The mathematical model, which takes into account the effects of the Fe vapours in the arc welding of stainless steel workpieces in closed volume are proposed. The physical phenomena in arc plasma and molten pool are considered in coupled unified MHD model. The system is solved in the variables vorticity-stream function for five variables: stream function, vorticity, current function, enthalpy and metal vapour concentration. The system of equations solved by the finite diference method on a rectangular non-uniform orthogonal grid using the five-point diference scheme.The efect of metal vapour from weld pool on the characteristics of the arc column was numerically investigated. Distribution of electric ifeld, current density and temperature field in the arc column and weld pool with and without consideration of the Fe vapor are shown.
Introduction Arc welding is characterized by high values of molten metal temperature gradients, with a significant portion of the surface of the weld pool metal is at a temperature close to the boiling temperature and generates a modest amount of metal vapor arc zone, which has a significant impact on the basic physical properties of the arc, energy efficiency, impact the size and shape of the weld pool. The atoms of metal have a lower ionisation energy compared with inert gases such as argon and helium. For example, the ionization energy of argon is 15,755 eV and the ionization energy of iron is 7.8 eV. This increases the radiation and electric conductivity of the plasma and causes a change in composition and properties of the plasma arc in the anode region and a portion of the arc column. In turn, evaporation of workpieces impurities changes the composition of the molten pool, which can cause changes in the microstructure of the metal and mechanical properties of the alloys. 2
Governing equations
In this paper, we propose a mathematical model of the joint consideration of
the electric plasma arc and the workpiece where their mutual influence on each
other, taking into account the influence of metal vapor evaporated anode.
Physical processes in electric arc column and interacting with the discharge of the
liquid metal are described by a single system of magnetohydrodynamics
equations [
+ ) ( ) − + The radial momentum conservation equation: The axial momentum conservation equation: ℎ + (︂ + = − ︂( + 3 + )︂ (︂
ℎ = − (︂ 2 (︂ 1 + +
(︂ 2 (︂ 1 ( ) − + 2 (︂ 3 3
1 (︂ ℎ )︂ − + 2 )︂ ︂( (︂ 1 − 2 + 2 + + )︂
− + = ︀( 2 + 2)︀ − + 1
(1) (2) (3) (4) (5) (6) (7) The energy conservation equation: 1 (︂
ℎ )︂ − Maxwell’s equations:
Ohm’s law:
vapor [
︂) 1 + (︂
︂) 1 where , are axial and radial flow velocity; is pressure; is temperature; is current density, is intensity of electric field, is intensity of the magnetic field, is magnetic induction, 1 is mass concentration of metal vapor, is diffusion coefficient, is density of the plasma, is specific heat, is viscosity, is thermal conductivity, is radiation, is electrical conductivity, ℎ is enthalpy.
In the momentum conservation equation: = ︂{ 0
for arc plasma −
( − 0) for weld pool where is coefficient of thermal expansion, is acceleration of gravity.
In the energy equation for the weld pool the effective heat capacity is used: where
is specific heat of melting of the anode material.
The weld pool liquid fraction varies linearly with temperature:
= + = ⎨ −
− where is solid phase temperature, is liquid phase temperature of the metal anode. Term 1 = [︂(
)︂ − (ℎ − ℎ ) 1 1 on the right side of the law of conservation of energy determines the enthalpy change due to the mixing of the metal vapor and the plasma gas, ℎ is enthalpy of the metal vapor.
The interaction between the plasma and metal vapor, their mutual
influence on each other is determined by the thermal properties of the medium as a
function of temperature and concentration of metal vapor in the plasma:
= (,
1), = (,
1), = (,
1), = (,
1)
= (,
1), ℎ = ℎ (,
1), = (,
1)
For determine the diffusion coefficient used the approximation of viscous
approximation [
− 3 − + ︁] 1 [︁ ︀( ︀) 2 + (︀ ︀) 2]︁ − + 1 − where 1, 2 are molar weight of the metal and the plasma gas, 1, 2, 1 1 are density and viscosity of the metal and gas, respectively; 1 = 2 = 1.385 based on experimental data.
MHD system of equations is solved in the variables ”vorticity-stream function”, the introduction of the following variables: is the intensity of the vortex, is stream function, is the function of the electric current, which in the case of a cylindrical coordinate system defined by the relations with axial symmetry:
Single entry form allows for solving the system of equations to use the same calculation algorithm. To solve the resulting system of differential equations is necessary to set the boundary conditions for these functions. Since the system equations are of elliptic type, the boundary conditions must be given around the contour surrounding the computational domain. The computational domain is shown in Fig. 1. Real non-consumable cathode plasma torch is a cylinder with a flat end, as the anode serves a workpiece, the system is in a confined space, limited by side walls at a distance of R.
; − − = ;
= ; = ; =
; 2 [︂ −
0 0 = 0;
= 0; = 1) Boundary conditions for the all solid walls are set as follows: Condition of impermeability for the stream function = 0. The function is determined from the condition of sticking. The temperature is assumed to be 0 = 300, thus determined ℎ
= ℎ ( 0). Electric current function is defined as vapor concentration equal zero 1 = 0.
2) The boundary conditions at the cathode are defined as follows: ( ) = ( − 0) 1 − )︁ (︁ 1 + ︁) + 0; = 2 . Metal
Here is the boiling point of the cathode, m is the degree of filling of the temperature profile.
3) In the arc column axis of symmetry conditions are implied: 4) At the weld pool surface the boundary conditions are defined as follows:
= = ; 1 = 1+(
−
; 1 −
) 2 − (︀ 4 − 0)︀ − ℎ ; . Here the index ”p” refers to the plasma arc, ︁( 1 −
1 )︁
the index ”a” refers to the material of the anode; to Stefan-Boltzmann
coefficient; to the emissivity of the anode; ℎ
to latent heat of evaporation;
to evaporation rate, which is obtained from the following approximation [
= 0;
In the area of anode, the equation of convective diffusion of metal vapor is not solved. The boundary conditions for the vorticity were set at a point at one step from the solid boundaries, thus avoiding the ambiguity of the boundary conditions at the corners to ensure sustainable convergence and bridge solutions on a rectangular grid for the boundary of any shape. 4
Numerical methods
The canonical equation was solved using integro-interpolation method based on
finite difference approach [
− 12 − 21 + (︂ − ︂)] (︂ −
︁∫ + 12 + 21
︁∫ − 12 − 12
We will call the first term as the convective term , second as the diffusion term , the third - source term . The first-order derivatives are approximated ︂[ (︂ ︂) + + = 0 (10) by backward difference, the second-order derivatives are approximated by central difference scheme. After integration of the convective term we have: Thus, the differential equation is transformed into a system of nonlinear , = ︀∑ [( , + , , ) , ] − , ( +1− −1)( +1− −1)
4 ︀∑ ( ,
+ , , ) This system of nonlinear algebraic equations solved by an iterative method
After integration of the diffusion term we have: After integration of the source term, we have: where where algebraic equations: of Gauss-Seidel: , where = ︁∑ ,
( , − , ) , = , 8
( , + | , |) −1, = −1, +1 + , +1 − −1, −1 − , −1 +1, = +1, −1 + , −1 − +1, +1 − , −1 , −1 = +1, +1 + +1, − −1, +1 − −1, = ︁∑ , ( , ,
− , , ) 1 , = 4 , ( +1 − −1) ( +1 − −1)
Over relaxation method was used to improve the convergence of the iterative process and stopping criterion was: max ⃒⃒⃒⃒ , − , −1 ⃒⃒ , ⃒ max ⃒⃒ , −1⃒⃒ ⃒⃒⃒ < ≈ 10−3
⃒ , ⃒ 5
Results and discussion
Based on the properties of the pure components, with the help of software
ASTRA and TERRA transfer coefficients for mixtures of Ar + 1% Fe, Ar + 3%
Fe were calculated. The data are in good agreement with the data given in [
The calculations used the following data: The melting point of steel =1773 K; The boiling point of steel =3133 K; The specific heat of fusion = 2.47 * 105 J/kg; The molar weight of argon 1 = 55 * 10−3 kg/mol; Molar mass of steel 2 = 27 * 10−3 kg/mol; Molar heat of vaporization of steel = 340 * 103 J/mol; Specific heat of vaporization of steel ℎ = 6.2 * 106 J/kg; Steel surface tension is determined according to the data given in Fig. 3.
The calculations were performed for the current I = 150 A and 200 A. To current I=150 A maximum concentration of iron vapor on the surface of the weld pool on the basis of the boundary conditions was 0,6%, which does not affect the transport coefficients argon arc.
Fig. 4 shows graphs of the distribution of iron vapor concentration on the anode surface and within the scope of the electric arc at a current I = 200 A. The maximum concentration of iron vapor with a current of 200 A is 1.05% on the axis of the arc. The distribution of the concentration of metal vapor is determined by convective and diffusive fluxes. Axial gas flow rate directed to the anode 5 times the radial velocity of the anode surface, so the metal vapor in the axial part concentrated mainly near the surface of the anode, and the metal vapor expansion region occurs outside of the arc axis. Also, this is due to the nature of the diffusion coefficient, the maximum value of which falls on the periphery of the nucleus of the arc where the metal atoms easily diffuse into the arcing region. Thus, the axial part of the convection is predominant, so the metal vapor are drawn into a radial motion of the gas flow and flow over the anode surface.
Fig. 5 shows the temperature fields with and without taking into account the
metal vapor in the argon plasma at a current I = 200 A. The presence of metal
vapor in the anode part narrow arc in radial direction, cooling the arc column at
the edges, and heating the arc core. This is because the emissivity of a mixture
of argon with significantly higher metal vapor in a temperature range of 5000 to
13000 K, which leads to an increase in radiation loss in the given interval and
a narrowing of the arc. Another cooling mechanism of the arc on the periphery
is to increase the thermal conductivity at temperatures below 8000 K, caused
by greater diffusion of heat in the vicinity of the plasma arc. This arc cooling
effect in the presence of metal vapor is consistent with the experimental and
theoretical results of [
Fig. 6 shows graphs of current density in the arc column. The current density at the anode surface in the presence of iron vapor is reduced, it is because the presence of vapor increases the electrical conductivity at temperatures below 10000 K, and electric current flows in the colder regions of the arc. Changing the conductivity a mixture of argon and metal vapor in the anode part is formed by two mechanisms. On the one hand, the presence of iron should increase plasma vapor conductivity. On the other hand, cooling of the arc due to the higher radiation losses and increase the thermal conductivity in the peripheral portion decreases the overall electrical conductivity of the mixture. As a result, the contribution of the electromagnetic component on the penetrating ability of the arc is reduced.
Fig. 7 illustrates the heat flux from arc column to the anode. Despite the fact that the core temperature of the arc to above metal vapor with argon, the heat flow toward the anode member to pure argon, due to the higher thermal
Fig. 8. The surface temperature of the anode. conductivity coefficient in this temperature range. Thus, the temperature of the weld pool surface in the presence of metal vapor is reduced (Fig. 8).
The properties of the workpiece produce a noticeable effect on the hydrodynamic conditions in the weld pool. The steel has a relatively low coefficient of thermal conductivity and high heat capacity ratio, which should lead to a shallow depth and radius of the weld pool (Fig. 9).
The plasma flow spreads radially from the surface of the molten metal and involves radial movement of the upper layers of the liquid metal due to shear stress of plasma convective flow and Marangoni convection and causes the formation of vortex in the weld pool volume. At the edges of the weld pool Marangoni force generated an additional reverse vortex involving in motion the same amount of metal, as in the main vortex (Fig. 10). Since the intensity of mixing of metal in a small vortex is very high, this strong vortex flow carries heat deep into the pool, which leads to additional melting of the base metal at the edges of the bath. The above phenomena are formed similar to the form of the weld pool.