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Quantitative prediction of microporosity in aluminum alloys
Journal of Materials Processing Technology 191 (2007) 265–269
Quantitative prediction of microporosity in aluminum alloys E. Escobar de Obaldia, S.D. Felicelli ∗ Department of Mechanical Engineering and Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA
Abstract A continuum model of dendritic solidification of multicomponent alloys is used to predict the volume fraction of porosity caused by precipitation of hydrogen gas during solidification of A356 aluminum plate castings. The conservation equations of mass, momentum, energy and species transport are solved to simulate the transient solidification process, including thermosolutal convection and segregation during the formation of the mushy zone. The amount of hydrogen supersaturation is calculated based on the transport of dissolved hydrogen and Sievert’s law. An innovative aspect of this work is the extension of the model to make quantitative predictions of the volume fraction of porosity. Although the hydrogen supersaturation has been used in previous works to estimate the amount of porosity, these predictions are usually grossly overestimated, because the barrier of pore nucleation is not considered. In this work it is shown that it is possible to make good estimations of the volume fraction of porosity by combining the hydrogen supersaturation with the calculated local solidification times. Computer simulations are performed in plate castings, showing good qualitative and quantitative agreement with experimental data. © 2007 Elsevier B.V. All rights reserved. Keywords: Modeling; Solidification; Porosity; Aluminum
1. Introduction The properties of cast parts are known to be primarily determined by the microstructure which develops during solidification. The formation of microporosity in particular is known to be one of the primary detrimental factors controlling fatigue lifetime and total elongation in cast aluminum components [1]. Microporosity refers to pores which range in size from micrometers to hundreds of micrometers and are constrained to occupy the interdendritic spaces near the end of solidification. The micropores can form due to microshrinkage produced by the pressure drop of interdendritic flow or because of the presence of dissolved gaseous elements in the liquid alloy. In this work, we focus on the calculation of gas microporosity and, unless noted otherwise, we use the term porosity to mean gas-induced microporosity. In the case of aluminum alloys, hydrogen is the most active gaseous element leading to gas porosity [2]. Many efforts have been devoted to the modeling of porosity formation in the last 20 years, particularly in aluminum alloys [3–8] and, in a lesser degree, to nickel superalloys [2,9] and steels [10,11]. More recently, rather sophisticated models have been developed to include the effect of pores on fluid flow (three∗
Corresponding author. Tel.: +1 662 325 1201; fax: +1 662 325 7223. E-mail address: [email protected] (S.D. Felicelli).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.072
phase transport) [12], multiscale frameworks that consider the impingement of pores on the microstructure [13] and new nucleation mechanisms based on entrainment of oxide bifilms [14]. A recent review on the subject of computer simulation of porosity and shrinkage related defects has been published by Stefanescu [15]. 2. Solidification model The model presented here is based on a robust and welltested multicomponent solidification model which calculates macrosegregation during solidification of a dendritic alloy with many solutes [16]. The model solves the conservation equations of mass, momentum, energy and each alloy component within a continuum framework in which the mushy zone is treated as a porous medium of variable permeability. In order to predict whether microporosity forms, the solidification shrinkage due to different phase densities, the concentration of gas-forming elements and their redistribution by transport during solidification were later added to the model [2]. In this form, the model was able to predict regions of possible formation of porosity by comparing the Sievert’s pressure with the local pressure; however, no capability was developed to calculate the amount of porosity. The model has already been presented in detail in previous works and will not be repeated here. In the following, the
266
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
model extension to calculate the volume fraction of gas porosity is described. This extension is based on the calculation of two new variables, the amount of hydrogen supersaturation and the local solidification time. The calculation of the volume fraction of hydrogen microporosity is based on the method of Poirier et al. [17], in which the mass of gas is assumed to be given by the supersaturation of hydrogen in both the liquid and the solid phases: 100mH = (ClH − SlH )gl + (CsH − SsH )gs
(1)
where mH is the mass fraction of hydrogen, gl and gs the volume fractions of liquid and solid, respectively, ClH and CsH the concentrations of hydrogen in wt.% and SlH and SsH are the solubilities. The solubility of hydrogen in the liquid is calculated with the Sievert’s law [18] while for the solid we take SsH = kH SlH , where kH is the partition coefficient of hydrogen. We assume complete diffusion of hydrogen in the local solid, hence CsH = kH ClH , and Eq. (1) can be rewritten as: 100mH = (ClH − SlH )(gl + kH gs )
(2)
Assuming ideal gas behavior, we can then calculate the volume fraction of hydrogen due to supersaturation as: gH =
¯ mH ρRT MPH2
(3)
where R is the universal gas constant, T the temperature in K, PH2 the pressure of hydrogen gas in atm, M the molecular weight
of hydrogen gas and ρ¯ is the density of the solid plus liquid mixture. The problem with the above approach is that Eq. (1), from which Eq. (3) is derived, assumes that all the hydrogen exceeding the solubility will become gas pores, which of course is an overestimation because the pores need to overcome nucleation barriers in order to become actual pores. Most of the mechanisms that have been proposed for the nucleation of pores are based on the size of interdendritic cavities and the theory of heterogeneous nucleation on non-wetted surfaces. After nucleation, pore growth occurs by diffusion of hydrogen into the pore. These ideas have been challenged by Campbell and co-workers [14] and others, who propose a nucleation-free mechanism for pore formation based on the concept of double oxide films or bifilms. In this scenario, during pouring in a casting process, the liquid surface of the alloy can fold upon itself. Because the liquid surface is covered by an oxide film, the folding action leads to bifilms, which are entrained into the bulk melt as a pocket of air enclosed by the bifilm. In effect, the bifilm with its air pocket is the beginning of a pore. After entrainment, the turbulence causes the bifilm to convolute and contract. Posterior pore growth can occur by the simple action of unfurling of the bifilms, without the aid of hydrogen diffusion. Although the identification of the mechanisms of pore formation and growth is still a subject of active research, the following two observations are supported by a large number of experimental and modeling works with aluminum alloys:
Fig. 1. Solidification of A356 plate casting at 400 s for 0.31 cm3 /100 g of hydrogen content. (a) Isotherms in K; (b) volume fraction of liquid and velocity vectors; (c) total concentration of H in wt.%.
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
(a) The amount of porosity increases for higher initial hydrogen content in the alloy. (b) The amount of porosity decreases for higher cooling rate. In view of these observations, we propose to modify the calculated volume fraction of porosity given by Eq. (3) according to: gP = agH tfb + gk
(4)
where gP is the corrected volume fraction of porosity, gH the volume fraction of hydrogen porosity assuming complete precipitation of the supersaturation (calculated in Eq. (3)), tf the local solidification time (calculated from the solidification history), a and b the experimental constants and gk is the volume fraction of porosity due to interdendritic shrinkage. In most cases with non-negligible hydrogen content, this last term is small compared with gas porosity; shrinkage porosity will not be considered in this work. Note that in Eq. (4), gH is implicitly also a function of the freezing time through the convection and segregation that occurred during solidification. The constant a in Eq. (4) is linked to the origin mechanism of the pores. From the heterogeneous nucleation perspective, it can be viewed as the fraction of inclusions or sites that overcame the nucleation bar-
267
riers in a H-supersaturated environment; from the double oxide bifilm perspective, it can be viewed as the fraction of bifilms that became active by the unfurling mechanism. The constant b in Eq. (4) is linked to the pore growth mechanism and carries information on the time that the pores need to grow by hydrogen diffusion and/or unfurling. An exponent different than one in gH was also tried in Eq. (4), with no significant improvement in results. A relation similar to Eq. (4), but without the supersaturation and shrinkage terms, was used by Anyalebechi [19] to analyze experimental results with alloy A356. 3. Results and discussion The solidification model is discretized in space and integrated in time using a finite element algorithm that is described by Felicelli et al. [2,16]. Aluminum A356 alloy is solidified by simulation in a bottom-cooled two-dimensional mold. The twodimensional simulated casting has dimensions of 2.6 cm in width and 30 cm in height. Gravity acts downwards. In addition to the alloy solutes in A356 (Si and Mg), the gas-forming element, hydrogen, is considered. The computational domain is the casting; the top boundary is left open in order to allow for liquid flow to feed shrinkage. A no-slip condition is used for velocity
Fig. 2. Experimental [3] and calculated porosity for different initial hydrogen contents.
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E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
at the bottom and two vertical boundaries, and a stress-free condition is used on the top open boundary. Solute diffusion flux is set to zero on the boundaries, with the exception of hydrogen, for which a dehydrogenation flux condition is used. The thermal boundary conditions utilized in Poirier et al. [7] are used here, which are extracted from a measured thermal history in the plates cast by Fang and Granger [3]. The simulations start with an all-liquid alloy of the nominal composition initially at a uniform temperature of 958 K, which is 70 K of superheat. The thermodynamic and transport properties of the alloy, including the alloy elements and hydrogen, are the same as the ones in ref. [7], with the exception of the partition coefficient of hydrogen, for which we used the developments of Poirier and Sung [20] to include the effect of the high eutectic fraction in A356. We performed simulations for the following values of initial hydrogen content: 0.11, 0.25 and 0.31 cm3 /100 g (note: 1 wt.% = 1.12 × 104 cm3 /100 g), for which measured volume fraction of porosity were reported in the experiments by Fang and Granger [3]. Fig. 1a–c shows iso-contour plots of temperature, volume fraction of liquid and total concentration of hydrogen after 400 s of simulation time. Approximately half of the casting has solidified completely and the mushy zone is about 40 mm thick. For clarity, the liquidus and eutectic isotherms are indicated with solid lines in Fig. 1a. Fig. 1b also shows the velocity field produced by thermosolutal convection, which is noticeable only in the all-liquid region. In the top 10% of the mushy zone, the velocities reduce to less than 10−6 m/s and their effect on the pressure can be neglected compared with the metallostatic head. The total concentration of hydrogen is shown in Fig. 1c, where a thin layer depleted in hydrogen can be observed near the surface of the casting due to dehydrogenation occurred during solidification and cooling. The volume fraction of porosity for each of the initial hydrogen contents is shown in Fig. 2. To facilitate comparison with the experimental data of ref. [3], the cooling rate is used as the x-variable, instead of solidification time. The dashed line labeled “Best Fit” is the porosity calculated with Eq. (4) using the Best Fit of the constants a and b for a given hydrogen content. The solid line labeled “Common Fit” is the porosity predicted using Eq. (4) with a common set of constants a and b for the three levels of hydrogen content: a = 0.003, b = 0.97. It is observed that when the experimental constants are fit for an individual hydrogen content, the predicted porosity (dashed line) agrees very well with the experimental data, except at the highest cooling rate. For the three values of initial hydrogen content considered, the Best Fit prediction underestimated the porosity at the highest cooling rate. This discrepancy is expected because of the contribution of shrinkage porosity (not considered in these calculations): at high cooling rate (short freezing time) there is not enough time for the gas pores to evolve and shrinkage porosity becomes relatively more important. If we attempt to predict the amount of porosity using a common set of constants in Eq. (4) for all values of hydrogen content, it is observed (solid lines in Fig. 2) that the accuracy of prediction is varied, with a good capture of the cooling rate trend but less accuracy in the level of porosity. This might indicate that the number of originated pores is not directly related to the level of hydrogen supersaturation,
but could arise from alternative mechanisms of pore formation like the entrainment of oxide bifilms. 4. Conclusions A finite element model of dendritic solidification was extended to allow the calculation of the volume fraction of hydrogen microporosity. The amount of porosity is calculated from the local hydrogen supersaturation and solidification time. Computer simulations are performed in plate castings of aluminum alloy A356, showing good qualitative and quantitative agreement with experimental data. Acknowledgement This work was partly funded by the National Science Foundation through Grant Number CTS-0553570. References [1] J.F. Major, Porosity control and fatigue behavior in A356T61 aluminum alloy, AFS Trans. 105 (1997) 901–906. [2] S.D. Felicelli, D.R. Poirier, P.K. Sung, A model for prediction of pressure and redistribution of gas-forming elements in multicomponent casting alloys, Metall. Mater. Trans. B 31B (2000) 1283–1292. [3] Q.T. Fang, D.A. Granger, Porosity formation in modified and unmodified A356 alloy castings, AFS Trans. 97 (1989) 989–1000. [4] Q. Han, S. Viswanathan, Hydrogen evolution during directional solidifications and its effect on porosity formation in aluminum alloys, Metall. Mater. Trans. A 33A (2002) 2067–2072. [5] P.D. Lee, J.D. Hunt, Hydrogen porosity in directional solidified aluminium– copper alloys: a mathematical model, Acta Mater. 49 (2001) 1383– 1398. [6] M. M’Hamdi, T. Magnusson, C. Pequet, L. Amberg, M. Rappaz, Modeling of micropososity formation during directional solidification of an Al–7% Si alloy, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 311–318. [7] D.R. Poirier, P.K. Sung, S.D. Felicelli, A continuum model of microporosity in an aluminum casting alloy, AFS Trans. 109 (2001) 379–395. [8] J.D. Zhu, S.L. Cockcroft, D.M. Maijer, R. Ding, Simulation of microporosity in A356 aluminium alloy castings, Int. J. Cast Met. Res. 18 (2005) 229–235. [9] J. Guo, M.T. Samonds, Microporosity simulations in multicomponent alloy castings, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 303– 311. [10] K.D. Carlson, L. Zhiping, R.A. Hardin, C. Beckermann, Modeling of porosity formation and feeding flow in steel casting, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 295–302. [11] P.K. Sung, D.R. Poirier, S.D. Felicelli, Continuum model for predicting microporosity in steel castings, Modell. Simul. Mater. Sci. Eng. 10 (2002) 551–568. [12] A.S. Sabau, S. Viswanathan, Microporosity prediction in aluminum alloy castings, Metall. Mater. Trans. B 33B (2002) 243–255. [13] P.D. Lee, A. Chirazi, R.C. Atwood, W. Wang, Multiscale modeling of solidification microstructures, including microsegregation and microporosity, in an Al–Si–Cu alloy, Mater. Sci. Eng. A A365 (2004) 57–65. [14] X. Yang, X. Huang, X. Dai, J. Campbell, J. Tatler, Numerical modeling of entrainment of oxide film defects in filling of aluminium alloy castings, Int. J. Cast Met. Res. 17 (2004) 321–331.
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269 [15] D.M. Stefanescu, Computer simulation of shrinkage related defects in metal castings—a review, Int. J. Cast Met. Res. 18 (2005) 129–143. [16] S.D. Felicelli, J.C. Heinrich, D.R. Poirier, Finite element analysis of directional solidification of multicomponent alloys, Int. J. Numer. Meth. Fluids 27 (1998) 207–227. [17] D.R. Poirier, K. Yeum, A.L. Maples, A thermodynamic prediction for microporosity formation in aluminum-rich Al–Cu alloys, Metall. Mater. Trans. A 18A (1987) 1979–1987.
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[18] M.C. Flemings, Solidification Processing, McGraw-Hill, New York, 1974, pp. 203–210. [19] P.N. Anyalebechi, Effects of hydrogen, solidification rate, and Ca on porosity formation in as-cast aluminum alloy A356, in: P.N. Crepeau (Ed.), Light Metals 2003, TMS, Warrendale, PA, 2007, pp. 971–981. [20] D.R. Poirier, P.K. Sung, Thermodynamics of hydrogen in Al–Si alloys, Met. Mater. Trans. A 33A (2002) 3874–3876.
Quantitative prediction of microporosity in aluminum alloys E. Escobar de Obaldia, S.D. Felicelli ∗ Department of Mechanical Engineering and Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA
Abstract A continuum model of dendritic solidification of multicomponent alloys is used to predict the volume fraction of porosity caused by precipitation of hydrogen gas during solidification of A356 aluminum plate castings. The conservation equations of mass, momentum, energy and species transport are solved to simulate the transient solidification process, including thermosolutal convection and segregation during the formation of the mushy zone. The amount of hydrogen supersaturation is calculated based on the transport of dissolved hydrogen and Sievert’s law. An innovative aspect of this work is the extension of the model to make quantitative predictions of the volume fraction of porosity. Although the hydrogen supersaturation has been used in previous works to estimate the amount of porosity, these predictions are usually grossly overestimated, because the barrier of pore nucleation is not considered. In this work it is shown that it is possible to make good estimations of the volume fraction of porosity by combining the hydrogen supersaturation with the calculated local solidification times. Computer simulations are performed in plate castings, showing good qualitative and quantitative agreement with experimental data. © 2007 Elsevier B.V. All rights reserved. Keywords: Modeling; Solidification; Porosity; Aluminum
1. Introduction The properties of cast parts are known to be primarily determined by the microstructure which develops during solidification. The formation of microporosity in particular is known to be one of the primary detrimental factors controlling fatigue lifetime and total elongation in cast aluminum components [1]. Microporosity refers to pores which range in size from micrometers to hundreds of micrometers and are constrained to occupy the interdendritic spaces near the end of solidification. The micropores can form due to microshrinkage produced by the pressure drop of interdendritic flow or because of the presence of dissolved gaseous elements in the liquid alloy. In this work, we focus on the calculation of gas microporosity and, unless noted otherwise, we use the term porosity to mean gas-induced microporosity. In the case of aluminum alloys, hydrogen is the most active gaseous element leading to gas porosity [2]. Many efforts have been devoted to the modeling of porosity formation in the last 20 years, particularly in aluminum alloys [3–8] and, in a lesser degree, to nickel superalloys [2,9] and steels [10,11]. More recently, rather sophisticated models have been developed to include the effect of pores on fluid flow (three∗
Corresponding author. Tel.: +1 662 325 1201; fax: +1 662 325 7223. E-mail address: [email protected] (S.D. Felicelli).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.072
phase transport) [12], multiscale frameworks that consider the impingement of pores on the microstructure [13] and new nucleation mechanisms based on entrainment of oxide bifilms [14]. A recent review on the subject of computer simulation of porosity and shrinkage related defects has been published by Stefanescu [15]. 2. Solidification model The model presented here is based on a robust and welltested multicomponent solidification model which calculates macrosegregation during solidification of a dendritic alloy with many solutes [16]. The model solves the conservation equations of mass, momentum, energy and each alloy component within a continuum framework in which the mushy zone is treated as a porous medium of variable permeability. In order to predict whether microporosity forms, the solidification shrinkage due to different phase densities, the concentration of gas-forming elements and their redistribution by transport during solidification were later added to the model [2]. In this form, the model was able to predict regions of possible formation of porosity by comparing the Sievert’s pressure with the local pressure; however, no capability was developed to calculate the amount of porosity. The model has already been presented in detail in previous works and will not be repeated here. In the following, the
266
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
model extension to calculate the volume fraction of gas porosity is described. This extension is based on the calculation of two new variables, the amount of hydrogen supersaturation and the local solidification time. The calculation of the volume fraction of hydrogen microporosity is based on the method of Poirier et al. [17], in which the mass of gas is assumed to be given by the supersaturation of hydrogen in both the liquid and the solid phases: 100mH = (ClH − SlH )gl + (CsH − SsH )gs
(1)
where mH is the mass fraction of hydrogen, gl and gs the volume fractions of liquid and solid, respectively, ClH and CsH the concentrations of hydrogen in wt.% and SlH and SsH are the solubilities. The solubility of hydrogen in the liquid is calculated with the Sievert’s law [18] while for the solid we take SsH = kH SlH , where kH is the partition coefficient of hydrogen. We assume complete diffusion of hydrogen in the local solid, hence CsH = kH ClH , and Eq. (1) can be rewritten as: 100mH = (ClH − SlH )(gl + kH gs )
(2)
Assuming ideal gas behavior, we can then calculate the volume fraction of hydrogen due to supersaturation as: gH =
¯ mH ρRT MPH2
(3)
where R is the universal gas constant, T the temperature in K, PH2 the pressure of hydrogen gas in atm, M the molecular weight
of hydrogen gas and ρ¯ is the density of the solid plus liquid mixture. The problem with the above approach is that Eq. (1), from which Eq. (3) is derived, assumes that all the hydrogen exceeding the solubility will become gas pores, which of course is an overestimation because the pores need to overcome nucleation barriers in order to become actual pores. Most of the mechanisms that have been proposed for the nucleation of pores are based on the size of interdendritic cavities and the theory of heterogeneous nucleation on non-wetted surfaces. After nucleation, pore growth occurs by diffusion of hydrogen into the pore. These ideas have been challenged by Campbell and co-workers [14] and others, who propose a nucleation-free mechanism for pore formation based on the concept of double oxide films or bifilms. In this scenario, during pouring in a casting process, the liquid surface of the alloy can fold upon itself. Because the liquid surface is covered by an oxide film, the folding action leads to bifilms, which are entrained into the bulk melt as a pocket of air enclosed by the bifilm. In effect, the bifilm with its air pocket is the beginning of a pore. After entrainment, the turbulence causes the bifilm to convolute and contract. Posterior pore growth can occur by the simple action of unfurling of the bifilms, without the aid of hydrogen diffusion. Although the identification of the mechanisms of pore formation and growth is still a subject of active research, the following two observations are supported by a large number of experimental and modeling works with aluminum alloys:
Fig. 1. Solidification of A356 plate casting at 400 s for 0.31 cm3 /100 g of hydrogen content. (a) Isotherms in K; (b) volume fraction of liquid and velocity vectors; (c) total concentration of H in wt.%.
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
(a) The amount of porosity increases for higher initial hydrogen content in the alloy. (b) The amount of porosity decreases for higher cooling rate. In view of these observations, we propose to modify the calculated volume fraction of porosity given by Eq. (3) according to: gP = agH tfb + gk
(4)
where gP is the corrected volume fraction of porosity, gH the volume fraction of hydrogen porosity assuming complete precipitation of the supersaturation (calculated in Eq. (3)), tf the local solidification time (calculated from the solidification history), a and b the experimental constants and gk is the volume fraction of porosity due to interdendritic shrinkage. In most cases with non-negligible hydrogen content, this last term is small compared with gas porosity; shrinkage porosity will not be considered in this work. Note that in Eq. (4), gH is implicitly also a function of the freezing time through the convection and segregation that occurred during solidification. The constant a in Eq. (4) is linked to the origin mechanism of the pores. From the heterogeneous nucleation perspective, it can be viewed as the fraction of inclusions or sites that overcame the nucleation bar-
267
riers in a H-supersaturated environment; from the double oxide bifilm perspective, it can be viewed as the fraction of bifilms that became active by the unfurling mechanism. The constant b in Eq. (4) is linked to the pore growth mechanism and carries information on the time that the pores need to grow by hydrogen diffusion and/or unfurling. An exponent different than one in gH was also tried in Eq. (4), with no significant improvement in results. A relation similar to Eq. (4), but without the supersaturation and shrinkage terms, was used by Anyalebechi [19] to analyze experimental results with alloy A356. 3. Results and discussion The solidification model is discretized in space and integrated in time using a finite element algorithm that is described by Felicelli et al. [2,16]. Aluminum A356 alloy is solidified by simulation in a bottom-cooled two-dimensional mold. The twodimensional simulated casting has dimensions of 2.6 cm in width and 30 cm in height. Gravity acts downwards. In addition to the alloy solutes in A356 (Si and Mg), the gas-forming element, hydrogen, is considered. The computational domain is the casting; the top boundary is left open in order to allow for liquid flow to feed shrinkage. A no-slip condition is used for velocity
Fig. 2. Experimental [3] and calculated porosity for different initial hydrogen contents.
268
E. Escobar de Obaldia, S.D. Felicelli / Journal of Materials Processing Technology 191 (2007) 265–269
at the bottom and two vertical boundaries, and a stress-free condition is used on the top open boundary. Solute diffusion flux is set to zero on the boundaries, with the exception of hydrogen, for which a dehydrogenation flux condition is used. The thermal boundary conditions utilized in Poirier et al. [7] are used here, which are extracted from a measured thermal history in the plates cast by Fang and Granger [3]. The simulations start with an all-liquid alloy of the nominal composition initially at a uniform temperature of 958 K, which is 70 K of superheat. The thermodynamic and transport properties of the alloy, including the alloy elements and hydrogen, are the same as the ones in ref. [7], with the exception of the partition coefficient of hydrogen, for which we used the developments of Poirier and Sung [20] to include the effect of the high eutectic fraction in A356. We performed simulations for the following values of initial hydrogen content: 0.11, 0.25 and 0.31 cm3 /100 g (note: 1 wt.% = 1.12 × 104 cm3 /100 g), for which measured volume fraction of porosity were reported in the experiments by Fang and Granger [3]. Fig. 1a–c shows iso-contour plots of temperature, volume fraction of liquid and total concentration of hydrogen after 400 s of simulation time. Approximately half of the casting has solidified completely and the mushy zone is about 40 mm thick. For clarity, the liquidus and eutectic isotherms are indicated with solid lines in Fig. 1a. Fig. 1b also shows the velocity field produced by thermosolutal convection, which is noticeable only in the all-liquid region. In the top 10% of the mushy zone, the velocities reduce to less than 10−6 m/s and their effect on the pressure can be neglected compared with the metallostatic head. The total concentration of hydrogen is shown in Fig. 1c, where a thin layer depleted in hydrogen can be observed near the surface of the casting due to dehydrogenation occurred during solidification and cooling. The volume fraction of porosity for each of the initial hydrogen contents is shown in Fig. 2. To facilitate comparison with the experimental data of ref. [3], the cooling rate is used as the x-variable, instead of solidification time. The dashed line labeled “Best Fit” is the porosity calculated with Eq. (4) using the Best Fit of the constants a and b for a given hydrogen content. The solid line labeled “Common Fit” is the porosity predicted using Eq. (4) with a common set of constants a and b for the three levels of hydrogen content: a = 0.003, b = 0.97. It is observed that when the experimental constants are fit for an individual hydrogen content, the predicted porosity (dashed line) agrees very well with the experimental data, except at the highest cooling rate. For the three values of initial hydrogen content considered, the Best Fit prediction underestimated the porosity at the highest cooling rate. This discrepancy is expected because of the contribution of shrinkage porosity (not considered in these calculations): at high cooling rate (short freezing time) there is not enough time for the gas pores to evolve and shrinkage porosity becomes relatively more important. If we attempt to predict the amount of porosity using a common set of constants in Eq. (4) for all values of hydrogen content, it is observed (solid lines in Fig. 2) that the accuracy of prediction is varied, with a good capture of the cooling rate trend but less accuracy in the level of porosity. This might indicate that the number of originated pores is not directly related to the level of hydrogen supersaturation,
but could arise from alternative mechanisms of pore formation like the entrainment of oxide bifilms. 4. Conclusions A finite element model of dendritic solidification was extended to allow the calculation of the volume fraction of hydrogen microporosity. The amount of porosity is calculated from the local hydrogen supersaturation and solidification time. Computer simulations are performed in plate castings of aluminum alloy A356, showing good qualitative and quantitative agreement with experimental data. Acknowledgement This work was partly funded by the National Science Foundation through Grant Number CTS-0553570. References [1] J.F. Major, Porosity control and fatigue behavior in A356T61 aluminum alloy, AFS Trans. 105 (1997) 901–906. [2] S.D. Felicelli, D.R. Poirier, P.K. Sung, A model for prediction of pressure and redistribution of gas-forming elements in multicomponent casting alloys, Metall. Mater. Trans. B 31B (2000) 1283–1292. [3] Q.T. Fang, D.A. Granger, Porosity formation in modified and unmodified A356 alloy castings, AFS Trans. 97 (1989) 989–1000. [4] Q. Han, S. Viswanathan, Hydrogen evolution during directional solidifications and its effect on porosity formation in aluminum alloys, Metall. Mater. Trans. A 33A (2002) 2067–2072. [5] P.D. Lee, J.D. Hunt, Hydrogen porosity in directional solidified aluminium– copper alloys: a mathematical model, Acta Mater. 49 (2001) 1383– 1398. [6] M. M’Hamdi, T. Magnusson, C. Pequet, L. Amberg, M. Rappaz, Modeling of micropososity formation during directional solidification of an Al–7% Si alloy, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 311–318. [7] D.R. Poirier, P.K. Sung, S.D. Felicelli, A continuum model of microporosity in an aluminum casting alloy, AFS Trans. 109 (2001) 379–395. [8] J.D. Zhu, S.L. Cockcroft, D.M. Maijer, R. Ding, Simulation of microporosity in A356 aluminium alloy castings, Int. J. Cast Met. Res. 18 (2005) 229–235. [9] J. Guo, M.T. Samonds, Microporosity simulations in multicomponent alloy castings, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 303– 311. [10] K.D. Carlson, L. Zhiping, R.A. Hardin, C. Beckermann, Modeling of porosity formation and feeding flow in steel casting, in: D.M. Stefanescu, J. Warren, M. Jolly, M. Krane (Eds.), Modeling of Casting, Welding and Advanced Solidification Processes, vol. X, The Minerals, Metals & Materials Society, 2003, pp. 295–302. [11] P.K. Sung, D.R. Poirier, S.D. Felicelli, Continuum model for predicting microporosity in steel castings, Modell. Simul. Mater. Sci. Eng. 10 (2002) 551–568. [12] A.S. Sabau, S. Viswanathan, Microporosity prediction in aluminum alloy castings, Metall. Mater. Trans. B 33B (2002) 243–255. [13] P.D. Lee, A. Chirazi, R.C. Atwood, W. Wang, Multiscale modeling of solidification microstructures, including microsegregation and microporosity, in an Al–Si–Cu alloy, Mater. Sci. Eng. A A365 (2004) 57–65. [14] X. Yang, X. Huang, X. Dai, J. Campbell, J. Tatler, Numerical modeling of entrainment of oxide film defects in filling of aluminium alloy castings, Int. J. Cast Met. Res. 17 (2004) 321–331.
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