=Paper=
{{Paper
|id=Vol-1839/MIT2016-p28
|storemode=property
|title= Computational modelling of metal vapor influence on the electric arc welding
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p28.pdf
|volume=Vol-1839
|authors=Amanbek Jainakov,Rena Sultangazieva,Bubusara Medralieva
}}
== Computational modelling of metal vapor influence on the electric arc welding==
https://ceur-ws.org/Vol-1839/MIT2016-p28.pdf
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Computational Modelling of Metal Vapor
Influence on the Electric Arc Welding
Amanbek Jainakov, Rena Sultangazieva, and Bubusara Medralieva
Kyrgyz State Technical University after I.Razzakov,
Mira ave. 66, 720044 Bishkek, Kyrgyzstan
{jainakov-41,renasultangazieva,medralieva}@mail.ru
http://www.kstu.kg
Abstract. The mathematical model, which takes into account the ef-
fects 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 concen-
tration. The system of equations solved by the finite difference method on
a rectangular non-uniform orthogonal grid using the five-point difference
scheme.The effect of metal vapour from weld pool on the characteristics
of the arc column was numerically investigated. Distribution of electric
field, current density and temperature field in the arc column and weld
pool with and without consideration of the Fe vapor are shown.
Keywords: electric arc plasma, MHD equations, metal vapours, vortic-
ity, stream function, weld pool, Marangoni effect
1 Introduction
Arc welding is characterized by high values of molten metal temperature gra-
dients, 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 radia-
tion 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.
312
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
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. Phys-
ical processes in electric arc column and interacting with the discharge of the
liquid metal are described by a single system of magnetohydrodynamics equa-
tions [1]. When recording MHD equations in the simplest form it is assumed the
following conditions: the plasma column is assumed to be in local thermody-
namic equilibrium (LTE), the plasma is a Newtonian fluid, flows are the steady
and laminar. MHD system of equations in cylindrical coordinates is as follows [2]:
The mass continuity equation:
1 π (πππ’) π (ππ£)
+ =0 (1)
π ππ ππ§
The radial momentum conservation equation:
(οΈ )οΈ
ππ£ ππ£ ππ 2 π ππ£ π£
ππ£ + ππ’ =β β ππ§ π΅π + ππ β 2π 2 +
ππ ππ§ ππ 3 ππ ππ π
(οΈ (οΈ )οΈ)οΈ (οΈ (οΈ )οΈ)οΈ
π ππ’ ππ£ π 2 1 π(ππ£) ππ’
π + β π + (2)
ππ§ ππ ππ§ ππ 3 π ππ ππ§
The axial momentum conservation equation:
(οΈ (οΈ )οΈ)οΈ
ππ’ ππ’ ππ 1 π ππ’ ππ£
ππ£ + ππ’ =β + ππ π΅π + ππ + β
ππ ππ§ ππ§ π ππ ππ ππ§
(οΈ (οΈ )οΈ)οΈ (οΈ )οΈ
π 2 1 ππ£π ππ’ π ππ’
π + +2 π + ππ’ (3)
ππ§ 3 π ππ ππ§ ππ§ ππ§
The energy conservation equation:
(οΈ )οΈ (οΈ )οΈ
1 π π πβ π π πβ 1 (οΈ 2 )οΈ
πππ£β β + ππ’β β = ππ + ππ§2 β π + ππΆ1 (4)
π ππ ππ ππ ππ§ ππ ππ§ π
Maxwellβs equations:
ππΈπ ππΈπ§ 1 πππ»π ππ»π
β = 0, = ππ§ , β = ππ (5)
ππ§ ππ π ππ ππ§
Ohmβs law:
ππ = ππΈπ§ , ππ§ = ππΈπ (6)
The system is supplemented by the equation of convective diffusion of metal
vapor [3]:
(οΈ )οΈ (οΈ )οΈ
1 π π 1 π ππΆ1 π ππΆ1
(πππ£πΆ1 ) + (ππ’πΆ1 ) = πππ· + ππ· (7)
π ππ ππ§ π ππ ππ ππ§ ππ§
313
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
where π’, π£ are axial and radial flow velocity; π is pressure; π is temperature; π
is current density, πΈ is intensity of electric field, π» is intensity of the magnetic
field, π΅ is magnetic induction, πΆ1 is mass concentration of metal vapor, π· is
diffusion coefficient, πis density of the plasma, ππ is specific heat, π is viscosity, π
is thermal conductivity, π is radiation, π is electrical conductivity, β is enthalpy.
In the momentum conservation equation:
{οΈ
0 for arc plasma
ππ’ =
ππ β πππ½(π β π0 ) for weld pool
where π½ is coefficient of thermal expansion, π is acceleration of gravity.
In the energy equation for the weld pool the effective heat capacity is used:
πππ
πππ
π = ππ + π₯π»π
ππ
where π₯π»π is specific heat of melting of the anode material.
The weld pool liquid fraction ππ varies linearly with temperature:
β§
β¨1 π > ππ
ππ = πππβπ π
π π < π < ππ
β© βππ
0 π < ππ
where ππ is solid phase temperature, ππ is liquid phase temperature of the metal
anode. Term
[οΈ(οΈ )οΈ ]οΈ [οΈ(οΈ )οΈ ]οΈ
π π ππΆ1 π π ππΆ1
ππΆ1 = ππ· β (βπ β β) + ππ· β (βπ β β)
ππ§ ππ ππ§ ππ ππ ππ
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, βπ is enthalpy
of the metal vapor.
The interaction between the plasma and metal vapor, their mutual influ-
ence 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:
π = π(π, πΆ1 ), π = π(π, πΆ1 ), π = π(π, πΆ1 ), π = π(π, πΆ1 )
π = π(π, πΆ1 ), β = β(π, πΆ1 ), ππ = ππ (π, πΆ1 )
For determine the diffusion coefficient used the approximation of viscous ap-
proximation [4]. Diffusion coefficient in the approximation is calculated by the
formula:
β (οΈ )οΈ0,5
2 2 π11 + π12
π·π΄πβπΉ π = (οΈ(οΈ )οΈ0,25 (οΈ )οΈ0,25 )οΈ2 (8)
π21 π22
π½ 2 π 2 π1
+ π½ 2 π 2 π2
1 1 2 2
314
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
where π1 , π2 are molar weight of the metal and the plasma gas, π1 , π2 , π1 π1
are density and viscosity of the metal and gas, respectively; π½1 = π½2 = 1.385
based on experimental data.
MHD system of equations is solved in the variables βvorticity-stream func-
tionβ, the introduction of the following variables: π is the intensity of the vortex,
π is stream function, π is the function of the electric current, which in the case
of a cylindrical coordinate system defined by the relations with axial symmetry:
(οΈ )οΈ
1 ππ£ ππ’ ππ ππ
π= β ; = ππ’π; β = ππ£π;
π ππ§ ππ ππ ππ§
ππ ππ
= πππ§ ; β = πππ ;
ππ ππ§
Then the original system can be written in the following canonical form:
[οΈ (οΈ )οΈ (οΈ )οΈ]οΈ [οΈ ]οΈ
π ππ π ππ π π
π π β π β π (ππ) β
ππ§ ππ ππ ππ§ ππ§ ππ§
[οΈ ]οΈ
π π
π (ππ) + ππ = 0 (9)
ππ ππ
where π is desired function, taking values π, π, β, π and πΆ1 ; a, b, c, e are
nonlinear coefficients corresponding to each of the equations. The values of these
ratios are presented in the Table 1.
Table 1. Values of the coefficients of the canonical equations
π π π π [οΈ (οΈ 2 )οΈ (οΈ 2π )οΈ ]οΈ
π’ +π£ 2 ππ π’ +π£ 2 ππ
π π 2
π 3
π βπ 2 π
ππ§ 2 ππ
π
β ππ 2 ππ§
β ππ3 ππ
ππ§
β π ππ
ππ
+ ππ€
1
π 0 1 ππ
ππ [οΈ(οΈ )οΈ (οΈ )οΈ2 ]οΈ
ππ 2
h 1 π
ππ
π 1 1
ππ ππ
+ ππ
ππ§
β ππ + ππΆ1
1
π 0 1 ππ
0
πΆ1 1 πππ· 1 0
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.
315
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 1. Scheme of the computational domain
3 Boundary conditions
1) Boundary conditions for the all solid walls are set as follows: Condition of
impermeability for the stream function π = 0. The function π is determined
from the condition of sticking. The temperature is assumed to be π0 = 300, thus
πΌ
determined β = β(π0 ). Electric current function is defined as π = 2π . Metal
vapor concentration equal zero πΆ1 = 0.
2) The boundary conditions at the cathode are defined as follows:
π = 0;
ππ
ππ§ = 0; (οΈ )οΈπ (οΈ )οΈ
ππ (π) = (ππ β π0 ) 1 β π
ππ 1 + π
ππ π + π0 ;
πΌ
β«οΈπ
ππ = π
π
ππππ;
β«οΈ
2π ππππ 0
0
πΆ1 = 0.
316
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Here ππ 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:
ππ πβ ππΆ1
π = 0; = 0; = 0; π = 0; = 0;
ππ ππ ππ
4) At the weld pool surface the boundary conditions are defined as follows:
π = 0;
ππΌ ππ
ππ ππ π = ππ ππ π β ππ ;
πππ πππ
(οΈ 4
ππ )οΈ
ππ ππ§ = ππ ππ§ β ππ π ππ β π0 β ππ£ βπ π ;
πππ πππ
ππ§ = ππ§ ;
π π1
πΆ1 = ππ£ππ π1 +(ππ£ππ
ππ‘π βππ£ππ )π2
. Here the index βpβ refers to the plasma arc,
the index βaβ refers to the material of the anode; ππ to Stefan-Boltzmann co-
efficient; π to the emissivity of the anode; βπ π to latent heat of evaporation;
ππ£ to evaporation rate, which is obtained from the following approximation [5]:
log ππ£ = π΄π£ + log πππ‘π(οΈβ 0, 5π(οΈ, π΄π£ -constant )οΈ)οΈ depending on the workpiece ma-
βπ»π£ππ 1 1
terial; ππ£ππ = πππ‘π exp π
π β ππππ - the partial vapor pressure of the
metal, which is a function of the molten metal weld pool temperature
5) At the lower boundary of the workpiece conditions are stated:
π = 0; π = 0; β = β(π0 ); ππ ππ§ = 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 [2]. Computational domain is covered by a rectangular
orthogonal non-uniform grid. We are integrating the equation (9) for the area
which is bounded by the dotted line (Fig. 2):
π§π+ 1 ππ+ 1 π§ 1 π 1
β«οΈ 2 β«οΈ 2 [οΈ (οΈ )οΈ (οΈ )οΈ]οΈ β«οΈπ+ 2 β«οΈπ+ 2 [οΈ (οΈ )οΈ
π ππ π ππ π πππ
π π β π ππππ§ β π +
ππ§ ππ ππ ππ§ ππ§ ππ§
π§πβ 1 ππβ 1 π§πβ 1 ππβ 1
2 2 2 2
π§ 1 π 1
(οΈ )οΈ]οΈ β«οΈπ+ 2 β«οΈπ+ 2
π πππ
+ π ππππ§ + ππππππ§ = 0 (10)
ππ ππ
π§πβ 1 ππβ 1
2 2
We will call the first term as the convective term πΌπ , second as the diffusion
term πΌπ , the third - source term πΌπ . The first-order derivatives are approximated
317
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 2. Difference grid fragment
by backward difference, the second-order derivatives are approximated by central
difference scheme. After integration of the convective term we have:
βοΈ
πΌπ = π΄π,π (ππ,π β ππ,π )
where π
π΄π,π = π,π
8 (πΉπ,π + |πΉπ,π |)
πΉπβ1,π = ππβ1,π+1 + ππ,π+1 β ππβ1,πβ1 β ππ,πβ1
πΉπ+1,π = ππ+1,πβ1 + ππ,πβ1 β ππ+1,π+1 β ππ,πβ1
πΉπ,πβ1 = ππ+1,π+1 + ππ+1,π β ππβ1,π+1 β ππβ1,π
After integration of the diffusion term we have:
βοΈ
πΌπ = π΅π,π (ππ,π ππ,π β ππ,π ππ,π )
where
ππβ 1 ,π π ππ+ 1 ,π π
π+1 βππβ1 π+1 βππβ1
π΅πβ1,π = 2
2 π§π βπ§πβ1 , π΅π+1,π = 2
2
π§πβ1 βπ§π
ππ,πβ 1 ππ,π+ 1
2 π§π+1 βπ§πβ1 2 π§π+1 βπ§πβ1
π΅π,πβ1 = 2 ππ βππβ1 , π΅π,π+1 = 2 ππ+1 βππ
After integration of the source term, we have:
ππ+1 β ππβ1 π§π+1 β π§πβ1
πΌπ = ππ,π ππ
2 2
Thus, the differential equation is transformed into a system of nonlinear
algebraic equations:
βοΈ π π (π βπ )(π§π+1 βπ§πβ1 )
[(π΄π,π + ππ,π π΅π,π ) ππ,π ] β π,π π π+1 πβ1 4
ππ,π = βοΈ
(π΄π,π + ππ,π π΅π,π )
This system of nonlinear algebraic equations solved by an iterative method
of Gauss-Seidel:
(οΈ )οΈ
πππ,π = πΌ ππβ1,π πππβ1,π + ππ+1,π ππβ1 π πβ1 πβ1
π+1,π + ππ,πβ1 ππ,πβ1 + ππ,π+1 ππ,π+1 + π·π,π +(1βπΌ)ππ,π
where
π΄π,π + ππ,π π΅π,π
ππ,π = βοΈ
(π΄π,π + ππ,π π΅π,π )
318
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
1
π·π,π = ππ,π ππ (ππ+1 β ππβ1 ) (π§π+1 β π§πβ1 )
4
Over relaxation method was used to improve the convergence of the iterative
process and stopping criterion was:
β β
β π β
β ππ,π β ππβ1
π,π ββ
max ββ β πβ1 β β < π β 10β3
π,π β max βπ ββ
π,π
π,π
5 Results and discussion
Based on the properties of the pure components, with the help of software AS-
TRA 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 [6].
When the content of iron vapors is about 1% electrical conductivity and radia-
tion have a noticeable difference in the temperature range from 5000 to 10000 K.
In this area isotherm of 8000 K lies, which usually take the visible border of arc.
The calculations used the following data:
The melting point of steel πππππ£ =1773 K;
The boiling point of steel ππππ =3133 K;
The specific heat of fusion π₯π»π = 2.47 * 105 J/kg;
The molar weight of argon π1 = 55 * 10β3 kg/mol;
Molar mass of steel π2 = 27 * 10β3 kg/mol;
Molar heat of vaporization of steel π»π£ππ = 340 * 103 J/mol;
Specific heat of vaporization of steel βπ π = 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.
319
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 3. Surface tension gradient of steel
a) b)
Fig. 4. Distribution of Fe vapors concentration a) on the anode surface b) in a column
of the electric arc.
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 [7].
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
320
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 5. The temperature distribution of arc, I = 200 A, a) for mixture Ar+1%Fe, b)
for pure Ar
Fig. 6. Current density in the electric arc column, I = 200 A, a) for mixture Ar+1%Fe,
b) for pure Ar
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
321
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 8. The surface temperature of the an-
Fig. 7. Heat flux to the anode.
ode.
conductivity coefficient in this temperature range. Thus, the temperature of the
weld pool surface in the presence of metal vapor is reduced (Fig. 8).
Fig. 9. The temperature distribution of weld pool, I = 200 A, a) for mixture Ar+1%Fe,
b) for pure Ar
The properties of the workpiece produce a noticeable effect on the hydro-
dynamic 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 in-
volves 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
322
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 10. The distribution of fluid flow and vector fields in the weld pool, I = 200 A a)
for mixture Ar+1%Fe, b) for pure Ar
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.
References
1. Jainakov A., Usenkanov J., Sultangazieva R.. On joint modeling of pro-
cesses in electric arc plasma and melted metal//6 general assembly of federation of
engineering institutions of Islamic countries. 27-30 June-1999, Almaty. P. 11β21.
2. Engelsht V.S. The mathematical modeling of electric arc.
Frynze: Ilim, 1983. 364p. (In Russ.)
3. Yamamoto K., Tanaka M., Tashiro S., Nakata K., Murphy A.B. Numer-
rical Simulation of metal vapor behavior in argon TIG welding //Transactions of
JWRI. 2007. V. 36, No. 2. P. 1β4.
4. Wilke C.R. A viscosity equation for gas mixtures The Journal of Chemical
Physics. 1950. Vol. 18, No. 4. P. 517β519.
5. Zacharia T., David S.A., Vitek J.M. Effect of evaporation and temeperature-
dependent material properties on weld pool development //Metall. Trans. 1991.
V. 22B. P. 233β241.
6. Schnick M., Fuessel U., Hertel M., Haessler M., Spille-Kohoff A.,
Murphy A.B. Modelling of gas-metal arc welding taking into account metal
vapour//Journal of Physics D: Applied Physics. 2010. V. 43. P. 4340β4348.
7. Lag-Lago F., Gonzalez J.-J., Freton P., Gleizes A. A numerical modelling
of an electric arc and its interaction with the anode: Part I. The two-dimensional
model//Journal of Physics D: Applied Physics. 2004. V. 37, No. 6. P. 883β888.
323