 Advances in Ma terials Physics and Che mist ry, 2012, 2, 270-273 doi:10.4236/ampc.2012.24B069 Published Online December 2012 (htt p://www.SciRP.org/journal/ampc) Copyright © 2012 SciRes. AMPC Simulation of Thermophysical Processes at Laser Welding of Alloys Containing Refractory Nanoparticle s Anatoly N. Cherepanov1, Vasily P. Shapeev1, Guangxun Liu2, Lamei Cao2 1Khristianovich Institute of Theoretical and Applied Mechanics SB RAS; Novosibirsk State University, Novosibirsk, Russia 2Beijing Institute of Aeronautical Materials, Beijing, China Received 2012 ABSTRACT Mathematical model is formulated for description of thermophysical processes at laser welding of metal plates for the case when modifying nanoparticles of refractory compounds (nanopowder inoculators – NPI) are introduced into the weld pool. Specially pre-pared n anoparti cles of refractor y compo und s serve here as cr ystallizatio n cent ers, i.e. in fact th ey are exogeno us ino culants o n which surface clu st ers are gro uped . This can be us ed to contro l the melt cryst alli zatio n proces s and formatio n of its stru ctur e, and, ther efore, properties of the weld seam. As an example, calculation results of the butt welding of aluminum alloy and steel plates are p resent ed. The results of calculation and experimen tal data compariso n are shown. Keywords: Mathematical Model; Laser Welding; Nanopowder Inoculators; Crystallization Process; Calculation Results 1. Introduction In the last years increasing attention has been paid to develop-ment of technology for laser welding of metal products. In this connection, development of appropriate mathematical models and numerical algorithms for their implementation is a pressing problem [1,2]. A mathe matical model is developed in this work for descrip-tion of thermophysical processes at laser welding of metal plates for the case when modifying nanoparticles of refractory compounds (nitrides, oxides, etc) are introduced in the molten pool. The mathematical model proposed is based on non-equilibrium birth and growth of crystal phase on the in-oculants, which are the nanoparticles, with use of the Kolmo-gorov’s theory for calculation of the solid phase fraction. At that, the homogeneous nucleation can be neglect ed. 2. Physicomathe ma tical Mod el o f the Process Let us consider a steady-state process of two plates butt weld-ing. We introduce the Cartesian coordinate system with x axis lying on the plates upper surface, z axis coinciding with the symmetr y axis an d direction of the laser beam operation, and y axis perpendicular to the joint. The coordinates origin lies on intersection of the beam axis and the plates upper surfaces (Figure 1). The beam moves along the joint in negative direc-tion of x axis with constant welding speed v=const. The metal of the welded plates is protected from oxidation by an inert gas blow that carries away a part of the metal vapors. We assume the thermophysical parameters to be constant and equal to their average values in the temperature range under consideration. Due to the small concentration of the dispersed nanoparticles (~0.05% by mass), their influence on the physical parameters of the alloy can be neglected. The alloy is considered to be a bi-nary system. With these assumptions, the three-dimensional equation of heat transfer in the weldpool and solid metal in the moving coordinate system takes the form: 2 2212 22sii iTT TTfcv vx xyzxρ λδκρ∂∂ ∂∂∂=++ −∂ ∂∂∂∂ (1) where ci, λi, ρi are specific heat, heat conductivity, and density of the i-th phase, respectively (indices i = 1, 2, 3 denote para-meters of solid, two-phase, and liquid states of the metal); δ = 0 in the melting zone; δ = 1 in the crystallization zone; fs is a share of the solid phase; fl =1–fs is a share of liquid phase in the two-phase zone. In variant of the model used in this paper, all particles of the product in movable coordinate system move parallel to x axis with the welding speed. The process of melt-ing is considered in Stefan problem approximation, considering the phase transition boundary to be a smooth surface, on which the conditions of thermodynamic equilibrium and heat balance are fulfilled: 21 121,( ),TTTTmnnλλ κρ∂∂=−+=⋅∂∂ vn where 0( )/2ml sTTT= + is fictitious melting temperature, 0,lsTT are temperatures of equilibrium values of liquidus and Figure 1. T he weld zone la yout: 1 – laser beam, 2 – st eam c hannel, 3 – liquid phase (molten pool), 4 – two-phase zone , 5 – solid phase. A. N. CHEREPANOV ET AL. Copyright © 2012 SciRes. AMPC 271 solidus, respectively, n is unit normal to the phase transition boundary, κ is the melting heat (heat of crystallization), ( )11/Tλ∂∂n and ( )22/Tλ−∂∂n are the heat flu xes calculated on the sides of the solid and liquid phases, respectively. This approximation is reasonable, because at high temperature gra-dients, zone of the phase transition at melting is a thin layer, which thickness is much less than the weld pool characteristic size. Considering 0( )/2ml sTTT= + to be the melting temper-ature, we can uninterruptedly proceed to modeling of nonequi-librium heterogeneous crystallization on active superdispersed seeds in region of the alloy solidification (0elTTT≤<). Here, Te is temperature of solidification completion, determined from kinetic equation ( )/lT TCKe eeeξ=−  where ,,CKeeeξ are velocity of the solidification boundary movement, impurity concentration on i t, and the kinetic co effi-cient, respectively. For the problem's simplification, the kinetic overcooling (/eeKξ) is neglected due to its smallness, and the temperature of solidification completion, similarly to the theory of quasi-equilibrium two-phase zone , is considered equal to the eutectic temperature (()e leT TC=, where eC is impurity concen tration in the eutectic po int). In th e area of solid ification (i = 2), the crystallization rate is defined by the processes of origin and growth of solid phase in overcooled alloy on seeds, which are refractory activated nanoparticl es . Con s id ering all th e nanoparticles to be centers of crystallization, we can define section of the solid phase, using the formula of Kolmogorov N.E. [4,5]: 1,sfe−Ω= − (2) where ( )034,, 3lxupp xKxy zNrTdvπξΩ =+∆∫ is volu me o f the cr ystal p hase for med in o vercoo led melt. H ere, pN is the number of nanoparticles in volume unit; xl0 is the coordinate of point x on isotherm with the temperature of li-quidus Tl0 , 0()llfT=4 /31pNeπ−≈, pr is the radius of nanopar-ticles; Ku is the constant of crystal growth in kinetic law [4-6] u =Ku [Tl(C)−T]n, where u is the growth rate, n is a physical constant (n = 1 at normal mechanism of growth, n = 2 at dislocation one); Tl(C) is the liquidus temperature, which is approximated by linear de-pendenc e on di s sol v e d (a lloy ing) component С: Tl(C) = Tl0 – β (C – C0 ), where C0 is initial concentration of the dissolved component, β is coefficient module of the liquidus line slope on the state dia-gram of corresponding binary alloy, T∆ is local overcooling, defined by equation ( ) – lT TCT∆= (3) According to (1), (2) and (3), in the crystallization zone (0elTTT≤<) appear s a source o f heat, connected with heat gen-eration during the melt crystallization. Due to nonlinear depen-dence of ()lfT, contribution of this heat generation can be taken into account by solving the equation of heat conduction iteratively, specifying on iterations lf and, consequently, T in the zo ne of crystallizat ion. Boundary conditions. At infinite distance from the radiation source , xy→∞ →∞ we assume ( )0,, .T xyzT= On the plates upper and lower surfaces (0, )z zh= = blown by the inert gas, outside the steam channel, are fulfilled conditions of complex con vective and radiat ion heat exchange with th e envi-ronment 0,0,( ).iт zh gzhTTTzλαΣ==∂−∂= (4) Here, Tg is the gas temperature, mαΣ is the total coefficient of heat transfer defined by expression ( )( )2200 0,mkmmzg zgTTTTαα εσΣ===+ ++ mε is reduced emissivity of the heat transfer surf aces, m=0, 1 for the upper and lower surfaces, respectively; 0σ is Ste-phan–Boltzmann constant, kmα is coefficient of convective heat transfer  1/2 1/30,646Re Pr,/kmmg lαλ= where Re/; P/;rmgmgg gvaνν== gmv is the gas flow ve-locity; l is the characteristic length of the cooling zone; νg, ag, λg are the gas kinematic viscosity, temperature conductivity, and heat conductivity, respectively. In the laser radiation impact zon e (on the steam channel su r-face z = Zc(x y) is fulfilled the heat balance condition: .3T Lmλ− ∇⋅=⋅− n qn (5) Here, m is mass velocity of the substance evaporation from the surface unit, related to the vapors excessive pressure P(z) necessary for keeping the channel walls from collapse. It is defined by relation v()m Pzρ=, vρ is the vapor density; L is specific heat of the alloy evaporation, q is the absorbed heat flux with the re-reflecti on taken int o account, n is unit normal to the ch annel surface. In numerical simulat ion , the last relat ion determines t he h eat flu x on the stea m chann el su rface and is the boundary condition for equation (1). We assume, that the welding is realized by CO2laser radiation with wavelength λ0 = 10,6 μm. The radiation intensity is described by Gaussian normal distribution ( )220,,exp( 2)zI xyzIrr= −/ where ( )202/ ;zIWrπ= W is the las er power, rz is the radius of the laser beam at depth z of the steam chann el, det er min ed by relation  220 , FzFFzZrr rλπ−=+  where rF is the laser beam radius i n focal plane; ZF is lo cation of the focus relative t o the upper surface s of the details welded. For density distribution of the absorbed radiation power on the surface of the steam channel with coordinates z=Zc(x, y) in the zo ne of direct i nteraction we have expressi on ( )( )( )2222, ,,exp2/efzzAWqxyZ xyrrrπ=− Here, (1 )efA AAAe=+− is the absorption effective coeffi-cient. The first member in this coefficient takes into account A. N. CHEREPANOV ET AL. Copyright © 2012 SciRes. AMPC 272 absorption of the radiation fallin g onto th e surface di rectl y fro m the laser beam, while the second member takes into account absorption of the radiation reflected repeatedly from the channel walls; ( )11neb bcnA AASSS∞==−/ (+)∑value is the equivalent absorption coefficient [ 9], S b, Sc are areas of lateral surface of the channel and inlet, respectively, A is coefficient of the laser radiation absorption by the steam channel surface. In the area reached by the reflected radiation only, efA lacks the first member. 3. Model of Steam Channel Formation If the laser power exceeds some critical value, then in the area of the b eam interactio n with the metal a stea m channel emerges. Even in the case of homogeneous material of the plates, and constant values of the radiation power and welding rate, its walls oscillate chaotically due to the hydrodynamic instability. The present model assumes that that channel walls oscillate by their time-averaged location. There are various models de-scribed in papers for considering the heat transfer from the beam to metal as well as ap pro aches to calculat ion of the stea m channel shape. Here was used a slightly modified method de-scribed in . Note that the channel wall shape turns out to be similar to that of the channel front wal l proposed i n . 4. Numerical Method For numerical solution of the problem about temperature dis-tribution in the calculation domain, a known iterative fi-nite-difference scheme of steadying for 3D heat conduction equation is used . As stated above, the heat balance equa-tions on different surfaces of the plates (4) were used as the boundary conditions. Relation (5) was taken into account on the steam channel surface. At a significant distance from the mol-ten pool, i.e. on a remote boundary of the calculation domain, the temperat ure was set equal to that of the environment. 5. Some Results of Numerical Simulation Some results of the temperature fields calculations in the mol-ten pool and surrounding layers of the solid alloy are shown in the Figure 4. Boundaries of the pool, two-phase crystall ization zone were determined from characteristic temperature values. The beam radius Fr in the focal plane was chosen as the length scale. Figure 4 shows dimensionless temperature nor-malized to liquidus temperature Tl. The picture in a small part of the calculation domain is shown in the figures for the pur-poses of obviousness. The numerical calculations were carried out for alloy Аl + 10% Si (% of mass) at the same thermophys-ical paramet ers as in [ 10,1 3]. For exampl e, laser p ower W=3.18 kW, welding rate v=4.7 m/min, plate thickness h=1.5mm. One can see in Figure 2 that the most warmed-up areas as well as the largest temperature gradients in calculation domain are located n ear th e steam chan nel surface. One can also n ot ice that the temperature on a sufficiently large part of front surface of the steam channel near the laser beam axis is close to the boil-ing-point temperature, that correlates with the channel con-struc ti o n method des c r ibe d i n [1 0 ]. Figure 2. Temperature field and isotherms in calculation domain (cross-section y=0): 1 – steam channel, 2 – T = 2629.1 K (boil-ing-point temperature), 3 – T = 21 55 K , 4 – T = 1724 K, 5 – Tl0 = 862 K (equilibrium liquidus temperature), 6 – T = 420 K. Figure 3. Temperature field and isotherms in calculation domain (view from above, plane z = 0): 1 – ste am cha nnel, 4 – T = T = 1293 K, 5 – Tl0 = 862 K (equilibium liquidus tem p eratu re) , 6 –T=560. 3 K . Figure 4. Non-dimensional temperature profile in the crystalliza-tion zone. The x axis begins at xl0. Curve 1 corresponds to Np = 1018m-3, curve 2 Np =1019m-3. Particularity of alloy crystallization with a modifying nano-powder is overcooling taking place in the region of emergence and growth of crystal phase, that causes non-monotony of the temperature variation along the x coordinate (Figure 4). Max-imal value of overcooling is ~5 K and it depends on nanopar-ticles quantity pN in the melt volume unit. Welding simulation of carbon steel plates was carried out as well. The steel chemical composition: С:0.17-0.24; Mn:0.35- 0.65; Si:0.17-0.37; S:0.04; P:0.35; Ni:0.3; Cu:0.3; Cr: 0.25. The welding was performed with CO2 laser with po wer W = 5. 2 kW and 2.3 kW, the welding rate Vw = 2 m/min and 0.6 m/min, ZF = 0 mm. Figure 5(a) shows photo of the weld seam cross-section obtained in an experiment, Figure 5(b) shows shape of this section obtained from the numerical simulation. The comparison shows satisfactory agreement between the results of calculation and experiment. A. N. CHEREPANOV ET AL. Copyright © 2012 SciRes. AMPC 273 (a) (b) (c) (d) Figure 5. Morphology comparison of cross-sections of weld seam obtained by the laser welding (a), (c) and by the numerical simula-tion (b), (d) at various powers of the laser radiation and the welding rate: ( а), (b) W = 5.2 kW, V = 2 m/min; ( с), (d) W = 2,3 kW , V = 0.6 m/ mi n . 6. Conclusions A mathematical model of crystallization of a multi-component alloy modified by active nano-dispersed inoculators is built. It allows of analyzing influence of the inoculators concentration and t hei r size o n the stru ctu re formati on p rocesses. Part icu larit y of heterogeneous crystallization of alloy with modifying nanop owder is appearance of overcooling in the area of birth and growth of the crystal phase. Comparison of the numerical si-mulation and experiment data shows satisfactory agreement of the results, that demonstrates a sufficient adequacy of the ma-thematical model proposed. 7. 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