Journal of Materials Processing Technology 102 (2000) 221±229

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1 Journal of Materials Processing Technology 102 (2000) 221±229 Application of mathematical simulation and the factorial design method to the optimization of the atomization stage in the spray forming of a Cu±6% Zn alloy M.M. Pariona a,*, C. Bolfarini b, R.J. dos Santos b, C.S. Kiminami b a Departamento de MatemaÂtica e EstatõÂstica, Universidade Estadual de Ponta Grossa, Campus Uvaranas, Bloco L, R. Nabuco de Araujo s/n, CEP , Ponta Grossa-PR, Brazil b Departamento de Engenharia de Materiais, Universidade Federal de SaÄo Carlos, Via Washington Luiz, Km 235, CP 676, CEP , SaÄo Carlos-SP, Brazil Received 2 March 1999 Abstract Process parameters optimization for the spray forming of a Cu±6% Zn alloy was developed by a combination of mathematical simulation and the factorial design method (FDM). The nite difference method was used for the mathematical simulation of the atomization stage and two levels FDM was applied considering the initial gas velocity, the axial distance between the atomizer and the substrate, the superheating of the melt and the melt ow rate. The validation of the optimized process parameters was done by processing the Cu±Zn alloy, a highdensity deposit, as high as 98.5% of the theoretical density with a very homogeneous microstructure being produced. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Simulation; Optimization; Atomization; Spray forming; Cu±6% Zn alloy 1. Introduction In the spray forming process (SF), a liquid stream of metal from a crucible is atomized by the impact of a high-energy gas stream. The formed droplets are propelled away from the atomization region towards the substrate [1]. By the relative movement between the substrate and the atomizer during deposition, near-net shape can be produced, such as strip, cylinder and tube geometries, at high production rates [2]. Due to the rapid solidi cation process established by the high cooling rates (in the range 10 3 ±10 4 Ks 1 ) the microstructure is formed typically by ne grains with low segregation, metastable phases and extended solid solubility of the alloying elements [3]. In this process one of the critical parameters for controlling the nal microstructure of the deposit is the amount of solid, liquid and semi-solid droplets that impinge on the * Corresponding author. addresses: moisespariona@convoy.com.br (M.M. Pariona), cbolfa@power.ufscar.br (C. Bolfarini) substrate surface. If the solid fraction is too low at the time of deposition, splashing and whipping up of liquid material by the gas produces pores of entrapped gas in the nal billet, whilst if the solid fraction is very high, pores will be formed in the deposit due to the low uidity of material to ll up pores and interstices [3,4]. In general, this parameter depends in a complex way upon the dynamic and thermal behavior of the variously sized droplets, which in turn is controlled by a set of several process parameters and the characteristics of the alloy: these include the atomizing gas pressure, the melt ow rate, the superheating of the melt, the gas ow rate, the diameter of the liquid metal discharge tube and the composition of the alloy [4,5]. The factorial design method (FDM) is very important in the optimization of experimental results and in numeric simulation [6]. This method is used extensively in all of the applied sciences, because it enables the saving of time and cost. The FDM in two levels [6], applied in the case of the SF process, allows the characterization of the combination of the optimal process parameters for the minimization, for instance, of the porosity, by analyzing a reasonable number /00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S (00)

2 222 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221±229 of parameter combinations, e.g., by selecting the lower- and upper-range levels for each of the parameters: T s (the degree of superheating), v gi (the initial velocity of the gas), L (the axial distance between the atomizer and the substrate), and v di (the initial velocity of the droplets) that intervene in the SF process. According to this selection, it is possible to combine all of these parameters in the different levels and each one of the combinations can be used in the numerical simulation of the process, for instance, in the solution of the differentials equations, resolving by the nite difference method. Through the FDM and the results of the differential equations solutions, it is possible to make feasible the optimization, control and to foresee the behavior of the parameters that have more in uence in the formation of the microstructure in the SF process. The FDM is a powerful tool that allows the optimization of experimental results or of simulation, reducing the time and cost of production. This method is still little reported in literature. The objective of this paper is the optimization of the atomization stage of the SF process for a Cu±6% Zn alloy by means of the numerical simulation associated with the FDM. Simulation of the velocity, the ight time, the heat-transfer coef cient between the droplets and the gas, the thermal history on cooling of the droplets for the case of heterogeneous nucleation, were considered. These results are compared with experimental data, in order to verify the validity of the mathematical simulation. 2. Application of mathematical simulation and factorial design 2.1. Application of mathematical simulation Grant et al. [3], and Lee and Ahn [4] formulated a mathematical model to describe the droplet/gas interaction, which predicted the velocity and temperature pro le of the droplets in ight. In the present work, the same model was applied with slight modi cation, followed by the FDM to describe the droplet solidi cation before deposition on a Cu±6% Zn alloy atomized with nitrogen gas. The model that is proposed assumes individual spherical droplets and a linear trajectory during ight. Considering F as a force acting on the droplets, Grant et al. [3] proposed the following equation: dv d F ˆ m d ˆ C D p d 2 r dt 2 g v g v d 2 (1) The symbols and constants used in this work being listed in Table 1. The initial velocity of the droplets (v di ) and the initial velocity of the gas (v gi ) were calculated by Pariona et al. [5] through the use of the Bernoulli equation [5], the values for the lower and higher limits of the process parameters being Table 1 Symbols and physical constants used in this work a 1ˆ (m 0.24 s 1.24 ) Coefficient of fit a 2ˆ Catalytic function coefficient a 3ˆ (m) Catalytic function coefficient A g (m 2 ) Effective area of the atomizer nozzle gas outlet C D Drag coefficient C m (J kg 1 K 1 ) Specific heat of metal C g (J kg 1 K 1 ) Specific heat of gas d 0 (mm) Diameter of the liquid metal discharge tube d (mm) Diameter of droplet F (N) Force acting on the droplets f Solid fraction g(y)ˆa 2 a 3 /d Catalytic efficiency h (J m 2 s 1 K 1 ) Heat-transfer coefficient I (m 3 s 1 ) Nucleation frequency K iˆ (m s 1 K 1 ) Solid/liquid interface mobility K g (J m 1 s 1 K 1 ) Thermal conductivity of gas K bˆ (J K 1 ) Boltzmann's constant L (mm) Axial distance m d (kg) Mass of droplet nˆ1.24 Eq. (5) Prˆn g /a g Prandtl's number P m (Pa) Over-pressure of liquid metal P g (Pa) Pressure of atomizing gas T s (K) Superheating degree T fˆ K Fusion temperature T lˆ1330 K Liquidus temperature T E (K) Eutectic temperature DH fˆ (T l T) (Jm 3 ) Latent heat of copper T a (K) Final recalescence temperature T g (K) Temperature of gas T N Nucleation temperature DTˆT l T N Undercooling v d (m s 1 ) Droplet velocity v di (m s 1 ) Droplet initial velocity v g (m s 1 ) Gas velocity v gi (m s 1 ) Gas initial velocity v rˆv g v d (m s 1 ) Relative velocity a gˆk g /r g C g (m 2 s 1 ) Thermal diffusivity y Contact angle Z m (N s 1 m 2 ) Dynamic viscosity of the metal r mˆ7886 (kg m 3 ) Alloy density r g (kg m 3 ) Gas density l vˆ0.1 (m) Decay coefficient s mˆ T (J m 2 ) Surface energy of the alloy n g (m 2 s 1 ) Kinematic viscosity of the gas presented in Table 2. The drag coef cient, C D, of the drops in the gaseous medium is related to the Reynold's number (Re) [2,4] by Table 2 Process parameters affecting the mean droplet size Parameters Low ( ) High ( ) 1 P g Pa Pa 2 P m Pa Pa 3 A g m m 2 4 ` m m 5 d m m

3 " # 6 C D ˆ 0:28 Re 1=2 where 21 Re Re ˆ vrd (3) v g In this work the exponential decay of the gas velocity proposed by Grant et al. [3] is used v g ˆ v gi exp y (4) l v where l v ˆ a 1 v n gi (5) The equation of the convective heat-transfer coef cient (h) between the drops and the gas was proposed by Ranz and Marshall [7] as h ˆ Kg d 2 0:6Re0:5 Pr 0:33 (6) Radiation is not considered as a heat-transfer phenomena in this equation. This approximation was used elsewhere in the literature [1±4] because it can be considered negligible when compared to the convective heat removal. During atomization the heat removed by convection at the droplet surface boundary layer can be equated to the change in droplet temperature, including a source term for the release of latent heat when the droplet is solidifying [3] 6h T T g ˆr m d M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221± C m dt dt DH f r m df dt Fig. 1 presents a schematic diagram of the thermal history of the atomized droplets during the atomization stage as proposed by Grant et al. [3]. According to this diagram, the (2) (7) droplets pass through regions (i)±(v) until complete solidi- cation. Prediction of the temperature pro le requires knowledge of the amount of undercooling (DT) and recalescence. The original proposal by Grant et al. [3] for heterogeneous nucleation was modi ed in the present work in order to include parameters that are available in the literature, such as Z m and s m : I ˆ 1030 exp 16ps3 m gg y Z m 3K b T N DHf 2 (8) The magnitude s mˆ TJm 2 in Eq. (8) was calculated according to the suggestion of Kaufman and Frase [8]. The dynamic viscosity of the metal (Z m ) was proposed by Ramachandrarao et al. [9] Z m ˆ 0:408 exp 0:0822 f T ˆ 0:029 exp 0:539 T f T T f T In Eq. (8), y is the solid/liquid/nucleant contact angle, and g(y) the catalytic ef ciency for heterogeneous nucleation. In the case of heterogeneous nucleation, Levi and Mehrabian [10] showed that droplet undercooling increases with a decrease in droplet diameter, and assuming an inverse relationship between the catalytic ef ciency and the droplet diameter, they proposed the following relationship: g y ˆ 2 3 cos y cos3 y 4 ˆ a 2 a 3 d (10) The nucleation temperature (T N ) is determined by Eq. (8). In region (ii) of Fig. 1, when nucleation starts at high undercooling, the initial nucleation is assumed to be segregation-free, with a solid/liquid interface velocity given by Eq. (12) proposed by Cahn et al. [11]: v i ˆ K i DT (11) The undercooling and recalescence control the initial nucleation and the interface velocity given by Eq. (11), which in their turn result in dendritic growth in this zone. The droplet temperature and solidi cation behavior during recalescence are described in Eq. (7). In this equation, the convective external heat removal is assumed to be negligible [3] Fig. 1. Schematic representation of the thermal history of the cooling of the atomized droplets: (i) cooling in the liquid state, (ii) nucleation and recalescence, (iii) segregation solidi cation, (iv) eutectic solidi cation, and (v) cooling in the solid state [3]. dt dt ˆ DH fk i DT r m C m d (12) df dt ˆ KiDT (13) d Eq. (13) is valid for plane-front solidi cation and Newtonian conditions. Segregation-free solidi cation occurs in the recalescence region. Consequently, Eqs. (12) and (13) determined the

4 224 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221±229 droplet temperature and solidi cation behavior up to the nal recalescence temperature (T a ). At this point, the internal heat release should be equal to the external heat removal 6h T T g df ˆ d DHf dt (14) When recalescence is complete, solidi cation continues by a segregation mechanism described by the Scheil equation [10], where the equilibrium partition coef cient for Cu±6% Zn (K e ) was considered to be Gas temperature equation assumed an exponential increase with axial distance [3]. In this work, the value of the constants of that equation were maintained equal to those measured by Grant et al. [3]. The effect of the following parameters has been investigated: the initial gas velocity, the axial distance between the atomizer and substrate, the degree of superheating of the liquid metal, and the melt ow rate. The droplet sizes used in the simulation were 20, 40, 80, 130 and 180 mm Factorial design Pariona et al. [5] undertook a study on the size distribution of the droplets of a Cu±6% Zn alloy during the atomization stage. In this study the critical process parameters that affected the mean droplet diameter were: P m (the overpressure of the liquid metal), P g (the pressure of the atomizing gas), `0 and d 0 (the length and diameter of the liquid metal discharge tube, respectively), A g (the effective area of the atomizer nozzle gas outlet), and T s (the degree of superheating of the liquid metal) were considered [5]. These parameters and the lower and upper levels pertaining to the range of variation of each one of these parameters in the SF process are shown in Table 2. By means of the FDM in two levels, it was veri ed that the main parameters determining the mean droplet size are the pressure of the atomizing gas and the diameter of the liquid metal discharge tube, the other parameters and their interactions only exerting a slight effect [5]. To obtain smaller droplets the gas pressure should be maintained high and/or the diameter of the liquid metal discharge tube should be small [5]. This result served to Table 3 Low and high values for superheating degree, T s, gas initial velocity, v gi, axial distance, L, and droplet initial velocity, V di, used in the factorial design method Parameters Low ( ) High ( ) 1 T s 508C 1508C 2 v gi 230 m s m s 1 3 L 0.2 m 0.5 m 4 v di 4ms m s 1 enable the extending of the present study to provide details of the atomization stage for same alloy. The simulation of the droplet velocity during ight, the droplet ight time, the heat transfer between the droplets and the gas and the individual droplet cooling behavior during ight were realized in this work. This simulation was performed by the nite difference method coupled with the FDM [6] considering the following process parameters: T s (the degree of superheating), v gi (the initial velocity of the gas), L (the axial distance between the atomizer and the substrate) and v di (the initial velocity of the droplets). The initial velocity of the liquid droplets was assumed to be equal to the velocity of the melt exiting the liquid metal discharge tube. Through FDM in two levels, the lower and upper levels pertaining to the range of variation of each one of the parameters were selected previously. According to this selection, it was possible to combine all of these parameters in the different levels and each one of the combinations was used in the solution of the differential equations, resolved by the nite difference method. The limits of the critic parameters of the SF process are shown in Tables 2 and 3. The parameters P m, A g, d 0, T s and L are characteristic of the SD equipment available within the authors' laboratory and the type of alloy used. The choice of the limits of `0 was based on the contraction phenomenon occurring in the metallic stream during ow: for the lower values of `0 this phenomenon takes place, but for greater lengths does not. Table 4 shows the results of analysis by the nite difference method, considering the different combinations of the Table 4 Results of the nucleation and recalescence (region ii) and segregation solidi cation (region iii) stages of simulation during the ight of the 180 mm droplet (the different combinations of the process parameters were established by the factorial design method, the characterization here concerning the fraction of solid formed) T s v gi L Droplet cooling 180 (mm) Region (ii) Region (iii) 1 Complete No 2 Almost complete No 3 Very little No 4 No No 5 Complete Complete (rapid cooling) 6 Complete Complete (rapid cooling) 7 Complete Complete (slow cooling) 8 Complete Complete (slow cooling)

5 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221± Fig. 2. Process parameters that correspond to the extreme conditions of the FDM in relation to the cooling rate of the biggest droplet (row 4 of Table 4) showing the behavior of droplet cooling during axial movement. process parameters T s, v gi and L established by the FDM, the characterization of this result being concerning with the fraction of solid formed. The factor v di is shown to exert a negligible effect on the FDM results of the droplet behavior during ight, probably because the values of this parameter are very low compared to those of v gi. Consequently, the parameter v di was not taken into account in the FDM. Through the FDM and the results of the solutions of the differential equations, it is possible to make feasible the optimization and control, and to foresee the behavior of the parameters that have greater in uence in the formation of the microstructure of the SF process. The FDM is a powerful tool that allows the optimization experimental results or of simulation, reducing the time and cost of production. The analysis of the mathematical simulation and the FDM will be presented in Section Results and analysis of the mathematical simulation and factorial design In order to discuss the results of the FDM and the simulation, the cooling of the biggest droplet (180 mm) was chosen. ThecoolingofthisdropletwasbasedonFig.1, regions (ii) and (iii). This droplet has interesting physical characteristics compared to the smaller drops that have a greater chance to solidify before reaching the substrate. Rows 4 and 5 of Table 4 correspond to the extreme conditions of the FDM in relation to the cooling rate of the biggest droplet, whilst Figs. 2 and 3 correspond to these rows, respectively. These gures show simulation results of the behavior of individual droplets during ight in the atomization stage. The following parameters were simulated: (a) the velocity, (b) the ight time, (c) the heat-transfer coef cient between the droplets and the gas, and (d) the thermal history of the cooling of the

6 226 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221±229 Fig. 3. Process parameters that correspond to the extreme conditions of the FDM in relation to the cooling rate of the biggest droplet (row 5 of Table 4) showing the behavior of droplet cooling during axial movement. droplets, considering the nucleation and recalescence for heterogeneous nucleation. Table 4 shows that L is the main parameter controlling the process, followed by v gi. It is observed that at low gas velocities and large axial distances, which correspond to the extreme conditions of the FDM in relation to the cooling rate of the biggest droplet, the bigger droplets arrive solid at the substrate. Consequently, there is no latent heat release and the cooling rate is high. On the other hand, for high gas velocities and large axial distances (rows 7 and 8), the bigger droplets arrive semi-solid at the substrate and, as a consequence of the latent heat release, their cooling rates are low. For small distances and high or low gas velocities (rows 1±4), the bigger droplets arrived liquid at the substrate. Figs. 2a and 3a show the result of the simulation of droplet and gas velocities as a function of axial distance. One may note that the smallest drop, having a diameter of 20 mm, is accelerated more rapidly than the others. At the beginning of the atomization stage, this droplet attains a maximum velocity, being decelerated faster than the bigger droplets. On the other hand, the biggest droplet, 180 mm, presents a slower acceleration in relation to the smaller droplets. In general, the droplet velocity attains a maximum when it equals the gas velocity. With an increase of distance, all droplets tend to accelerate similarly to the gas. Figs. 2b and 3b present the ight time (from the atomizer to the substrate) of the droplets as a function of axial distance. Table 5 shows the results of the ight times for each droplet diameter, which correspond to each row in Table 4. Each one of these results were obtained by integrating the inverse function of curves 2a and 3a. One observes that the ight time is longer for bigger droplets or large axial distances than for smaller droplets. Particularly, it is observed that the ight times corresponding to rows 5 and 6 in Table 5 are the longest and that those corresponding to rows 3 and 4 are the shortest. This is related to the velocity of the droplets: the velocities corresponding to rows 5 and 6 are lower than those corresponding to rows 3 and 4.

7 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221± Table 5 Flight time of the droplets from the atomizer of the substrate No. of Table 4 rows Flight time (10 3 ) (s) 20 mm 40mm 80 mm 130 mm 180 mm Figs. 2c and 3c shows the behavior of the convective heattransfer coef cient (h). Three characteristic regions can be observed: a very steep drop of h down to a minimum, an increase in h up to a second maximum and nally a smooth drop in h. The results of h are related to the corresponding gures denoted as (a), which in their turn are directly related to the Reynold's number (Re) of Eq. (3). In the region where h drops quickly in the gures denoted as (c), the heat-transfer coef cient at the beginning (a small axial distance) shows a maximum for all droplets, being higher for the smaller droplets and for high v gi and L. This means that in the beginning the droplets lose heat rapidly, transferring it to the surrounding gas. This rapid heat transfer is related to Re, where in the beginning there is a large difference between the velocities of the gas and of the droplets, resulting in a high Re value which decreases as the axial distance increases. This coef cient (h) attains a minimum value of 2K g /d, at the same position as where the droplets attained their maximum velocity, where the droplet velocity equals that of the gas, i.e., Reˆ0. These gures show that the minimum value of h, for the smaller droplets is shifted upwards more than for the bigger droplets. Following this region of strong h decrease, the second maximum value of h appears, being higher for the smaller droplets. At the beginning of this region, the droplet/gas relative velocities start to increase with axial distance up to a maximum. In this region, the heat-transfer coef cient is low. After the second maximum value of h, the gas/droplet relative velocity starts to decrease, attaining a behavior that becomes approximately constant with axial distance. In this region, the heat-transfer coef cient is low. Simulation of the thermal history of the droplets based on the diagram of Fig. 1 was performed assuming constant h, the minimum chosen being hˆ2k g /d. Furthermore, K iˆ ms 1 K 1 was chosen, as for other values T a presents inconsistent values. The results for the thermal behavior of the individual droplets considering heterogeneous nucleation are presented in Figs. 2d and 3d as a function of axial distance. In these gures, one may observe that the 20 mm droplet does not display recalescence, indicating that high levels of undercooling are established and if the glass transition could be reached, the viscosity would be very high, consequently, the solidi cation of these droplets could result in an amorphous structure. Droplets bigger than 20 mm exhibited recalescence and solidi ed in region (ii) and/or (iii). The undercooling for the bigger droplets is not so pronounced, and nucleation takes place at higher temperatures. This is expected when taking into account Eq. (8). Analysis of the results of the gures corresponding to the rows in Table 4, referring to the bigger droplets, shows that for high gas velocities (v gi ) the adequate axial distance lies in the range 250±300 mm and for low gas velocities this range is from 200 to 250 mm. These axial distance ranges ensure that the bigger droplets arrive at the substrate as liquid or semi-solid, improving the microstructure quality of the deposited material. These ranges are related to the amount of solid, liquid and semi-solid droplets that reach the substrate's surface. If the solid fraction is very low at the time of deposition, splashing and whipping up of the liquid material will occur due to the gas stream on the deposit, resulting in the formation of pores of entrapped gas. If the solid fraction is very high, pores will be formed in the deposit due to lack of uidity of the material to ll up pores and interstices [3]. Consequently, the gas velocity and the axial distance control the amount of solid or semi-solid droplets that arrive at the substrate. 3. Cu±Zn spray deposition A slight modi cation of the mathematical model proposed by Grant et al. [3], and Lee and Ahn [4] was introduced in this work, where coupled with FDM, it can be used in conjunction with the results of experiments, to obtain a better understanding in the role of the processing on the properties. Notwithstanding, in order to be sure that it is reliable, some validation experiments were conducted. Alloy of nominal composition Cu±6 wt.% Zn was processed by SF. Charges of about 3 kg of the alloy were melted under an argon cover. Nitrogen was used as the atomizing gas at pressures of 0.35±0.75 MPa. Depositions were carried out on ceramic substrates positioned 250 and 300 mm below the atomization zone, with superheating assuming the values of 50 and 150 K. These process parameters were selected in order to be close to those described in Table 4 for rows 4 and 5. Table 6 resumes the process parameters used in the experiments. The results for experiments 1 and 2 are shown in Figs. 4 and 5, respectively. From these results it would be expected that all particles smaller than 125 mm will arrive at the deposition substrate in a completely solid-state condition. This represents a mean solid fraction of about 63% for experiment 1 (mass median particle sizeˆ90 mm) and 55% for experiment 2 (mass median particle sizeˆ105 mm). The masses mean particle size of the distribution was calculated using the Lubanska [12] empirical equation. In this equation the mass mean particle size means the opening of a screening mesh that lets through 50% of the powder. The solid fraction was calculated using Eq. (12) proposed by Flemings [13].

8 228 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221±229 Table 6 Validation experiments: process parameters Experiment Row a Gas pressure (MPa) Superheating (K) V gi (m s 1 ) L (m) a The parameters are similar to those of Table 4, except the ight distances that are adjusted to produce optimal conditions. Fig. 4. The largest solidi ed droplet versus ight distance for the conditions of experiment 1. The solidi ed deposits were examined by scanning electron microscopy coupled to an image analysis system to determine the microstructure and the porosity. Samples near to the geometrical center of the deposit, i.e., between 15 and 20 mm from the bottom along the spray axis, were cut out and prepared for micrographic analysis. Representative micrographs of the samples are shown in Figs. 6 and 7, for experiments 1 and 2, respectively. One can see that the microstructures are very homogeneous for both the experiments and comprise individual recrystallized grains with no prior particle boundaries. The porosity is ne, isolated, homogeneously dispersed in the matrix and assumes values in the range 1.4±1.6% for experiments 1 and 2, respectively. Due to the mean grain sizes are very similar, i.e., 33.7 mm for experiment 1 and 31.8 mm for experiment 2, one can assume that the enthalpy of the system during the deposition Fig. 6. Micrograph correspondents to experiment 1 (row 1 of Table 6) (grain size 33.7 mm and porosity 1.6%; scale: 1 cmˆ25 mm). Fig. 7. Micrograph correspondents to experiment 2 (row 2 of Table 6). (grain size 31.8 mm and porosity 1.4%; scale: 1 cmˆ25 mm). process was quite the same, which is in accordance with the predictions of the model. 4. Conclusions Fig. 5. The largest solidi ed droplet versus ight distance for the conditions of experiment 2. The results allowed the selection of the best range for the distances between the atomizer nozzle and the substrate, i.e., 250±300 mm for high gas velocities and between 200 and 250 for low gas velocities, in order to optimize the nal deposit under the criterion of solid fraction, of 60%, and heterogeneous nucleation. Further, these results are compared with experimental data in order to verify the validity of the mathematical model. The performed experiments for

9 M.M. Pariona et al. / Journal of Materials Processing Technology 102 (2000) 221± Cu±6% Zn alloy with process parameters close to those of the optimal calculated conditions showed a high-density deposit (about 98.5% of the theoretical density) with a very homogeneous microstructure. The experiments showed that the application of a mathematical model associated with the FDM could be a good approach for optimizing the process parameters. Acknowledgements The authors acknowledge Jan Hendrik Schaay for his aid in the computational part of this work, and CNPq and PADCT-FINEP for nancial support. References [1] P. Mathur, S. Annavarapu, D. Apelian, A. Lawley, Mater. Sci. Eng. A 142 (1991) 261±276. [2] P. Mathur, D. Apelian, A. Lawley, Acta Metall. 37 (1989) 429± 443. [3] P.S. Grant, B. Cantor, Katgerman, Acta Metall. Mater. 41 (1993) 3097±3108. [4] E. Lee, S. Ahn, Acta Metall. Mater. 42 (1994) 3231±3243. [5] M.M. Pariona, C. Bolfarini, C.S. Kiminami, Zeitschrift fuèr Metallkunde 89 (1998) 494±497. [6] G.E. Box, W.G. Hunter, J.S. Hunter, Statistics for Experimenters, Wiley, New York, [7] W.E. Ranz, W.R. Marshall, Chem. Eng. Progr. 48 (1952) 173. [8] M.J. Kaufman, H.L. Fraser, in: E.W. Collings, C.C. Koch (Eds.), Undercooled Alloy Phases: Undercooling and Microstructural Evolution in Glass Forming Alloys, The Met. Soc., PA, 1987, pp. 249±268. [9] P. Ramachandrarao, B. Cantor, R.W. Cahn, Non-Cryst. Solids 24 (1977) 109±120. [10] C.G. Levi, R. Mehrabian, Metall. Trans. A 13A (1982) 221±234. [11] J.W. Cahn, W.B. Hilling, G.W. Sears, Acta metall. 12 (1964) 1421± [12] H. Lubanska, J. Met. 32 (1970) 45±49. [13] M.C. Flemings, Solidi cation Processing, McGraw-Hill, New York, 1974.