6.5 Numerical Examples - Optimization Problem
6.5.2 Optimization problem for 2D horizontal solidification of an alloy138
The first optimization considered is the design of time history of the external mag-netic field for convection damping during a two-dimensional horizontal solidification of a Pb-Sn alloy, which was previously addressed in Section 6.4.2. The problem is characterized by strong thermosolutal buoyancy forces and severe convection driven macrosegregation in the solidifying alloy. The optimization problem was solved us-ing 4 and 5 design variables that use third and fourth degree Bezier-Bernstein curves, respectively. The tolerance during the optimization process for terminating the non-linear CG iterations was chosen as 10−3. The time interval under consideration is given by t ∈ [0, 120]. ˆt obeys ˆt ∈ [0, 1] with tmax in Eq. (6.23) being 120 seconds.
Figs. 6.9(a) and 6.9(b) show the cost functional variation for both 4 and 5 design variables. Figs. 6.10(a) and 6.10(b) show the variation of the optimal magnetic field under the time interval considered here for both cases. The optimal magnetic field is captured well by both design space discretizations. It is higher in the initial stages
CG Iterations
Figure 6.9: Cost functional versus CGM iteration for Example 6.5.2 (a) 4 design variables (b) 5 design variables.
Time (s)
Figure 6.10: Plot of the optimal magnetic field for Example 6.5.2 (a) 4 design variables (b) 5 design variables.
577.2
Figure 6.11: (a) Temperature (K) (b) Solute concentration (c) Liquid volume frac-tion and velocity at t = 120 s (optimal time varying magnetic field) for Example 6.5.2.
to counter the strong thermosolutal buoyancy force, prevalent initially. Under the combined influence of thermosolutal and Lorentz forces the convection weakens with time and the magnetic field decreases with time accordingly. The magnetic field ap-proaches a final asymptotic value, lower than its starting magnitude, towards the end of the time interval that is required to suppress weaker thermosolutal convec-tion in the later stages. This eliminates the need to maintain a constant high field throughout the solidification process.
Table 6.9 shows the difference between the maximum and minimum solute con-centrations, ∆C, at two different times. Figs. 6.11(a)-6.11(c) show different field variables under the influence of the optimal magnetic field at t = 120 s. From these figures, it is clear that the application of the optimal magnetic field helps in damp-ing out thermosolutal convection significantly. From Fig. 6.11(b) and Table 6.9, it is evident that macrosegregation is suppressed to a large extent and the solute concentration profile in the solidifying alloy is more uniform in the presence of the
Table 6.9: Comparison of ∆C at two different times for Example 6.5.2 with no and optimal magnetic fields (∆C = Cmax− Cmin).
∆C (%Sn) ∆C (%Sn)
(t = 60 s) (t = 120 s)
No magnetic field 13.35 17.57
Optimal magnetic field 0.92 1.52
z
Figure 6.12: Problem domain for Example 6.5.3 involving 3D directional solidifica-tion of a Pb-Sn alloy. G = 1000 K/m, r = 0.0167 K/s, C0 = 10 wt. % Sn and T0 = Tliq.
optimal field than that observed in Fig 6.5(i)(b) in the absence of any field.
6.5.3 Optimization problem for 3D directional solidification of an alloy
The second optimization problem considered involves designing the time history of the applied magnetic field during the three-dimensional directional solidification of a Pb-Sn alloy addressed previously in Section 3.4.3 of Chapter 3. The time interval
CG Iterations
Costfunctional
0 1 2 3 4 5 6 7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 6.13: Cost functional versus CG iteration for Example 6.5.3.
considered here is given by t ∈ [0, 800] with tmax being 800 seconds. The tolerance during the optimization process for terminating the non-linear CG iterations was chosen as 10−3. Fig. 6.13 shows the cost functional and Fig. 6.14 shows the variation of the optimal magnetic field under the time interval considered. The time variation is very similar to that observed in the previous example. The magnitude of the applied field, which is higher in the initial stages to suppress stronger thermosolutal buoyancy forces, decreases with time. Towards the end, there is a marginal increase in magnetic field to suppress residual solutal convection in the melt, but it is lower than the starting magnitude. This marginal increase is required to counter the residual solutal convection that is responsible for freckle formation and growth at later times. Figs. 6.15(a)-6.15(b) show the solute concentration and liquid volume fraction fields for the optimal magnetic field. In the absence of a magnetic field, channel formation occurs in the bulk leading to the formation of freckle defects where solute concentration differs greatly compared to the bulk as observed from Figs.
6.16(a)-6.16(b). The optimal magnetic field is successful in suppressing channel formation and inhibiting macrosegregation that were observed in Figs.
6.16(a)-Time (s)
Optimalmagneticfield(T)
0 200 400 600 800
3 3.5 4 4.5 5 5.5 6 6.5
Figure 6.14: Plot of the optimal magnetic field for Example 6.5.3.
6.16(b) and also in Table 6.10, where ∆C values at two different times are tabulated, in the absence of any magnetic field. The need to maintain a constant high magnetic field throughout the solidification process is also eliminated.
6.6 Summary
A numerical study of the effect of magnetic fields on solidification of metallic alloys with significant mushy zones was presented. The direct problem was based on a single domain volume-averaged model for alloy solidification under the influence of magnetic fields. Based on this, a continuum sensitivity based finite dimensional optimization problem was formulated for designing the time history of externally imposed magnetic fields. The main objective here was to use an optimal magnetic field to suppress thermosolutal convection and inhibit macrosegregation during alloy solidification. The orientation of the applied magnetic field was fixed and only the magnitude was chosen to vary with time. For metallic alloys considered here, the Lorentz force arising due to the application of magnetic fields on moving melts
X Y
Figure 6.15: (a) Solute concentration (b) Liquid volume fraction at t = 800 s (op-timal time varying magnetic field) for Example 6.5.3.
X Y
Figure 6.16: (a) Solute concentration (b) Liquid volume fraction at t = 800 s (no magnetic field) for Example 6.5.3.
Table 6.10: Comparison of ∆C at two different times for for Example 6.5.3 with no and optimal magnetic fields (∆C = Cmax− Cmin).
∆C (%Sn) ∆C (%Sn)
(t = 400 s) (t = 800 s)
No magnetic field 5.97 7.40
Optimal magnetic field 0.40 1.65
is the primary damping force. The cost functional minimization was carried out using a non-linear conjugate gradient method that utilized finite element solutions of the continuum direct and sensitivity problems. The magnetic field was tailored so as to capture variations in thermosolutal convection during solidification of the alloy. The magnitude of the optimal magnetic field was higher during the initial stages of solidification and decreased with time before approaching an asymptotic value for the 2D example in Section 6.5.2 or increasing slightly towards the end of the time interval considered for the 3D example in Section 6.5.3. During 3D directional solidification, this increase is required to counter the solutal convection that is responsible for freckle formation and growth at later times. The use of an optimal magnetic field alleviates the need to use a magnetic field with a high magnitude throughout the solidification process, which in turn translates into power and energy savings. Optimization examples in both two and three dimensions were considered to highlight the efficacy and dimension independent nature of the design simulator.
Chapter 7
Conclusions and suggestions for the future research
A stabilized finite element based numerical methodology has been developed in this thesis to address alloy solidification processes based on a single domain volume av-eraged continuum mathematical model. The main aim here is to model alloy solidi-fication processes characterized by strong convection that leads to macrosegregation or large scale solute redistribution in the solidifying alloy. Macrosegregation leads to the presence of various kinds of defects in solidifying alloys. The numerical scheme is first validated using an example involving solidification of ammonium chloride and water mixture where thermosolutal buoyancy forces are very strong. It is then extended to model the combined effect of buoyancy and shrinkage driven flows. The developed numerical scheme is general in nature and suitable for modeling a wide class of solidification problems in both two and three dimensions. The formation of segregation defects like freckles and channels, and surface defects in metallic al-loys due to different types of convection is of special interest here. Towards this end, we explore certain techniques to control macrosegregation or related defects in
146
solidifying alloys.
The role of uneven surface topography on transport phenomena occurring during alloy solidification is explored next. We focus on solidification of an aluminum-copper alloy from sinusoidal cavities and the effect of uneven surface topography on convection and macrosegregation during horizontal and vertical alloy solidification.
The volume-averaged solidification model is then coupled with a hypoelastic rate dependent small deformation model to study stresses developing in the mushy zone and solid as solidification occurs from a sinusoidal cavities. Non-uniform contact at the metal-mold interface leads to air-gap formation that results in a dynamic variation of the heat flux at the metal-mold junction. Inverse segregation that occurs at the casting bottom due to shrinkage driven flows is also affected by changes in topography. A parametric analysis is then performed to determine the effect of various process variables like mold surface wavelength, melt superheating, solute concentration and mold materials on air-gap evolution at the metal-mold interface and stress development in the mushy and solid zones. With maximum equivalent stress and growth front unevenness (a surface defect) as the criteria, an optimal wavelength is determined from the parametric analysis.
The role of magnetic fields in damping convection and suppressing macrosegre-gation during solidification of metallic alloys is examined. The application of a mag-netic field on a moving fluid produces Lorentz force that opposes fluid motion. An finite dimensional optimization problem is formulated based on the continuum sen-sitivity method (CSM) to design the time history of the externally applied magnetic field to damp thermosolutal convection and suppress macrosegregation. Both two and three dimensional optimization problems are addressed. In particular, freckle suppression during directional solidification of a lead-tin alloy in three dimensions
using time varying optimal magnetic fields is demonstrated.
The model described in this thesis is valid only on the macroscale and does not take into account nucleation and microstructure evolution. The underlying microstructure in a solidifying alloy has profound influence on properties on the macroscale. A more complete picture will emerge when the continuum model on the macro scale is coupled to a dendritic evolution model on the microscale to study the effects of microstructure on macroscopic variables. This aspect is addressed briefly next.
7.1 Multi-length scale framework for alloy solidi-fication processes
A multi-length scale model that takes into account nucleation and microstructure evolution on the microscale will give a more realistic picture of the alloy solidification process. The microstructure evolution model can be further coupled with a proba-bilistic nucleation models to predict grain growth and orientation in the solid phase [103]. The envisaged multi-length scale framework will involve the current single domain model on the macroscale with a dendritic evolution model based on ex-tended finite element (XFEM) method [104] on the micro scale. Effective averaging techniques are required to transfer information from the micro to the macro scale.
Towards this end, homogenization based upscaling methods can be used as effective tools for property predictions [105]. These methods are superior compared to simple averaging techniques for obtaining property estimates from smaller scales. Effective micro-macro coupling strategies need to be developed so as to represent physics of scales smaller than the computational mesh in a consistent manner [105]. The use
of homogenization upscaling methods will enable us to study the fluctuation of field variables on different length scales [105]. Development of this framework will allow us to address a multi-length scale design problem where the main objective will be to obtain a desired microstructure by controlling macroscopic process parameters like boundary heat flux or magnetic fields.
Appendix A
Material properties
Important material properties used in various numerical examples in Chapters 2-6 are listed in Tables A.1-A.3.
150
Table A.1: Important physical parameters for ammonium-chloride water system [8].
Symbol Value
Solid thermal conductivity ks 3.93 × 10−4 kW m−1K−1 Liquid thermal conductivity kl 4.68 × 10−2 kW m−1K−1 Solid specific heat cs 1.87 kJkg−1K−1
Liquid specific heat cl 3.249 kJkg−1K−1 Latent heat of fusion hf 313.8 kJkg−1 Equilibrium partition ratio κp 0.30
Thermal expansion coefficient βT 3.832 × 10−4 K−1 Solutal expansion coefficient βC 0.257 (kg/kg)−1
Solid density ρs 1078.0 kgm−3
Liquid density ρl 1078.0 kgm−3
Viscosity of the liquid µ 1.3 × 10−3 kgm−1s−1 Eutectic temperature Te 257.75 K
Melting point Tm 633.59 K
Acceleration due to gravity g 9.81 ms−2 Slope of the liquidus line mliq −4.68 Kwt%−1 Liquid solute diffusivity Dl 1.5 × 10−9 m2s−1
Table A.2: Important physical parameters for lead-tin alloys [28, 56].
Symbol Value
Solid thermal conductivity ks 1.855 × 10−2 kW m−1K−1 Liquid thermal conductivity kl 1.855 × 10−2 kW m−1K−1 Solid specific heat cs 0.167 kJkg−1K−1
Liquid specific heat cl 0.167 kJkg−1K−1 Latent heat of fusion hf 37.6 kJkg−1 Equilibrium partition ratio κp 0.31
Thermal expansion coefficient βT 1.2 × 10−4 K−1 Solutal expansion coefficient βC 0.515 (kg/kg)−1
Solid density ρs 1.01 × 104 kgm−3
Liquid density ρl 1.01 × 104 kgm−3
Viscosity of the liquid µ 2.495 × 10−3 kgm−1s−1 Eutectic temperature Te 456.0 K
Eutectic concentration of Sn Ce 61.9 wt.%
Melting point of lead Tm 600.0 K
Initial temperature Ti 580.0 K
Ambient temperature T∞ 298 K
Acceleration due to gravity g 9.81 ms−2 Slope of the liquidus line mliq −2.33 Kwt%−1 Liquid solute diffusivity Dl 3 × 10−9 m2s−1 Liquid electrical conductivity σe 1.5 × 106 Ohm−1m−1
Table A.3: Important physical parameters for aluminum-copper alloys,[41, 43].
Symbol Value
Solid thermal conductivity ks 0.1925 kW m−1K−1 Liquid thermal conductivity kl 0.0826 kW m−1K−1 Solid heat capacity cs 1.06 kJkg−1K−1 Liquid heat capacity cl 1.06 kJkg−1K−1 Latent heat of melting hf 397.5 kJkg−1 Equilibrium partition ratio κ 0.17
Thermal expansion coefficient βT 4.95 × 10−5 K−1 Solutal expansion coefficient βC −2.0 (kg/kg)−1 Shrinkage coefficient βsh 0.10417
Solid density ρs 2650 kgm−3
Liquid density ρl 2400 kgm−3
Liquid viscosity µ 0.003 kgm−1s−1
Eutectic temperature Te 821.0 K Melting temperature of Al Tm 933.0 K Ambient temperature Tamb 298.0 K Eutectic concentration of Cu Ce 33.2 wt.%
Acceleration due to gravity g 9.81 ms−2
Slope of the liquidus line mliq −3.3735 Kwt%−1 Liquid solute diffusivity Dl 3 × 10−9 m2s−1
Bibliography
[1] M. C. Flemings, Solidification Processing, Materials Science and Engineering Series, McGraw Hill Publications, 214-262:1974.
[2] W. Kurz and D. J. Fisher, Fundamentals of Solidification, (Third Edition), Trans Tech Publications, Switzerland, 1989.
[3] W. Shyy, H. S. Udaykumar, M. M. Rao and R. W. Smith, Computational Fluid Dynamics with Moving Boundaries, (First Edition), Tailor and Francis, Washington, D.C., 1996.
[4] M.G. Worster, The dynamics of mushy layers, Interactive Dynamics of Con-vection and Solidification, S.H. Davis et al. editors, Kluwer Academic Pub-lishers, Netherlands, 1992.
[5] V. C. Prantil, P. R. Dawson, Application of mixture theory to continuous casting, American Society of Mechanical Engineers, 29:47-54, 1983.
[6] R. N. Hills, D. E. Loper and P. H. Roberts, A Mathematically consistent model of a mushy zone, Quaterly Journal of Mechanics and Applied Mathematics, 36:505-539, 1983.
[7] F. P. Incropera and W. D. Bennon, A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems - I. Model
154
formulation, International Journal of Heat and Mass Transfer, 30:2161-2170, 1987.
[8] F. P. Incropera and W. D. Bennon, A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems. II. Appli-cation to solidifiAppli-cation in a rectangular cavity, International Journal of Heat and Mass Transfer, 30:2171-2187, 1987.
[9] V. R. Voller, A. D. Brent and C. Prakash, The modeling of heat, mass and solute transport in solidification systems, International Journal of Heat and Mass Transfer, 32:1718-1731, 1989.
[10] P. J. Prescott and F. P. Incropera, Numerical simulation of a solidifying Pb-Sn alloy : The effects of cooling rate on thermosolutal convection and macroseg-regation, Metallurgical Transactions B, 22B:529-540, 1991.
[11] P. J. Prescott, F. P. Incropera and W. D. Bennon, Modeling of dendritic solidification Systems: reassessment of the continuum momentum equation, International Journal of Heat and Mass Transfer, 34(9):2351-2359, 1991.
[12] P. J. Prescott and F. P. Incropera, Convective transport phenomenon and macrosegregation during solidification of a binary metal alloy-I. ASME Journal of Heat Transfer, 116:735-741, 1994.
[13] F. P. Incropera and J. Ni, Extension of the continuum model for transport phenomenon occurring during metal alloy solidification-I. The conservation equations, International Journal of Heat and Mass transfer, 38(7):1271-1284, 1995.
[14] F. P. Incropera and J. Ni, Extension of the continuum model for transport phenomenon occurring during metal alloy solidification-II. Microscopic con-siderations, International Journal of Heat and Mass transfer, 38(7):1285-1296, 1995.
[15] C. Beckermann, R. Viskanta, Double-diffusive convection during dendritic so-lidification of a binary mixture, PhysicoChemical Hydrodynamics 10:195-213, 1988.
[16] C. Prakash, Two phase model for binary solid-liquid phase change. Part I:
Governing equations, Numerical Heat Transfer B 1990; 18: 131-145.
[17] J. Ni and C. Beckermann, A volume-averaged two-phase model for transport phenomena during solidification, Metallurgical Transactions B, 22B:349-361, 1991.
[18] S. D. Felicilli, J. C. Heinrich, D.R. Poirier, Numerical model for dendritic solidification of binary alloys, Numerical Heat Transfer B, 23:461-481, 1993.
[19] E. Mcbridge , J. C. Heinrich and D. R. Poirier, Numerical simulation of incom-pressible flow driven by density variations during phase change, International Journal of Numerical Methods in Fluids, 31:787-800, 1999.
[20] J. C. Heinrich and D. R. Poirier, Convection modelling in directional solidifica-tion, Comptes Rendus Academie Sciences Serie I - Mechanique, 332:429-445, 2004.
[21] C. Beckermann, Modeling of macrosegregation: application and future needs, International Materials Review, 47:243-261, 2002.
[22] A. V. Reddy and C. Beckermann, Modeling of macrosegregation due to ther-malsolutal convection and contraction-driven flow in direct chill continuous casting of an Al-Cu round ingot, Metallurgical Transactions B, 28B:479-489, 1997.
[23] M. C. Schneider and C. Beckermann, Formation of macrosegregation by mul-ticomponent thermosolutal convection during the solidification of steel, Met-allurgical Transactions A, 26 A:2373-2388, 1995.
[24] J. P. Gu and C. Beckermann, Simulation of macrosegregation in a large steel ingot, Metallurgical Transactions A, 30 A:1357-1366, 1999.
[25] S. D. Felicelli, D. R. Poirier and J. C. Heinrich, Macrosegregation patterns in multicomponent alloys, Journal of Crystal Growth, 177:145-161, 1997.
[26] S. D. Felicelli, J. C. Heinrich and D. R. Poirier, Finite element analysis of directional solidification of multicomponent alloys, International Journal for Numerical Methods in Fluids, 27:207-227, 1998.
[27] S. D. Felicelli, J. C. Heinrich and D. R. Poirier, Three-dimensional simulations of freckles in binary alloys, Journal of Crystal Growth, 191:879-888, 1998.
[28] J. Guo and C. Beckermann, Three dimensional simulations of freckle forma-tion during binary alloy solidificaforma-tion: effect of mesh spacing, Numerical Heat Transfer A, 44:559-576, 2003.
[29] S. D. Felicelli, D. R. Poirier and J. C. Heinrich, Modeling freckle formation in three dimensions during solidification of muticomponent alloys, Metallurgical Transactions B, 29B:847-854, 1998.
[30] C. Frueh, S. D. Felicelli and D. R. Poirier, Predicting freckle-defects in di-rectionally solidified Pb-Sn alloys, Materials Science and Engineering: A, 328:245-255, 2002.
[31] J. C. Ramirez and C. Beckermann, Evaluation of a Raleigh-number-based freckle prediction criterion for Pb-Sn alloys and Ni-base superalloys, Metallur-gical Transactions A, 34A:1525-1536, 2003.
[32] S. Tin and T. M. Pollock, Predicting freckle formation in single crystal Ni-base superalloys, Journal of Materials Science, 39:7199-7205, 2004.
[33] F. Yigit and L. G. Hector Jr.,Critical Wavelengths for Gap Nucleation in Solidification Part I: Theoretical Methodology, Journal of Applied Mechanics, 67:66-76, 2000.
[34] F. Yigit and L. G. Hector Jr., Critical Wavelengths for Gap Nucleation in Solidification Part II: Results for Selected Mold-Shell Material Combinations, Journal of Applied Mechanics, 67:77-86, 2000.
[35] L. G. Hector Jr., J. Howarth, O. Richmond and W. Kim, Mold surface wave-length effects on gap nucleation in solidification, Journal of Applied Mechanics, 67:155-159, 2000.
[36] O. Richmond, R. Tien, Theory of thermal stress and air-gap formation during the early stages of solidification in as rectangular mold, Journal of Mechanics and Physics of Solids, 19:273-284, 1971.
[37] N. Zabaras, Y. Ruan and O. Richmond, Front tracking thermomechanical model for hypoelastic-viscoplastic behaviour in a solidifying body, Computer Methods in Applied Mechanics and Engineering, 81:333-364, 1990.
[38] N. Zabaras, Y. Ruan and O. Richmond, On the calculation of deformations and thermal stresses during axially symmetric solidification, Journal of Applied Mechanics, 58:865-871, 1991.
[39] R. Lewis, R. Ransing, A Correlation to describe interfacial heat transfer during solidification simulation and its use in the optimal feeding design of castings, Metallurgical Transactions B 29:437-448, 1998.
[40] A. Mo, Mathematical modeling of surface segregation in aluminum DC cast-ing caused by exudation, International Journal of Heat and Mass Transfer, 36(18):4335-4340, 1993.
[41] E. Haug, A. Mo and H. J. Thevik, Mathematical modeling of surface segrega-tion in aluminum DC casting caused by exudasegrega-tion, Internasegrega-tional Journal of Heat and Mass Transfer, 38(9):1553-1563, 1995.
[42] H. J. Thevik and A. Mo, Macrosegregation near a cast surface caused by exudation and solidification shrinkage, International Journal of Heat and Mass Transfer, 40(9):2055-2065, 1997.
[43] J. H. Chen, H. L. Tsai, Inverse segregation for a unidirectional solidification of aluminum-copper alloys, International Journal of Heat and Mass Transfer, 36(12):3069-3075, 1993.
[44] M. J. M. Krane and F. P. Incropera, Analysis of the effect of shrinkage on macrosegregation in alloy solidification, Transport Phenomena in Solidifica-tion, ASME HTD-284/AMD 182, ASME, New York, 13-27, 1994.
[45] L. G. Hector Jr., N. -Y. Li and J. R. Barber, Strain rate relaxation effect on freezing front growth stability during planar solidification of pure metals. Part 1 - Uncoupled theory, Journal of Thermal Stresses, 17:619-646, 1994.
[46] N. -Y. Li, L. G. Hector Jr. and J. R. Barber, Strain relaxation effect on freezing front growth stability during planar solidication of pure metals. Part 2 - Coupled theory, Journal of Thermal Stresses, 18:69-85, 1995.
[47] M. Rappaz, J. Drezet and M. Gremaud, A new hot-tearing criterion, Metal-lurgical Transactions A, 30:449-455, 1999.
[48] I. Farup and A. Mo, Two-phase modeling of mushy zone parameters assosi-cated with hot tearing, Metallurgical Transactions A, 31:1461-1472, 2000.
[49] A. Strangeland, A. Mo, ∅. Nielsen, D. Elskin and M. Hamdi, Development of thermal strain in the coherent mushy zone during solidification of aluminum alloys, Metallurgical Transactions A, 35A:2903-15, 2004.
[50] G. M. Oreper and J. Szekely, The effect of a magnetic field on transport phenomena in a Bridgeman-Stockbarger crystal growth, Journal of Crystal
[50] G. M. Oreper and J. Szekely, The effect of a magnetic field on transport phenomena in a Bridgeman-Stockbarger crystal growth, Journal of Crystal