Fume emissions by electric arc during gas metal arc welding

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Introduction.
The arc welding is based on the electric discharge in gas with high amperage, which is called an electric arc.Electric arc causes ionization of the gas atoms and forms the plasma with temperature about 10000-20000 K, which allows to use the electric arc for melting and welding of metals.
The arc column is surrounded by shielding gas, which protects the molten metal in droplets and weld pool from oxidation.The shielding gas is evaporated from electrode covering in shield metal arc welding (SMAW) or is injected into welding workspace in gas metal arc welding (GMAW) and gas tungsten arc welding (GTAW), which are different by type of electrodes.The consumable or non-consumable electrodes can be used in arc welding with shielding gas.In the first case (GMAW), there is a melting of welding wire and transfer of metal droplets to weld pool, where the weld joint is formed.In the second case (GTAW), the electrode is used only for arc providing, and the welding wire is entered directly into the weld pool.
Arc welding create the serious ecology dangers, because it is the source of welding fumes, which are formed via metal vapors condensation, and has toxically hazard for human and environment.The study of these dangers and their minimization are the important problem, as the arc welding is used very widely.
The experimental studies of electric arc in the welding process are complicated because arc column is not stable, but continuously moves on the welded surface.The optical or probe techniques are inapplicable in these conditions.Only high-speed ЕЛЕКТРОФІЗИКА photography in the visible and infrared ranges allows to record the droplets transfer and temperature distribution in arc.
Therefore, search of the welding arc authentic computer model is the important problem.Many authors calculate the arc plasma parameters via numerical solving of the classical equations system [1][2][3][4][5][6][7][8].It is difficult to understand at such approach, how this or that process influence the arc parameters and welding fume properties.
The simplified sequential modeling of separate processes in boundary layers of plasma near anode and cathode is proposed in the present paper.Thus, modeling comes down to the solution of separate tasks, such as space distribution of temperature or particle number densities, in the anode and cathode regions.These tasks are united in the common problem, as processes influence at each other, which is solved by approximation method.The purpose of modeling is determination of vapor formation rate, which is flowed out from molten pool to environment, because this metal vapor is the source of welding fumes.
The temperature of welding wire tip is about of boiling temperature, when arc welding with consumable electrode is used.Therefore, metal vapor forms the layer near electrode tip surface with high number of the iron atoms [9], which provide the electron and ion number densities via thermal ionization.These free charge carriers provide the space charge regions near anode and cathode, which determine the electric current through arc.The space charge region is the main research object in the presented modeling.
The high-temperature metal vapor from arc mixes with the shielding gas and cools.At some temperature the vapor supersaturation reaches critical value, which provides the nucleation beginning.The heterogeneous ion-induced nucleation occurs, because vapor-gas mixture is ionized.The presented consideration of nucleation is took into account the exchange of charges and energy between ionized environment and nuclei.The droplets of nuclei grow via vapor condensation and coalescence up to solidification.Then coagulation of solid primary particles occurs.These processes are considered in the present paper for various values of welding current.
The consideration is focused on conditions of our experiments [10,11]: reverse polarity, when electrode wire is positive (anode) and the molten pool is negative (cathode); welding wire with 0.8 mm diameter; arc voltage of 18 to 30 V; arc current of 50 to 150 A; wire feed speed of 4 to 12 cm/s; shielding gas flow rate of 6 to 12 lpm.The plasma outside near-electrode layer is considered in the local thermodynamic equilibrium (LTE) with temperature T pl = 15000 K, for definiteness [6].

Temperature space distribution.
The traditional description of arc plasma adjacent to anode surface is based on division of the near-anode layer into several zone [8,12], such as sheath, presheath and arc column.The sheath zone is a space charge layer, where the plasma neutrality is disturbed and the potential drop between plasma and anode is formed.
The mean free path of iron atoms near consumable electrode (which is anode, when reverse polarity is used) with temperature T w ∼ 3000K is described by the following expression [8]: , A and B is the Antoine constants for iron vapor.The Eq. ( 1) is defined the Knudsen layer with collisionless plasma near anode tip surface.In the Langmuir sheath model [13] is noted that description of heat transfer from the plasma to the electrode using plasma thermal conductivities cannot be used in collisionless layer.Instead, heat transfer from the plasma to the electrode is determined by the charged particle fluxes through the layer.
The presheath zone is the region of non-isothermal quasineutral plasma, where the electrons and ions are generated via thermal ionization [12].This consideration assumes that Debye screening length r D is essentially smaller than the mean free path Eq. ( 1).However, the metal vapor from consumable anode influences the physical properties of arc plasma and, consequently, the processes of mass, energy and charge transfer [7,14].
In this case, as it demonstrated below (Fig. 3), screening length near anode also 1 D r m ∼ µ .Therefore, near-anode layer cannot be divided into sheath and presheath in this case, and for atoms, which are evaporated from electrode, Fourier's law with Newton's law of cooling can be used and allow defining the temperature gradient near electrode surface [15]: where h c is the heat transfer coefficient, k is the thermal conductivity, z is the coordinate directed from welding wire to molten pool, and radiation loss is neglected.The heat transfer coefficient for wire electrode 16], and thermal conductivity is ( ) where ρ = n a m a is the mass density; m a is the iron atom mass; c p = 5R/2µ is the isobaric heat capacity; µ is the iron molecular weight; is the atom thermal velocity, R is the ideal gas constant.
Thus, it is possible to defined the characteristic length for Eq.(2) as where ( ) / 3Q This equation has cumbersome solution that, however, can be approximated by function ( ) which is presented in Fig. 1 together with solution of Eq. ( 4).The electrode tip temperature is T w = 3000 K; the temperature in arc region is T pl = 15000 K.The definition T h is used for heavy component (ions and atoms).

Space distributions of gas particles.
The vapor mass flow from consumable electrode wire is 0 is the iron vapour mass flux from electrode in the Hertz-Knudsen-Langmuir approximation [17] and Aw is the tip surface area.This flow is mixed with shielding gas mass flow J sg : . The mixture flow divergence along coordinate z in the linear approach is where z 0 is the mixing length, which is defined as z 0 = kd w ; d w is the welding wire diameter; k = 0.4 is the Karman-Prandtl constant [18].From Eq. ( 7) follows The analysis of the experimental data [19] demonstrates that distribution of ratio g Fe /g sg along coordinate z is described well by Eq. (8), which allows to calculate the available for ionization atom number density ( ) where p atm is the atmospheric pressure.The atom ionization in the arc column occurs via electron-atom collisions and via photoionization by UV radiation.However, photoionization performs weak effect on the ionization balance [20], which, therefore, can be determined by Saha equation [21] ( )exp ( ) where n e , n i and n a are the average number densities of electrons, ions and atoms, respectively; n a = n A − n i ; Σ i and Σ a are the ion and atom statistical weights; ν e (T) =  ( ) m k T πℏ is the effective density of the electron states; E ion is the atom ionization energy; ℏ is the Plank constant; K S is the Saha constant.
In the unperturbed plasma the neutrality condition n e = n i = n 0 is valid, where n 0 is the unperturbed number density, which value follows from Saha equation for each component: and atom number density for each component is The space distribution of the unperturbed number density n0(z) is presented in Fig. 2. Distributions of the iron atoms' number density and the number density of shielding gas atoms are also presented for two types of shielding gas: argon and carbon dioxide.The carbon dioxide is dissociated in the arc plasma to CO, O and C; however, CO2, CO and O have ionization potential about 14 eV.Carbon ionization potential is 11.3 eV.Therefore CO2, CO and O atoms are combined in n aCO .The temperature dependency of carbon dioxide decomposition g j0 (T) from [22] was used.Almost all iron atoms are ionized at the distance from anode more than 1 mm, because they have lowermost ionization potential (7.9 eV) in this system.

Mean free paths of gas particles.
Mean free path of the particle is defined by its thermal velocity 8 / T B v k T m = π and collision frequency with target particles f:

The collision frequency is
where n 2 is the target particle number density; Q 1,2 is the particle collision cross section.
The collision cross section of the neutral particles is determined by their radiuses, for example, the atom-atom collision is described by cross section 2 2

(
) The collision cross section of the charged particles is determined by impact parameter b 0 = e 2 /3k B T e (for single charge) [24], for example, the electron-ion collision is described by cross section is the Debye screening length, where T e is the electron temperature [3,4].
Because the temperature varies along z in the boundary layer, the collision frequency also is changed, and those or other kinds of collision can prevail in the different layer areas.Therefore, resulting collision frequency is defined as sum of individual frequencies: f e = f ei + f ea for electrons and f i = f ii + f ia for ions (scattering on electrons is neglected because of a little momentum transfer).
Thus, the mean free paths for electrons and ions are 0 The space distributions of electron and ion mean free paths in the boundary layer are presented in Fig. 3 together with Debye screening length.In addition, the mean free paths for ion-ion and ion-atom collisions are presented in this figure, from which clear that ion-atom collisions dominate at electrode surface.However, the situation becomes reverse outside the Knudsen layer, where λ ia >> λ ii and, therefore, ion thermalization occurs, i.e. ions acquire common velocity.
The particular interest represents the distance from which electron or ion reaches the electrode surface (λ(z) = z): for electron this distance is 0 1.7 e m λ ≅ µ and for ion 0 1.4 i m λ ≅ µ in the system under consideration.
It should be taken into account that electron mean free path is greater than ion and atom ones.Therefore, cooling of electrons in the boundary layer must lag from the temperature of heavy components.The electron temperature can be calculated via consideration of energy transfer in collisions [25]: where ξ = z b − z is the coordinate which is digitized from the sheath boundary z b , where ξ = 0 and T e = T h ; λ e = λ e (ξ) and T h (ξ) is described by Eq. ( 5).
The result of numerical solving of Eq. ( 14) for z b = 50 µm is presented in Fig. 4.This solution is approximated well by following function: from which follows that electron temperature at the anode surface is T ew ∼ 5800 K.
For molten pool equations ( 4) and ( 14) should be solved with temperature T m = 2000 K. Then electron temperature at the cathode surface is T em ∼ 5000 K.
The Coulomb logarithm, which is necessary for calculation of electron and ion mean free pathes, contains the Debye screening length, which depends on the electron temperature.Meantime, the parameter λ e0 is necessary for calculation the electron temperature by equation (15).This problem was solved by approximation method, where λ e0 was chosen as a parameter for minimization of difference between input and output values.Thus, the dependencies ( 5) and (15) were taking into account for calculation of particle mean free paths λ(z) and Debye screening length r D (z), which are presented in Fig. 3.

Equilibrium electric potential.
The welding arc is the nonequilibrium system with high value of current.However, it can be considered as a hypothetical equilibrium system without current, where current causes the deviation from equilibrium values.The electric potential space distribution between two flat electrodes in the equilibrium plasma as solution of Poisson equation is presented in Jacobi elliptic functions [26].If the distance between electrodes L > 8r D , the potential space distribution with respect to the bulk plasma potential can be described in the following form: where Φ = eϕ/k B T is the dimensionless relative potential; x = z/r D is the dimensionless coordinate; s w Φ is the dimensionless relative potential of welding wire with coordinate z = 0; s m Φ is the dimensionless relative potential of molten pool with coordinate z = L; x inf is a value of the coordinate which approaching the potential asymptotically tends to infinity: Boundary relative potential distributions can be obtained from Eq. ( 16) in a semi-infinite approach for wire electrode ( ) and for molten pool ( ) The resulting total potential is the sum φ = ϕ+φ pl , where the bulk plasma potential φpl characterizes the size of operation that is necessary for the plasma to gain some volumetric charge [27].The bulk plasma potential φ pl for the plasma layer is determined by both boundary conditions [26], The equilibrium relative potential of electrode (without current) is determined via balance of the electron and ion fluxes on the electrode surface, and in the collisionless plasma it is represented as the floating potential of Langmuir probe with taken into account the thermionic emission from electrode, which is described by Richardson's equation ( ) ( ) where is the electrode work function W with Schottky correction; E w is the surface field strength [26]: Φ is the potential barrier in the plasma near electrode wire in the arc current absence.The Schottky correction describes the decrease in electron work function W as a result of the image force action [28] and only for negative potential barrier is applicable.The electron adsorption flux through the Knudsen layer [29] 0 and the ion flux to electrode through the potential barrier 0 exp , 0 4 , 0 The equilibrium potential barrier near molten pool is determined by following equation 0 ( ) ln exp where m eT v is the electron thermal velocity; m iT v is the ion thermal velocity; m B v is the Bohm velocity and 0 m n is the unperturbed number density at the molten pool.In the system under consideration the equilibrium relative potentials are: The total potential space distribution φ(z) = ϕ(z) + φ pl is presented in Fig. 5 together with field distribution, which strength is determined by following equation [26] ( ) ( ) ( ) ( ) ( ) Presented in figure 5 potential and field distributions are calculated for the plasma without arc current and correspond to modeling of authors [1] for low current and electrode temperatures of 3000 K and 4000 K.The electrode potential barrier decreases with increasing T w .

Electric current across arc.
The total arc voltage U c can be presented as sum of voltage drops at electrode wire δU w , at arc column δU arc and at molten pool δU m : U c = δU w +δU arc +δU m .In addition, the current I c through system allows defining all voltage drops.In present consideration, when electrode wire is anode, U c and I c are positive, when current directed from electrode wire to molten pool.
Ohm's Law determines the current across arc column c arc arc arc The potential barrier in the plasma near electrode wire is determined by equilibrium value and voltage drop: ( ) where 2 / 4 , d w is the diameter of anode spot.Thus, Eq. ( 30) allows determining the voltage drop on electrode wire as a function of arc current δU w (I c ) via numerical calculation, as a root of equation The current through molten pool is the sum: ( ) where 2 / 4 , where d m is the diameter of cathode spot on the molten pool and current components are defined by similar equations, but with others parameters and signs.Besides, it should be considered that high potential barrier near cathode provides the field emission with flux m eE j .The inward current in molten pool is positive and potential barrier remains negative: where s m ϕ is the equilibrium relative potential of molten pool, which is defined by Eq. (24).
The thermionic emission flux is The field emission is described by Fowler-Nordheim equation: 16 3 The electron flux through Knudsen layer and the ion flux are Then, Eq. ( 31) allows determining the voltage drop on molten pool as a function of arc current δU m (I c ).The dependencies of potential barrier heights ( 29) and ( 32) on the arc current are presented in Fig. 6 for electrode wire (V w ) and molten pool (V m ).The following parameters were used: diameter of wire anode spot d w = 1.2mm (on the tip of electrode wire the metal droplet is formed in the welding process, and diameter of this droplet can exceed wire diameter); arc column average diameter d arc = 5 mm; diameter of molten pool cathode spot d m = 6 mm; arc length L = 14 mm; T w = 3000K; T m = 2000K; T pl = 15000 K.
The total arc voltage is and potential distribution between electrode wire and grounded molten pool can be describe by following equations From calculations follows that basic voltage drop happens in the thin plasma layer near cathode (molten pool), which is caused by need to maintain the large field strength for effective electron emission.The potential barrier near anode surface remains negative at small currents.The increase in current leads to change of the barrier sign to positive.

Heat transfer in arc and metal vapor generation.
The electron and ion currents provide not only transport of charges, but also energy transfer through electrode surface.It should be considered that collision of ion with electrode wire (anode) leads to its neutralization and provides transfer of the potential energy, which is equal to (E ion −W), and the kinetic energy, which is defined by the average ion motion energy outside the Knudsen layer 3 / 2 and by the energy of acceleration in the sheath electric field −V w [3], if the potential barrier V w < 0. Thus, flow of energy transfer via ion-anode collisions is described by equation 3 , 0 2 3 , 0 2 In this case, the work function without Schottky correction is used, because the image-force concept, which is a source of Schottky effect, is inapplicable to the electron exchange between adsorbed ion and electrode surface.
The adsorbed electron provides transfer of energy, which is equal to sum of the average motion energy 3 / 2 , the energy of acceleration V w , if the potential barrier is positive, and the work function as a potential energy.The flow of energy transfer via electron adsorption is described by equation The electron leaving the anode via emission provides loss in energy, which is equal to the work function: , 0 , 0 The evaporation flux (6) provides loss in energy: where is the heat flux, which occurs via radiation, ε w is the emissivity of the welding wire, σ SB is the Stefan-Boltzmann constant; κ w ∇T w is the heat flux along the electrode wire with thermal conductivity κ w .
where 3 2 Under stationary conditions, the inflow of heat in molten pool Γ m should be counterbalanced with outflow of heat via metal evaporation and spraying to environment.Therefore, if spraying of metal from molten pool is neglected, flow of metal vapor Γ vap = Γ m .Fig. 7 demonstrates the dependency of vapor mass flow, which emitted by molten pool, on the arc current, which corresponds with experimental data [31,32].Growth of current leads to increase in the potential barrier −V m , that causes increase in heat flow Γ and reducing of m e Γ .These competing trends cause existence of the minimum in dependence, which is presented in Fig. 7.
The potential barrier near electrode wire Vw change its sign at some value of arc current.This change provides sharp increase in energy flow, which is transferred from the plasma to anode by conductivity electrons, because they are accelerated in positive potential barrier.This effect can be a reason for transition of metal transfer in welding process from globular to spraying regime [33].

Vapor ionization.
The high-temperature metal vapor from welding arc mixes with shielding gas under the welding torch nozzle.The numerical simulation [34] demonstrates that the gas average temperature in the boundary region at the torch outlet T ∼ 2500 K.It means that condensation of metal vapor occurs in the vaporshielding gas mixture, before mixing with air.
The high-temperature vapor contains several elements, which concentrations depend on compositions of electrode and welding material.However, for simplicity, it is possible to assume that vapor contains only iron atoms.The flow of iron vapor ( 37) is mixed with shielding gas, and the mixture flow divergence can be described by equation, similar to Eq. ( 7).Thus, it is possible to determine the available for ionization atom number density as a function of distance from arc boundary r in the following form ( ) where m a is the iron atom mass; v vap ∼ 10m/s is the vapor flow velocity; l mix isthe mixing length scale, which is determined experimentally and in the system under consideration l mix ∼ 1.7 cm [35,36].
The mixing of high-temperature vapor with cold shielding gas also determines the temperature of mixture, which can be described by following equation [36] ( )

(
) where T sg = 300K is the temperature of shielding gas; T vap = 3000K is the temperature of vapor.Ionization of atoms Eq. ( 38) can occur by electron impact or by UV radiation.In the paper [20] was demonstrated that relaxation time of UV ionization at the arc boundary τ UV ∼ 10 µs, and for thermal impact ionization such relaxation time τ T ∼ 0.02 µs, that is much less.Therefore, it is possible to describe atoms' ionization by Saha equation (10) and determine the unperturbed number density as with taken into account Eqs. ( 38) and (39).

Nucleation.
The cooling of vapor-gas mixture provides the increase in supersaturation S = p/p sat , where p = n A /k B T is the partial pressure of iron vapor, and at some distance from arc boundary rn condensation begins, when S ≥ 1.The conden-sation occurs in the plasma as a heterogeneous ion-induced nucleation and provides the change in Gibbs free energy ∆G, which is described in the papers [37,38,39], where was taken into account the exchange of charges and energy between nucleus and environment: where a is the nucleus radius; γ = γ 0 a/(a + 2δ) is the surface free energy of the nucleus; γ 0 is the surface free energy of the flat; δ is the Tolmen length; m a is the mass of the condensable atoms; ρ is the nucleus density; S is the supersaturation of the condensable material; Z n is the nucleus charge number; ε is the dielectric constant; r i is the radius of the single-charged positive ion, which induced the nucleation; E γ is the change in surface free energy as a result of the electrical double layer on the nucleus surface; E ex is the change in Gibbs free energy as a result of the interphase energy exchange; E q is the change in Gibbs free energy as a result of the nucleus charging.The temperature of nucleation start together with distance between nucleation zone and arc boundary r n in Fig. 8 is demonstrated as a function of arc current I c .This dependence is similar to dependence in Fig. 7, because vapor flow from arc J vap determines the iron partial pressure in vapor-gas mixture and, accordingly, the supersaturation.
It should be noted, that the plasma not contains atoms of alkali metals when GMAW is used.Therefore, the iron atoms execute both functions: a source of ionization and a source of nucleation, i.e. the plasma ionization balance is provided by ions of iron, which number density n i should be excluded from atom number density, which available for nucleation.This feature was described in paper [36], from which the equation for number density of nuclei follows: where is the number of atoms in the nucleus; ρ is the nucleus density; ∆G(a n ) is the change in Gibbs free energy under formation of equilibrium nucleus with radius an.
In the welding fume from GMAW equilibrium nucleus number density much greater equilibrium (Saha) electron and ion number densities [36].Because ions are the centers of ion-induced nucleation, they disappear from gas phase under nucleation and electron-ion recombination is replaced by the neutralization of nuclei, which have initial positive charge.As a result, the balance between ionization and recombination is broken in favor of ionization, and new electrons and ions appear in the plasma via impact atom ionization, until electrons are being captured by nuclei.Therefore, achievement of equilibrium nucleus number is the time-phased process, which includes additional atom ionization and formation of additional nuclei until equilibrium will be reached.
The appearance of nuclei creates a new ionization-recombination channel.Atom ionization and ion recombination occurs via interaction with nucleus surface.Surface atom ionization degree is described by well-known Saha-Langmuir equation [40].When the first nuclei are formed, their large number causes the capturing of electrons and leads to change of ionization mechanism.Initial thermal ionization is replaced by ionization via nuclei surface.Accordingly, the energy exchange term in Gibbs free energy is also replaced: the electron transition between nucleus Fermi level and valence electron level of the adsorbed atom should be used instead of transition between nucleus and gas.Surface atom ionization and ion recombination provide increase in ion number density.Thus, welding fume plasma in nucleation zone consist mainly of positive ions and nuclei with average negative charge if GMAW is used [41].
The two kinds of nuclei are formed by the heterogeneous ion-induced nucleation: the equilibrium nuclei with radius a n , which are in the equilibrium state with the environment; and the nonequilibrium critical nuclei with radius a cr , which appear as a result of fluctuations.The radius of the equilibrium nucleus is determined as a minimum of the function ∆G(a), and the radius of the critical nucleus is determined as the maximum of the function ∆G(a).The equilibrium nucleus is in the stable stage and some activation energy is necessary for the nucleus growth E act = ∆G(a cr ) − ∆G(a n ).This activation energy decreases down to zero (E act → 0) with the vapor-gas mixture cooling.After that, the unrestricted growth of nuclei begins, until the condensable materials do not deplete.The temperature, which corresponds to nucleation termination in Fig. 8 also is presented as a function of arc current.

Droplet growth.
The nuclei growth occurs via two mechanisms: condensation growth and coagulation, which is a coalescence, because nuclei are in the liquid state.Condensation growth of nucleus with radius an is determined by the increment in the mass due the adsorption and transpiration fluxes difference, and is described in the free molecular regime by following equation [35] 1 4 where α c is the eveporation-condensation coefficient [42]; n a = n A −n i −N an n n is the number density of iron atoms in the gas phase; S is the current supersaturation; S R = p sat (a n )/p sat (∞) is the change in vapor partial pressure at the account of the surface curvature and interphase interaction.
The large number density of nuclei n n ∼ 10 16 cm −3 causes their intensive Brownian collisions and coalescence.As a result, the coagulated droplets with radius a cd , which grow through coalescence and condensation are formed.However, it is necessary to take into account that the nucleation continues, and the system thermodynamics requires the presence of the nuclei with equilibrium number density (41).Therefore, number and size of nuclei in the nucleation zone cannot change via coalescence, because new nuclei appear.This requirement is removed after the nucleation termination.
Thus, already at the initial stage of nucleation the bimodal size distribution of the droplets occurs.The first mode contains the droplets of nuclei; coagulated drop-lets resulting from the long-term coalescence and condensation represent the second mode.It should be noted that in this case nucleation is the original "pump" for the transfer of the atoms from the gas phase into the coagulated droplets, and intensity of this process can be higher than the intensity of the vapor condensation (42) on the aggregated droplets [43].
Droplets can be described by a log-normal size distribution, which is based on the number of atoms N contained in the droplets, using probability density function for two modes: where index n is used for nuclei, index cd is used for coagulated droplets; The evolution of such a system can be described by the integral moments of the distribution (43) [43].The moments are described by the following equation: , and the Brownian coagulation can be described as where N and N′ are the numbers of atoms in the colliding droplets; β(N,N′) is the collision kernel.The Brownian collision kernel can be determined by the kinetic theory of gases, or by the diffusion theory according to droplet size.If the droplets are much smaller than mean free path of the gas particles, the gas-kinetic theory should be used to determine the collision kernel ( ) ( ) where Coulomb interaction is neglected.The evolution of the moments (44) can be determined for each mode (43).The zero moments represent the total number densities of the generated particles n n and n cd .The total numbers of atoms in these are determined by the first moments.As a result, the average number of atoms in the droplets of each mode is (1) (0) and the standard deviations are defined by the following equations: (0) (2) ln ln ( 1) Equations ( 43)-( 46) allow to describe the evolution of the coalescence in the bimodal system of droplets with taking into account both the intramodal coalescence and the intermodal association of the different droplet modes.
The growth of droplets occurs in the cooling environment and continues up to phase transition.The temperature of the phase transition depend on the droplet radius [44], i.e. the droplets of the coagulated mode (the melting point is about 1730 K) solidify earlier, than droplets of the nuclei mode (the melting point is about 1330 K); and exists the transient stage, when the coagulated mode is presented by solid particles, the nuclei mode is the liquid droplets.The bimodal coalescence terminates, when coagulated droplets become the solid particles.There is only coalescence of the nuclei mode.
Calculation of the droplets' growth gives the parameters of primary particles size distribution after solidification, which is presented in the table 1. Increase in the arc current leads to growth of particles' sizes, but their number densities are changed little, because vapor flow rate is increased with current.

Formation of welding fume via coagulation of primary particles.
The agglomerates generated from the solid primary particles are the irregular structures; therefore, the radius of agglomerate, which contains N primary particles (monomers), is described as , where a is the radius of the monomers; D f is the fractal dimension (D f = 3 for perfect sphere).
The collision kernel for the free molecule regime without electrical interaction between particles is described by the following equation [45]: where ρ is the monomer density; N and N′ are the numbers of monomers in the colliding agglomerates with fractal dimension D f .The collision kernel for charged particles is [46] ( ) ( ) , , , exp where U N,N′ is the interaction energy on the shortest distance between the interacting particles, which can be considered in the Coulomb approximation ( ) where Z N and Z′ N are the particle charges divided by the elementary charge, which are determined by following equation which under condition r ag V b ≪ r D k B T can be described in Coulomb approach: The calculation of coagulati method of moments while agglomerates' sizes less than mean free path of the particles.When sizes exceed the mean free path length, Eq. ( 49) becomes ble.In this case the collision kernel shoul ( ) where η is the viscosity; C(a

Conclusion.
Thus, modeling of processes, which are occur in the electric arc during arc ing, allows determining the rate of high to environment.It is very important parameter, because metal a source of welding fumes and inhalable particles in the The resulting collision kernel is The calculation of coagulation is carried out in the free molecular regime by method of moments while agglomerates' sizes less than mean free path of the particles.When sizes exceed the mean free path length, Eq. ( 49) becomes ble.In this case the collision kernel should be described in the diffusion ( ) ( ) The calculation results presented in Fig. 9 as the inhalable particle size tions for two values of the arc current, which correspond to minimum temperature vapor flow rates.
Thus, modeling of processes, which are occur in the electric arc during arc ng, allows determining the rate of high-temperature vapors efflux from to environment.It is very important parameter, because metal vapors' condensation is a source of welding fumes and inhalable particles in the breathing zone [35,39,49].
Inhalable particle size distribu-= 45A (solid) and (49) on is carried out in the free molecular regime by method of moments while agglomerates' sizes less than mean free path of the gas particles.When sizes exceed the mean free path length, Eq. ( 49) becomes inapplicad be described in the diffusion regime: ) is the Canningham slip correction [48] .9 as the inhalable particle size distributions for two values of the arc current, which correspond to minimum and maximum Thus, modeling of processes, which are occur in the electric arc during arc weldtemperature vapors efflux from molten pool vapors' condensation is breathing zone [35,39,49].
The equilibrium in contact of electrode wire or molten pool with plasma, which is produced by arc, provides the negative potential barrier in the plasma near electrodes.The electric current across the arc provides increase in negative barrier near cathode and its decreasing near anode (the electrode wire is anode in the system under consideration).At some value of current the change in barrier sign occurs and further current amplification leads to increase in the positive potential barrier near anode.The change in potential barrier sign provides sharp increase in energy flow, which is transferred from the plasma to anode by conductivity electrons, because they are accelerated in positive potential barrier.This effect can be a reason for transition of metal transfer in welding process from globular to spraying regime.
Increase in the arc current leads to growth of high-temperature vapor flow rate from welding arc to environment and provides increase in the size of solid primary particles, because occurs the shift of nucleation zone toward the arc, i.e. in the area with higher temperature; and, accordingly, increase in the time duration of droplets' growth.This tendency persists in the inhalable particle size distribution after coagulation of primary particles, what is correlated with experimental data [50].Thus, the fumes produced by welding with high current can easier be filtered.
Ключові слова: зварювальна дуга, плазма, пари металу, конденсація, коагуляція collision cross section; r a is the atom radius; number density near electrode tip surface; k B is the Boltzmann constant; p sat is the saturated vapor pressure over electrode surface, system under consideration 15 pl l m ≅ µ ).Accordingly, Eq. (2) can be presented in the following form

Fig. 1 :
Fig. 1: Heavy component temperature space distribution in electrode layer: grey solid curve is the solution of Eq. (4); black dashed curve is the function Eq. (5).

Fig 2 :
Fig 2: Space distributions of unperturbed number density n0, atom number density for iron n aFe for two kinds of shielding gas: pure argon (n aAr ) and pure carbon dioxide (n aC and n aCO ).

Fig. 3 :
Fig. 3: Space distributions of the mean free path for ion-ion collisions λ ii , ionatom collisions λ ia , ion total λ i , electron total λ e and Debye screening length r D .

Fig. 4 :
Fig. 4: Temperature space distribution in electrode layer: dashed curve is the electron temperature; solid curve is the temperature of heavy component.

Fig 5 :
Fig 5: Space distribution of the total potential φ (solid) and field E (dashed) in equilibrium boundary layer for electrode temperature 3000 K (black) and 4000 K (grey).

Fig 6 :
Fig 6: Dependencies of potential barrier height on the arc current for electrode wire (solid) and molten pool (dashed)..
velocity in arc region; E arc is the field, which is defined by the arc voltage drop and arc length: E arc = δU arc /L.Hence, the voltage drop at arc column as a function of current is c arc arc arc

ϕ
is the equilibrium relative potential (without current) of electrode wire, which is defined by Eq. (23):/ electrode wire surface is the sum of emission current and currents of electron and ion fluxes to electrode from the plasma through Knudsen layer:

Fig. 7 :Fig. 8 :
Fig. 7: Vapor formation rate.Fig. 8: Dependencies of the temperature of nucleation start and finish; and the distances from arc boundary to beginning and end of nucleation zone on the arc current the median of distribution; σ is the standard deviation; a N is the average number of atoms in the droplets.
is the mean free path.The calculation results presented in Fig tions for two values of the arc current, which correspond to minimum high-temperature vapor flow rates.

Fig 9 :
Fig 9: Inhalable particle size distrib tions for arc current I c I c = 155A (dashed) the mean free path.