NUMERICAL SIMULATION OF MODIFYING MATERIAL DISTRIBUTION DURING THE IMPULSE INDUCTION HEATING OF METAL SURFACE

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Introduction. The use of constructions and their components operated in extreme conditions excluding repair or replacement makes urgent the increase in their properties such as thermo - and wear-resistance, plasticity, durability etc. A perspective way of improvement of details operational properties is modifying of the melt at laser processing of their surfaces by the prepared nano- size particles of refractory compounds (carbides, nitrides, carbonitrides and etc.) that allows to increase the number of the centers of crystallization and to crush structural components of the solid metal, increasing its uniformity [1; 2]. At the same time along with the wide use of the laser during modifying and alloying of metal surface, the use of energy of the high-frequency induction field [3; 4] is also possible. However nowadays possibilities of such technology are insufficiently studied especially regarding the use of nano-size modifiers for material structure improvement [4]. As part of the study of the processes going on in a metal surface affected by a laser impulse it is defined that one of the main factors influencing the efficiency of modifying is thermocapillary convection which under certain conditions can promote homogeneous distribution of the insertion materials which penetrate during submelt- ing into melt [5; 6]. In [7; 8] the influence of surface- active substance dissolved in the melt on the nature of convection is considered and defined that the structure of flows in molten metal depends on composition and amount of the present impurities in it. In this regard pos- sible changes in technology of modifying of a surface make necessary a detailed research of the happening processes taking into account perhaps more influencing factors. In this work with the use of numerical modifying the impact of an impulse of high-frequency electromagnetic field on the distribution of the modifying material pene- trating into a substrate from its surface in the presence of surface-active substance in metal is considered. As an object of research iron alloy (Fe) + 0.42 % on carbon weight (C) + 0.001-0.02 % sulfur (S) is used. This system is chosen in view of the existence in literature [9; 10] the thermophysical parameters and physical constants neces- sary for calculations and analysis of the happening proc- esses. By means of the offered mathematical model describing thermo - and hydrodynamic phenomena, the non-stationary processes including metal heating, its melt- ing, convective heat transfer in the melt and further after the termination of an impulse its solidification are consid- ered. According to the results of numerical experiments the modes of impulse impact on a substrate connected with an amount of surface-active substance in metal and promoting homogeneous distribution of the modifying nano-size particles in the melt are defined. Mathematical model and algorithm of realization. The impact of an impulse of the high-frequency electro- magnetic field on a metal substrate is considered. The scheme of the process is illustrated in fig. 1. Duration of impulse tH . The cylindrical head of the electromagnetic field inductor is located over the flat surface of an iron alloy sheet. Induction influence is carried out through r0 radius spot. The distribution of electromagnetic energy in metal is described by the empirical formulas used in engineering thermal calculations of induction heaters [11]. The surface of a substrate is covered with a layer of spe- cially prepared modifying nano-size particles of refractory compounds which serve as active centers of crystalliza- tion after penetration into the melt [2]. Affected by energy metal is heated and melted. Phase transition happens at a substrate material melting temperature Tm. In the fluid cavity formed and increasing over time the convection under the influence of capillary and thermogravitational forces develops. The moistened particles under the influ- ence of convective flows penetrate into the melt depth. On the border of a phase transition sticking conditions are satisfied. After completion of an impulse due to heat ex- change to the environment and heat removal in not warmed material of a substrate the cooling of the melt and its volume-consecutive crystallization occurs in the as- sumption that all penetrated modifying particles are its centers. Fig. 1. Scheme of an inductive impulse influence: 1 - an inductor with the covering ferrite magneto- screen; 2 - a metal substrate and particles of the modifying material (3) on a surface; 4 - the melt; 5 - border of the melted zone; zg and rg - depth and radius of the considered area in a substrate Рис. 1. Схема воздействия индукционного импульса: 1 - индуктор с охватывающим ферритовым магнитоэкраном; 2 - металлическая подложка и частицы модифицирующего материала (3) на поверхности; 4 - расплав; 5 - граница про- плавленной зоны; zg и rg - глубина и радиус рассматриваемой области в подложке For simplification of the task it is considered that thermal characteristics of fluid, solid and two-phase mediums are identical and do not depend on temperature. The mass particles mp content penetrating into the melt is not enough (mp £ 0.05 %), their diameter dp is much less than reference size of the fluid cavity and influence of inclusions on physical parameters of the melt can be neglected. The quantity of the dissolved components in the melt is not considerable so it is possible not to consider their influence on phase transition temperature supposing Tm = const . Melting of metal is considered at the Stefan’s approximation with the use of efficient thermal capacity [12]. It is supposed that at the considered heating modes small rates of convective speeds cause flatness of the free surface of liquid [13]. Borders rg, zg of the considered area are chosen so that their position does not affect the studied processes. Distribution of a temperature profile in solid and fluid material of a substrate is described by the equations of convective heat transfer in a cylindrical coordinate frame (r, z) which have the following form for the dimensionless variables: where D1 - current penetration depth in material (m), i. e. skin layer thickness, D1 = 503 re1 / (m1 f ) , re1 - specific electrical resistance, m1 - the relative magnetic conduc- tivity when warming up of metal is lower than the temperature of magnetic transformations (Curie point), f - working frequency of the generator of the field, Hz. At the achievement of magnetic transformations temperature the relative magnetic conductivity decreases to value μ2 = 1 cef [qt + (Ñ × u)q] = 1 Dq + Ki c(r, z), Pr Pr (1) and specific electrical resistance increases up to size re2 , consequently the current depth of penetration in material 0 £ r £ rg , - zg £ z £ 0, 0 £ t £ tH , becomes D2 = 503 re2 / (m2 f ) and energy distribution at influence of an inductor; over the depth of a substrate is described by expressions q + 1 ìïez / D2 , - zK (r) £ z £ 0, t (Ñ × u)q = Pr Dq + St( fs )t , (2) c(r, z) = í ïîe - zK / D2 e ( z + zK )/ D1 , z < -zK (r), r £ r0 , 0 £ r £ rg , - zg £ z £ 0, t > tH , where zK (r) - distance from a substrate surface to a after completion of warming up. Here q - temperature; t - time; Pr =n / a - Prandtl’s number, n - kinematic vispoint with a temperature of magnetic transformations ( T = T ). cosity of the melt, а - thermal diffusivity; u - a speed vector in molten metal with components u, w in radial (r) K At transition to the dimensionless variables we get and vertical (z) directions respectively (in solid metal c(r, z) = ez/D1 , q(r, 0) £ qK , r £ 1 , z £ 0 ; u º 0 , w º 0 ); St = k / (cpTm ) - the Stefan’s number, k - specific heat of melting; cp - specific thermal capac- ìïez / D2 , c(r, z) = í - zK (r) £ z £ 0, ity; fs - a part of a solid phase in material (0 £ fs £ 1) ; ïîe- zK / D2 e( z + zK )/ D1 , z < -zK (r), Ki = ( p0 r0 ) / (lTm ) - Kirpichyov’s number, l - a thermal conductivity, p0 - the specific capacity absorbed by unit of a surface of the heated body. The efficient thermal capacity considering the latent heat of a phase transition of k is described by a ratio cef = 1+ d(q)St where d(q) = 1 q(r, 0) > qK , r £ 1 ; where D1 = D1 / r0 , D2 = D2 / r0 , zK = zK / r0 , qK = TK / Tm . Boundary conditions for the equations (1), (2) are the following. On an axis of symmetry at qÎ[1- Dq / 2, 1+ Dq / 2] and d(q) = 0 if ¶q qÏ[1- Dq / 2, 1+ Dq / 2] , Dq - a temperature band where the phase boundary “is smeared”. The dimensionless pa- rameters are defined with the help of the reference size r0, ¶r = 0 , r = 0, -zg £ z £ 0 . (3) On the external side border of calculated area speed v0 =n / r0 , time t0 = r0 / v0 and temperature Tm: ¶q r = r / r0, z = z / r0, u = u / v0 , w = w / v0 , t = t / t0 , q= T / Tm . At impact of a high-frequency electromagnetic field on a substrate an apparent density of internal sources of heat w0 according to [11] is described by a ratio ¶r = 0 , r = rg , -zg £ z £ 0 . (4) On the upper bound ( z = 0 ). When heating from a source t £ tH p ¶q = a(q - q) , 1 < r £ r . (5) 0 w (r, z) = 0 c(r, z). ¶z c g r0 Here c(r, z) - the dimensionless function depending After completion of an impulse ( t > tH ) only on coordinates, r0 - the reference size, in our case the radius of a spot of inductive impact on a substrate surface, p0 - the specific power absorbed by a unit of a heating body surface. Assuming that the central point of “the spot” of induc- tive influence of radius r0 is located in a point (0, 0) . ¶q = a(q - q) , 0 £ r £ r c g ¶z On the lower bound of calculated area ¶q = 0 , 0 £ r £ r , z = -z . (5а) . (6) c In this case the distribution of energy release over the sheet thickness can be described with the dependence: ¶z g g c(r, z) = ez/D1 , r £ r , z £ 0 , Here a = a0r0 / l , a0 = es0 (T 2 + T 2 )(T + Tc ) , s0 - 0 a Stefan-Boltzmann constant, e - degree of blackness of material of a substrate, Tc - environment temperature where s0 - the interfacial tension in clear metal at melt- (qc = Tc / Tm ) . ing point; ks = -¶s / ¶T - for clear metal; R - universal When melting material speed vn for each point of the gas constant; Cs - concentration of surface-active subborder “melt - solid phase” z(r, z, t) is defined by a condition stance impurity on a melt surface; Sl - the constant cor- Pr St vn = ¶q ¶n q=1- - ¶q ¶n q=1+ , responding to an entropy of segregation; DH 0 - standard absorption heat; Gs - excess of impurity in the surface where n - a unit vector of a normal. During solidification it is supposed that all nano-size particles are crystallization centers, then the portion layer per unit area, then s ¶s KCs GsDH 0 of a solid phase fs in the two-phase melt zone is defined ¶T = -ks - RGs ln(1+ KCs ) - 1+ KC T , (11) according to [14] of a ratio fs = 1- exp{- WNp } where from where are defined Mn = -ksTmr0 / (rna) - Maran- 3 4p é t ù goni’s number and h W(r, z, t - x) = êrp + Knò(Tm - T )d zú s s 3 êë x úû h= KCs GsDH 0 the volume of the growing nucleus which arose in an 1+ RGs ln(1+ KCs ) / ks + 1+ KC Tk . instant t =x , Np - number of nanoparticles in the unit Initial conditions for the heat transfer equation (1) of melt volume, rp = dp /2 , Kv - an empirical constant. fs = 0.95 corresponds to the end of solidification. When melting metal in the area q> 1 with the border “melt - solid phase” z(r, z, t) the flow in liquid is described by the Navier-Stokes equations in the Boussinesq approximation q = q0 , (12) for (7) in just melted metal u = 0 , w = 0 . (13) It is supposed that well moistened modifying nanout + (Ñ × u)u = -Ñp + Ñ2u + kGr(q -1), Ñ× u = 0. (7) size particles under the influence of convective flows penetrate into the depth of the melted cavity. Movement and distribution of nanoparticles in the melt was estimated by means of M markers which initial position is de- Here p - pressure ( p = p / P , P0 = rv2 , r - substrate scribed by coordinates (r0, z0 ) , 0 £ r0 £ 1 , z0 = -r , material density), Gr = br gr3T 0 0 / n2 - Grasgof’s number, m = 1, ..., M . m m m m p 0 m g - a free fall acceleration, br - coefficient of a cubic Markers move according to locally average speeds in thermal expansion of a melt, k - unit vector along a coordinate axis of z. Boundary conditions for the equations (7) are the fol- lowing. On the surface “melt - solid phase” z(r, z, t) the closest vicinity of everyone. At realization of model (1)-(13) which includes the equations of Navier-Stokes and convective heat transfer the finite-difference algorithm was applied. Discretization of dimentional calculated domain was carried out on u = 0 , w = 0 . (8) I ´ K cells. Step t along a temporary variable is con- On an axis of symmetry r = 0, -z f (t) £ z £ 0 stant. Distribution of temperature was described by values in grid clusters. Systems of difference equation were (-z f (t) - coordinate of the border of phase transition) u = 0, ¶w = 0 , (9) ¶r under construction by means of implicit approximation of the balance ratios received by an integration of the equa- tions (1), (2), (7) taking into account the corresponding boundary conditions. At approximation (7) in the field of the melted material, by analogy with MAC and SIMPLE On the surface of liquid z = 0 , 0 £ r < rf (t) methods [15; 16], components of velocity u, w, were defined in the middle of lateral faces of cells, and pressure ¶u = Mn ¶q h , ¶z Pr ¶r w = 0 , (10) p - in the centers of cells. The algorithm of each temporary step has the follow- ing operations procedure. Initially the temperature profile where rf (t) - radius of the cavity of molten metal on the in a substrate at influence of energy of a high-frequency electromagnetic field was calculated. After occurrence free surface. In the presence of surface-active substance in the melt, for the description of the interfacial tension the empirical formula [9] is used s = s0 - ks (T - Tm ) - TRGs ln (1+ KCs ) , K = Sl exp éë-DH 0 / (RT )ùû , of the fluid cavity on each temporary step the border of phase transition was established. Substitution of the found values of temperature in momentum equations made pos- sible the determination of the field of velocities compo- nents. Further, with the use of a method of simulated compressibility [17] pressure was culculated. Several iterations were performed to coordinate found pressure distribution and speeds before realization of inequality max Ñ× u £ g where γ - the given small number. The solution of the algebraic systems received at approxima- tion of motion equations and a heatmass transfer was car- ried out by iterative methods. m m m m m m By means of ratios rn = rn-1 + un t , zn = zn-1 + vn t , The considered sulphur load in iron is Cs = 0.001-0.04 % of weight. In fig. 2, 4 regarding calculated area adjacent to a zone of molten metal in the dimensionless coordinates, the structure of flows, trajectories of movement and distribu- tion of the modifying particles at surface-active substance in the melt 0.002 % of weight for various characteristics m = 1, ..., M , new coordinates of markers a n-temporary step were calculated. Here rn , m m un , n on z v m - n m of an impulse are displayed. Arrows in drawings specify the direction, and their length characterizes intensity of a flow. speeds of particles movement, defined according to locally average speeds in the nearest neighborhood of each of them. Calculations were carried out on spatial grids I ´ K from 120´150 to 240´300. Value of a temporary step t = 10-4, g = 10-4, Dq = 0.001, M = 25. After completion of an impulse the solution of a task proceeded until disappearance of overheat in molten metal of a substrate. Convective redistribution of markers occurred only in the melt, and in the solid area the last calculated coordinates were fixed. Results of numerical experiments. Numerical researches were conducted at the following parameters: r0 = 0.001 м, zg = 0.0012 м, rg = 0.0015 м; tH = 70 мс; Tс = 300 K; T0 = 300 K; p0 = (6.5-7.0)×108 Вт/м2; f = 1200 кГц; Кv = 2.5×10-5 м/(с×K); dр = 5·10-8 м; Np = 2.0×1015 1/м3. Properties of metal [9-11]: r = 7065 кг/м3, cp = 787 Дж/(кг×K), l = 27 Вт/(м×K), k = 2.77×105 Дж/кг, n = 8.5×10-7 м2/с, ks = 4.3·10-4 Н/(м×K), br = 1.18×10-4 1/K, R = 8314.3 Дж/(моль·K), Sl = 3.18 × × 10-3, Gs = 1.3·10-8 моль/м2, DH 0 = -1.88·108 Дж/моль, Tm = 1775 K, e = 0.7, σ0 = 5.7×10-8 Вт/(м2×K4), TK = 1141 K, m1 = 14, re1 = 5.2×10-7 Ом×м, m2 = 1, re2 = 1.0×10-6 Ом×м. Fig. 2 shows the results received at p0 = 6.5×108 W/sq.m, tH = 70 ms. Fig. 2, а illustrates the field of speeds in the melt at the time of completion of inductive influence. From the submitted data it appears that by the time of the impulse termination a big whirlwind with the direction of a flow near the free surface of liquid from border of a phase transition to the center of the cavity with the strong downflow in this area is formed. The similar structure of a flow exists from the moment of emergence of the melt to the completion of an impulse on condition of a small overheat of its surface regarding phase transition temperature. It leads to the fact that the modifying particles penetrate into the central part of the cavity in the considerable depth, and then are evenly distributed by the dispersing flow on all melt volume (fig. 2, b). A small local whirlwind in the central part of the cavity at the free surface does not have any influence on markers movement. The depth of melt penetration is about 450 microns that is 2-3 times more than at laser processing by a beam with the close values of radius and power [7]. It should be noted that slight overheat of a surface of molten metal promotes preservation of its flatness and allows to avoid the subsequent after-treatment. After completion of an impulse the melt cools down and solidifies that is promoted by heat removal from its surface and in not warmed material. Movement of nano-size particles happens as long as the molten metal exists. 0,0 0,0 Подпись: z-0,2 -0,2 Подпись: z-0,4 -0,4 -0,6 0,0 0,2 0,4 0,6 0,8 r -0,6 0,0 0,2 0,4 0,6 0,8 r а b Fig. 2. The field of speeds (а), trajectories and distribution of markers (b) in the melt at q0 = 6.5×108 W/sq.m , tH = 70 мs Рис. 2. Поле скоростей (а), траектории и распределение маркеров (б) в расплаве при q0 = 6.5×108 Вт/м2, tH = 70 мс Fig. 3 illustrates distribution of the dimensionless temperature and the c function over substrate thickness from its surface in a point of the maximal warming up (r = 0). It follows from the presented results that the depth of penetration of current does not exceed the thickness of the considered sheet, and the main energy release oc- curs in the metal layer heated higher than the temperature of magnetic transformations of z = -0.65. As a result, the temperature accepts maximal values on the processed surface and monotonically decreases in the process of removal from it, and the substrate warming up through all thickness does not occur. Fig. 4 shows the results received at p0 = 7.0×108 W/sq.m, tH = 70 ms, that is at higher specific power absorbed by a surface unit of the heated body. It follows from the submitted data that by the time of the termination of a laser impulse in the melt two toroidal-shaped whirlwinds appears at collision with the counter flow, an intensive downflow is formed. As a result chaotic movement of the particles which got into molten metal in the field of inten- sive whirlwinds appears. In the central part of the cavity the modifying additives penetrate into the considerable depth only at the beginning of melting, and further when the considerable overheat of the melt is formed, upward movement of liquid obstructs it. In general it turns out that particles are not distributed in all volume of molten metal, are absent in the central part and on peripheries of the melt and the considered processing mode of the sur- face layer is impossible to be called efficient. On completion of an impulse temperature gradients in liquid decrease, intensity of a convection decreases, metal cools down and solidifies (13-15 ms) that is promoted by the low initial temperature of material. After overheat decrease in the melt there takes place a volumedetermined by the parameter ¶s / ¶T which accepts both consecutive crystallization in the direction to a substrate positive, and negative values are formed. That is a surface temperature increase changes the structure of molten metal. Near the free surface a flow from the border of a phase transition to the center of the cavity occurs (fig. 4, a) promoting that the modifying particles do not penetrate into the depth of the melt on its periphery (fig. 4, b). Also near the surface a flow dispersing from the center which surface. By results of numerical calculations it is defined that homogeneous nucleation of crystals in the presence of highly activated nano-size particles in the melt does not happen and does not affect crystallization kinetics as at the existing cooling rates conditions necessary for this are not reached. Fig. 3. Change of temperature θ (1) and c (2) function in a substrate at r = 0 for q0 = 6.5×108 W/sq.m, tH = 70 ms Рис. 3. Изменение температуры θ (1) и функции c (2) в подложке при r = 0 для q0 = 6.5×108 Вт/м2, tH = 70 мс 0,0 -0,2 Подпись: z-0,4 0,0 -0,2 Подпись: z-0,4 -0,6 0,0 0,2 0,4 0,6 0,8 r -0,6 0,0 0,2 0,4 0,6 0,8 r а b Fig. 4. The field of speeds (a), trajectories and distribution of markers (b) in the melt at q0 = 7.0×108 W/sq.m, tH = 70 ms Рис. 4. Поле скоростей (а), траектории и распределение маркеров (б) в расплаве при q0 = 7.0×108 Вт/м2, tH = 70 мс Fig. 5. The maximal values of melting temperatures at various concentration surface-active substance in metal for receiving homogeneous distribution of the modifying particles Рис. 5. Максимальные значения температур расплава при различных концентрациях ПАВ в металле для получения гомогенного распределения модифицирующих частиц By results of the analysis of a formula (11) and numerical experiments for various concentration surface- active substance in metal (0.001 % < Cs ≤ 0.04 %) maxi- mum temperatures of warming up of the free surface of the melt are determined (fig. 5) at which the steady flow near it is formed with the direction from border of phase transition to the center of the cavity similar to the pre- sented in fig. 2, а. Conclusion. By results of numerical modeling the possibility of the use of inductive processing for modifi- cation of molten metal by the nano-size particles of re- fractory compounds allowing to improve its structural components when solidifying is determined. The results of calculations show that the existence of surface-active substance in the melt significantly influences physical processes during modifying and in particular the flow in the melt. To improve the processing quality of the metal layer surface it is necessary to take into consideration its structure and accounting it to choose the modes of an impulse of a high-frequency inductive field. The use of optimum modes can promote the homogeneous distribu- tion of the modifying particles in full volume.

作者简介

V. Popov

Khristianovich Institute of Theoretical and Applied Mechanics SB RAS

Email: popov@itam.nsc.ru
4/1, Institutskaya Str., Novosibirsk, 630090, Russian Federation

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