Abstract
In this article, an investigation of the interaction between gas porosity and a planar solid/liquid (SL) interface is reported. A two-dimensional numerical model able to accurately track sharp SL interfaces during solidification of pure metals and alloys is proposed. The finite-difference method and a rectangular undeformed grid are used for computation. The SL interface is described through the points of intersection with the grid lines. Its motion is determined by the thermal and solute gradients at each particular point. Changes of the interface temperature because of capillarity or solute redistribution as well as any perturbation of the thermal and solute field produced by the presence of non-metallic inclusions can be computed. To validate the model, the dynamics of the interaction between a gas pore and a solidification front in metal alloys was observed using a state of the art X-ray transmission microscope (XTM). The experiments included observation of the distortion of the SL interface near a pore, real-time measurements of the growth rate, and the change in shape of the porosity during interaction with the SL interface in pure Al and Al-0.25 wt pct Au alloy. In addition, porosity-induced solute segregation patterns surrounding a pore were also quantified.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- A 1, A 2, A 3, Q :
-
quantities defined in Eqs. [18] through [20] to calculate the normal growth velocity
- b E , b N , b S , b x , b y , b w :
-
coefficients in the discrete form of the transport equation of field variables
- C :
-
concentration
- C L , C S :
-
liquid and solid concentrations, respectively
- C P :
-
volumetric specific heat
- D L , D S :
-
solute diffusion coefficient in the liquid and solid phases, respectively
- Φ:
-
generalized field variable
- f :
-
coefficient that takes the value of 0.5
- G L , G S :
-
temperature gradient in the liquid and solid, respectively
- ΔH f :
-
latent heat of fusion
- h c :
-
distance between the planar interface and the center of the particle
- k :
-
partition coefficient
- k L , k S , k P :
-
thermal conductivities of the liquid, solid, and particle, respectively
- α L , α S :
-
thermal diffusivities of the liquid and solid, respectively
- K :
-
solid/liquid interface curvature
- Γ:
-
Gibbs-Thomson coefficient
- N 1, N 2, N 3, N 4 :
-
shape functions used for the interpolation of the field variables
- N:
-
unit normal vector
- Δn :
-
length of the probe projected along the normal direction
- m L :
-
liquidus slope
- R P :
-
particle radius
- T :
-
temperature
- T M :
-
melting temperature
- T L , T S :
-
temperature in the liquid and solid phases, respectively
- t, Δt :
-
time, time-step size
- V :
-
imposed growth velocity through external boundary conditions
- V n :
-
normal growth velocity
- x, z :
-
Cartesian coordinates
- Ξ:
-
generalized diffusivity
- L :
-
pertaining to the liquid phase
- S :
-
pertaining to the solid phase
- i :
-
at the solid/liquid interface
- o :
-
at the beginning of the time-step
- nL, nS :
-
on the normal direction in the liquid and solid phases, respectively
References
D.M. Stefanescu: The Science and Engineering of Casting Solidification, Kluwer Academic/Plenum Publishers, New York 2002.
H. Fredriksson and L. Svensson: Metall. Trans. B, 1976, vol. 7B, pp. 599–06.
J.M. Kim, D.G. Kim, H.W. Kwon, and C.R. Loper, Jr.: Scripta Mater., 1998, vol. 39 (7), pp. 969–75.
P.D. Lee and J.D. Hunt: Modeling of Casting, Welding, and Advanced Solidification Processes VII, M. Cross and L. Campbell, eds., TMS, Warrendale, PA, 1995, pp. 585–92.
H. Jamgotchian, R. Trivedi, and B. Billa: J. Cryst. Growth, 1993, vol. 134, pp. 181–95.
D. Shangguan and D.M. Stefanescu: Metall. Trans. B, 1991, vol. 22B, pp. 385–88.
T.S. Piwonka and M.C. Flemings: Trans. TMS-AIME, 1966, vol. 236, pp. 1157–65.
D.R. Poirier, K. Yeum, and A.L. Maples: Metall. Trans. A, 1987, vol. 18A, pp. 1979–87.
P.D. Lee and J.D. Hunt: Acta Mater., 1997, vol. 45 (10), pp. 4155–69.
V.R. Voller and S. Sundarraj: Int. J. Heat Mass Transfer, 1995, vol. 38 (6), pp. 1009–18.
P. Rousset, M. Rappaz, and B. Hannart: Metall. Mater. Trans. A, 1995, vol. 26A, pp. 2349–58.
L.N. Brush and R.F. Sekerka: J. Cryst. Growth, 1989, vol. 96, pp. 419–41.
M. Lacroix: Num. Heat Transfer B, 1989, vol. 15, pp. 191–210.
C.H. Kuo and W.C. Schreiber: HTD, 1994, vol. 275, pp. 57–64.
S.L. Wang, R.F. Sekerka, A.A. Weeler, B.T. Murray, S.R. Coriel, R.J. Braun, and G.B. McFadden: Physica D, 1993, vol. 69, pp. 189–200.
J.A. Warren and W.J. Boettinger: Acta Metall. Mater., 1995, vol. 43 (2), pp. 689–703.
W.J. Boettinger and J.A. Warren: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 657–69.
S. Osher and J. Sethian: J. Comp. Phys., 1988, vol. 79, pp. 12–49.
J.A. Sethian and J. Strain: J. Comp. Phys., 1992, vol. 98, pp. 231–53.
A. Karma and W.-J. Rappel: Phys. Rev. E, 1996, vol. 53 (4), pp. R3017–20.
A. Karma: Phys. Rev. Lett., 2001, vol. 87, 115701, pp.1–4
B. Merriman, J.K. Bence, and S.J. Osher: J. Comp. Phys., 1994, vol. 112, pp. 334–63.
M. Sussman, P. Smereka, and S. Osher: J. Comp. Phys., 1994, vol. 114, pp. 146–59.
T.Y. Hou, Z. Li, S. Osher, and H. Zhao: J. Comp. Phys., 1997, vol. 134, pp. 236–52.
S. Chen, B. Merriman, S. Osher, and P. Smereka: J. Comp. Phys., 1997, vol. 135, pp. 8–29.
D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang: J. Comp. Phys., 1999, vol. 155, pp. 410–38.
Yung-Tae Kim, N. Goldenfeld, and J. Dantzig: Phys. Rev. E, 2000, vol. 62 (2), pp. 2471–74.
W. Shyy, H.S. Udaykumar, M.M. Rao, and R.W. Smith: Computational Fluid Dynamics with Moving Boundaries, Taylor & Francis, Washington, DC, 1996.
H.S. Udaykumar and W. Shyy: Num. Heat Transfer B, 1995, vol. 27, pp. 127–53.
H.S. Udaykumar and W. Shyy: Int. J. Heat Mass Transfer, 1995, vol. 38 (11), pp. 2057–73.
A.V. Catalina: Ph.D. Dissertation, The University of Alabama, Tuscaloosa, AL, 2000.
A.V. Catalina and D.M. Stefanescu: Modeling of Casting and Solidification Processes 1999, C.P. Hong, J.K. Choi, and D.H. Kim, eds., Yonsei University pub., Korea, Seoul, 2000, pp. 3–11.
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery: Numerical Recipes in C, Cambridge University Press, Cambridge, United Kingdom, 1986, p. 113.
R.D. Cook, D.S. Malkus, and M.E. Plesha: Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, NY, 1989, pp. 98–99 and 166–67.
V.G. Smith, W.A. Tiller, and J.W. Rutter: Can. J. Phys., 1955, vol. 33, pp. 723–45.
A.V. Catalina, S. Mukherjee, and D.M. Stefanescu: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 2559–68.
W.W. Mullins and R.F. Sekerka: J. Appl. Phys., 1964, vol. 35 (2), pp. 444–51.
A.V. Catalina, S. Sen, D.M. Stefanescu, and P.A. Curreri: Modeling of Casting, Welding and Advanced Solidification Processes IX, P.R. Sahm, P.N. Hansen, and J.G. Conley eds., Shaker Verlag, Aachen, Germany, 2000, pp. 445–52.
A.V. Catalina, D.M. Stefanescu, and S. Sen: Modeling of Casting, Welding and Advanced Solidification Processes X, D.M. Stefanescu, J. Warren, M. Jolly, and M. Krane eds., TMS, Warrendale, PA, 2003, pp. 125–32.
S. Sen, W.F. Kaukler, P.A. Curreri, and B.K. Dhindaw: Proc. 6th Asian Foundry Congr., A.K. Chakrabarti, B.K. Dhindaw, G.L. Datta, and C.S. Sivaramakrishnan, eds., The Institute of Indian Foundrymen, Calcutta, 1999, pp. 303–08.
W. Kurz and D.J. Fisher: Fundamentals of Solidification, 2nd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1986, pp. 240–41.
CRC Handbook of Chemistry and Physics, 67th ed., R.C. West, ed., CRC Press Inc., Boca Raton, FL, 1986.
Binary Alloy Phase Diagrams, T.B. Massalski, ed., ASM, Metals Park, OH, 1987, p. 90.
Thermophysical Properties of Matter, Y.S. Touloukian, ed., The Macmillan Company, New York, NY, vol. 4.
J. Forsten and H.M. Miekkoja: J. Inst. Met., 1967, vol. 95, pp. 143–45.
Thermophysical Properties of High Temperature Solid Materials, Y.S. Touloukian, ed., The Macmillan Company, New York, NY, vol. 4.
D.R. Poirier and G.H. Geiger: Transport Phenomena in Materials Processing, TMS, Warrendale, PA, 1994, pp. 192 and 620.
Engineering Properties of Selected Ceramic Materials, J.F. Lynch, C.G. Ruderer, and W.H. Duckworth, eds., The American Ceramic Society, Inc., Columbus, OH, 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Catalina, A.V., Sen, S., Stefanescu, D.M. et al. Interaction of porosity with a planar solid/liquid interface. Metall Mater Trans A 35, 1525–1538 (2004). https://doi.org/10.1007/s11661-004-0260-z
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11661-004-0260-z