phenomena that arise are totally different from those typical to fusion welding conventional processes (Radaj, 1992). Electrons always penetrate matter. At the impact of the beam with the material there is a penetration of the beam to a certain depth and the kinetic energy of the electrons is transferred to the material that is penetrated and transformed into heat upon impact.

On the one hand, the penetration is influenced by the characteristics of the beam itself. On the other hand, it is influenced by the characteristics of the material, practically by the way electrons interfere with the material microstructure.

The maximum penetration depth (the electron range) depends upon a lot of factors: electron beam current, accelerating voltage, electron beam diameter, electron beam power, material density, etc.

Absorptivity is a variable which should be carefully analyzed as it greatly influences the thermal field. It directly influences the material transferred energy amount.

Power absorption per unit of volume can be expressed as:

g(y,t) = [[eta].sub.A] U x j(y,r) = [[eta].sub.A] P(t)/S x r f(y,r) (1)

where: j(t) is the electron current density, U the acceleration voltage, P(t) the power of EB, S the irradiation area of the EB, [[eta].sub.A] beam power absorption ratio (normal value for steels ranges: 0.7-0.8.). In the above equation, f stands for the distribution function of the absorbed output density.

As electrons interact with material, x-rays, secondary electrons, and backscattered electrons are generated, and these phenomena bring alterations (fractionally cuts off the heat). Due to this loss, the absorption ratio of the beam output is chosen to be 0.8. From experimental studies different empirical formulas for the penetration depth (r) of an EB into matter have been obtained. For 10keV [less than or equal to] eU [less than or equal to] 100keV , it can be expressed, with the formula:

r [approximately equal to] 2.1 x [10.sup.-5] [U.sup.2]/[rho], (2)

Experimental data analysis allows us to find out that the distribution function of the absorbed output density is influenced by various factors, like: the distance between the current point and the bombarded area (y), the maximum penetration depth, the atomic number and consistency, etc.

For the distribution function of the output density this paper operates with the dependence described by the following ratio:

f(y,r) = 1 - 9/4[(y/r - 1/3).sup.2] (3)

The absorbed power distribution per volume unit as being dependant on the electron track inside the material describes some specific curves. One can notice that its maximum value is reached at about 1/3 of the beam track (Ying et al., 2003).

3. MATHEMATICAL APPROACH

We suppose to be welding two identical parallelepiped plates made of the same material. So we consider the workpiece to be a parallelepiped plate. The EB is considered to be a cylinder, of radius R, whose axis is parallel to the united faces, and which penetrates the material. Experimentally it was observed that heat spreading within the material is more intense in the direction of the beam action.

[FIGURE 1 OMITTED]

Taking this fact into account we, will consider in this approach that the area in which the conduction heat transfer is studied is a two-dimensional one (the section of the workpiece with a plane parallel to the welding direction-Oy axis- passing through the projection of the beam centre on the superior face. The area under consideration is represented in Fig. 1

Considering so the 2D stationary heat transfer equation and an isotropic homogenous body with a constant thermal conductivity, the mathematical model is characterized by:

([[partial derivative].sup.2]T/[partial derivative][x.sup.2] + [[partial derivative].sup.2]T/ [[partial derivative][y.sup.2] = -1/[lambda] q(x,y), (x,y) [member of] [OMEGA] = [0,2] x [0,2] (4)

where: [lambda] is the thermal conductivity, q is a source-type term. The boundary condition is: [T(x,y)/.sub.[partial derivative][SIGMA]] = [T.sub.0], [T.sub.0]-temperature of the environment. q(x,y) depends on the energy generated by the EB and on the chosen material. For a uniform power distribution of EB, we model q(x,y) by the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In order to solve (4), we apply the finite difference method (Antia, 2002). To discretize the area, we use 33 equidistant nodes along both axes: [x.sub.i] = ih, [y.sub.j] = jh, 0 [less than or equal to] i, j [less than or equal to] 32. By using the central finite difference formula we get the following linear system of equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [T.sub.i,j] = T([x.sub.i], [y.sub.j]).

4. NUMERICAL RESULTS AND CONCLUSIONS

We have implemented the method described in this paper into a MathCAD application in order to obtain numerical solutions for the temperature field and to study the influence of the process parameters on the thermal field.

The numerical results obtained for a 13CrMo4 steel and process parameters: U = 60 kV, I = 80mA are performed in the following diagrams:

[FIGURE 2 OMITTED]

The first graph shows the nodal values of the temperature for R = 0.2cm, while the second graph shows the condition of the isothermal lines which appear inside the thermal field.

We notice that the temperature does not reach its top value at the surface the beam was incident to, but inside, i.e. at the point corresponding to the 1/3 of the electron penetration depth in agreement with the maximum value of the absorbed power distribution per volume unit (Ying et al., 2003).

The influence of the electron beam radius is studied considering different values of this parameter. For R=1.7, respectively 0.1 the numerical results are performed in Fig. 3, 4.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

One can notice that the point temperature reaches its peak value has not changed too much (as we have not changed the parameters influencing electron tracks within the material under consideration). A significant modification is connected to isothermal lines and temperature values within the area under discussion.

The results indicate the fact that in any parameter situation, maximum temperature level is under the surface which is directly exposed to the EB action. Temperature distribution and isothermal line form change and depend on the EB radius. These results are in concordance with others in literature, which are obtained by others techniques or experimentally.

The MathCAD application can also be used in the future to study the diagram of the process predicting melting bath formation, and it also offers the opportunity to study the effect of the material thermal properties may have upon the thermal field., and for making comparison studies. In a further work the non-stationary case will be considered and also the 3D case.

5. REFERENCES:

Antia H. M. (2002). Numerical Methods for Scientists and Engineers, Birkhauser, ISBN-10: 3764367156.

Eagar, T.W., Heat flow and laser/electron beam welding, MIT courses, October 2002.

Koleva, E. & Mladenov, G. (2000). Analysis of the thermal Processes and Shapes of Melted Zones at Electron Beam Welding and Melting, Proceedings of the 4th General Conference of the Balkan Physical Union, 22-25 August 2000, Veliko, Turnovo, Bulgaria, pp.83-96, ISSN 1310-0157.

Radaj, D. (1992). Heat effects of welding--Temperature Field Residual Stress Distorsion, Springer--Verlag Berlin, ISBN-10: 0387548203

Ying Qin & Chuang Dong & Xiaogang Wang & Shengzhi, Hao & Aimin Wu & Jianxin Zou, & yue Liu. (2003). Temperature profile and crater formation induced in high current pulsed electron beam, processing , Journal of Vacuum Science and Technology 2003, vol. 21, Issue 6, pp 1934-1938, ISSN: 0734-2101.
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