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ABSTRACT
A fully transient model of the direct-chill casting process is used to predict the macrosegregation development of aluminum alloy 7050. The ingot diameter, casting speed, superheat, secondary cooling, and thickness of pure Al at startup are varied. Predicted radial composition distributions are fit to Weibull probability density functions at each axial location, and the normalized standard deviation describes the macrosegregation level and the time when the process reaches steady state. The sump depth, steady-state height, and macrosegregation level were most affected by changes in casting speed and ingot diameter. The pure Al dilutes the alloy and delays compositional steady state.
Nomenclature
| a | = | Coefficients of conservation equations |
| B | = | Buoyancy force |
| C | = | Composition |
| c | = | Specific heat |
| D | = | Mass diffusion coefficient |
| d | = | Free-floating particle size |
| F | = | Fluid volume fraction |
| f | = | Phase mass fraction |
| g | = | Phase volume fraction |
| = | Gravity | |
| H | = | Enthalpy |
| h | = | Heat transfer coefficient |
| K | = | Permeability |
| k | = | Thermal conductivity |
| kp | = | Partition coefficient |
| Lf | = | Latent heat |
| Lh | = | Vertical thickness of rigid mushy zone |
| M | = | Macrosegregation number |
| ml | = | Slope of liquidus |
| ms | = | Slope of solidus |
| P | = | Pressure |
| Q | = | Volumetric water flow rate |
| q” | = | Heat flux |
| S | = | Source term |
| SD | = | Sump depth |
| T | = | Temperature |
| t | = | Time |
| u | = | Axial velocity component |
| ∀ | = | Volume |
| = | Velocity vector | |
| Vc | = | Casting speed |
| v | = | Radial velocity component |
| W | = | Normalized Weibull deviation |
| Z | = | Ingot axial height |
| α | = | Weibull shape parameter |
| βS | = | Solutal expansion |
| βT | = | Thermal expansion |
| Γ | = | Gamma function |
| ε | = | Function relating temperature to liquid fraction |
| λ2 | = | Secondary dendrite arm spacing |
| μ | = | Viscosity |
| ρ | = | Density |
| φ | = | Ingot diameter |
| Ψ | = | Stream function |
| ω | = | Weibull scale parameter |
| Subscripts | = | |
| ∞ | = | Ambient temperature |
| boil | = | Boiling water temperature |
| cv | = | Control volume |
| H20 | = | Cooling water temperature |
| IB | = | Incipient boiling criteria |
| in | = | Incoming/inlet |
| l | = | Liquid |
| m | = | Melting temperature |
| nb | = | Neighboring control volumes |
| p | = | Control volume of interest |
| s | = | Solid |
| s,crit | = | Critical solid fraction |
| ss | = | Steady state |
| wall | = | Wall temperature |
| Superscripts | = | |
| 1 | = | Fluid one |
| 2 | = | Fluid two |
| i | = | Alloying element of interest |
| n | = | Iteration level |
| o | = | Previous time step |
Nomenclature
| a | = | Coefficients of conservation equations |
| B | = | Buoyancy force |
| C | = | Composition |
| c | = | Specific heat |
| D | = | Mass diffusion coefficient |
| d | = | Free-floating particle size |
| F | = | Fluid volume fraction |
| f | = | Phase mass fraction |
| g | = | Phase volume fraction |
| = | Gravity | |
| H | = | Enthalpy |
| h | = | Heat transfer coefficient |
| K | = | Permeability |
| k | = | Thermal conductivity |
| kp | = | Partition coefficient |
| Lf | = | Latent heat |
| Lh | = | Vertical thickness of rigid mushy zone |
| M | = | Macrosegregation number |
| ml | = | Slope of liquidus |
| ms | = | Slope of solidus |
| P | = | Pressure |
| Q | = | Volumetric water flow rate |
| q” | = | Heat flux |
| S | = | Source term |
| SD | = | Sump depth |
| T | = | Temperature |
| t | = | Time |
| u | = | Axial velocity component |
| ∀ | = | Volume |
| = | Velocity vector | |
| Vc | = | Casting speed |
| v | = | Radial velocity component |
| W | = | Normalized Weibull deviation |
| Z | = | Ingot axial height |
| α | = | Weibull shape parameter |
| βS | = | Solutal expansion |
| βT | = | Thermal expansion |
| Γ | = | Gamma function |
| ε | = | Function relating temperature to liquid fraction |
| λ2 | = | Secondary dendrite arm spacing |
| μ | = | Viscosity |
| ρ | = | Density |
| φ | = | Ingot diameter |
| Ψ | = | Stream function |
| ω | = | Weibull scale parameter |
| Subscripts | = | |
| ∞ | = | Ambient temperature |
| boil | = | Boiling water temperature |
| cv | = | Control volume |
| H20 | = | Cooling water temperature |
| IB | = | Incipient boiling criteria |
| in | = | Incoming/inlet |
| l | = | Liquid |
| m | = | Melting temperature |
| nb | = | Neighboring control volumes |
| p | = | Control volume of interest |
| s | = | Solid |
| s,crit | = | Critical solid fraction |
| ss | = | Steady state |
| wall | = | Wall temperature |
| Superscripts | = | |
| 1 | = | Fluid one |
| 2 | = | Fluid two |
| i | = | Alloying element of interest |
| n | = | Iteration level |
| o | = | Previous time step |
Acknowledgments
Financial support for this work for K. Fezi and M. J. M. Krane was from Shandong Nanshan Aluminum Co., Beijing Nanshan Institute of Aeronautical Materials, and for A. Plotkowski was from Purdue’s School of Materials Engineering.