An Experimental Derivation of Transient Nonuniform External Boundary Conditions for the Solidification Process Modeling of Equiaxed Investment Castings
DOI:
https://doi.org/10.7494/jcme.2024.8.3.30Keywords:
solidification process modeling, casting simulation, nickel-based superalloy, equiaxed investment casting, large gas turbine hardware, transient nonuniform heat transfer, initial condition natural convection, surface condition natural convection, mixed convection, rotational convection, ProCAST validationAbstract
The external heat transfer mechanisms acting on the external mold surfaces for equiaxed casting processes are very complex. The mechanisms are multi-mode, transient, and nonuniform, consisting of very complex radiative and convective definitions. In this work, a real-life mold, SGT6-5000 FD 3/4 Blade 4 cast in Alloy-247, was instrumented with thermocouples and temperature readings were recorded throughout the entire casting sequence of events. Analytical models based on the first law of thermodynamics, Fourier’s law, Newton’s Law of Cooling, and diffuse gray radiation for an N-sided enclosure were developed to use the thermocouple data as input to back calculate the emissivity of the mold, as well as the spatially varying heat transfer coefficients for a number of local regions. The derived external heat transfer mechanisms are presented as transient Biot numbers. The derived emissivity and nonuniform heat transfer coefficients for these surfaces were then validated in ProCAST numerical simulation by comparing the external mold temperature profiles. An extensive iterative, curve fitting, extrapolating, and averaging procedure was exercised to derive an expression for emissivity across the entire temperature range associated with the casting process. The predicted temperatures on the nodes corresponding to the thermocouple locations agree within reasonable error with the experimental data. The model also qualitatively predicted the shrinkage porosity detected via x-ray imaging for this casting. The current study confirms the hypothesis of previous work by the current authors with respect to the transient nonuniform boundary condition concept. Unique values of heat transfer coefficients were observed at different vertical positions along the airfoil. The analytical models were also able to capture phenomena associated with specific sequences of the casting process. This work provides the analytical models, and procedure, needed to derive these spatially varying conditions. The current authors contribute to the intellectual know-how of the large gas turbine casting industry which by other foundries is considered highly proprietary and strictly confidential. This paper should be used to set the precedence for how foundries derive and validate the external boundary conditions used in solidification process modeling.
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Olson W., Stemmler M., Fernandez E. & Kapat J. (2024). Solidification process modeling of equiaxed investment castings with transient nonuniform boundary condition definition. Journal of Casting & Materials Engineering, 8(1), 1–10. Doi: https://doi.org/10.7494/jcme.2024.8.1.1.
Gebhart B. (1979). The 1978 Freeman scholar lecture: Buoyancy induced fluid motions characteristic of applications in technology. Journal of Fluids Engineering, 101(1), 5–28. Doi: https://doi.org/10.1115/1.3448735.
Sanders C.J. & Holman J.P. (1972). Franz Grashof and the Grashof number. International Journal of Heat and Mass Transfer, 15(3), 562–563.Doi: https://doi.org/10.1016/0017-9310(72)90220-7.
Ostrach S. (1953, January). An analysis of laminar free-convection flow and heat transfer about a flat plate paralled to the direction of the generating body force. NASA NACA-TR-1111
Sparrow E.M. & Gregg J.L. (1958). Similar solutions for free convection from a nonisothermal vertical plate. Transactions of the American Society of Mechanical Engineers, 80(2), 379–386.
Eckert E.R.G. (1950). Introduction to the Transfer of Heat and Mass. McGraw-Hill, New York.
Bayley F.J. (1955). An analysis of turbulent free-convection heat-transfer. Proceedings of the Institution of Mechanical Engineers, 169(1), 361–370. Doi: https://doi.org/10.1243/PIME_PROC_1955_169_049_02.
Vliet G.C. & Ross D.C. (1975). Turbulent natural convection on upward and downward facing inclined constant heat flux surfaces. Journal of Heat Transfer, 97(4), 549–554. Doi: https://doi.org/10.1115/1.3450427.
Raithby G.D. & Hollands K.G.T. (1976). Laminar and turbulent free convection from elliptic cylinders, with a vertical plate and horizontal circular cylinders as special cases. Journal of Heat Transfer, 98(1), 72–80. https://doi.org/10.1115/1.3450473.
Kuehn T.H. & Goldstein R.J. (1976). Correlating equations for natural convection heat transfer between horizontal circular cylinders. International Journal of Heat and Mass Transfer, 19(10), 1127–1134. Doi: https://doi.org/10.1016/0017-9310(76)90145-9.
Churchill S.W. & Chu H.H. (1975). Correlating equations for laminar and turbulent free convection from a horizontal cylinder. International journal of heat and mass transfer, 18(9), 1049–1053. Doi: https://doi.org/10.1016/0017-9310(75)90222-7.
Fand R.M., Morris E.W. & Lum M. (1977). Natural convection heat transfer from horizontal cylinders to air, water and silicone oils for Rayleigh numbers between 3 × 102 and 2 × 107. International Journal of Heat and Mass Transfer, 20(11), 1173–1184. Doi: https://doi.org/10.1016/0017-9310(77)90126-0.
Sparrow E.M. & Gregg J.L. (1956). Laminar-free-convection heat transfer from the outer surface of a vertical circular cylinder. Transactions of the American Society of Mechanical Engineers, 78(8), 1823–1828. Doi: https://doi.org/10.1115/1.4014194.
Ede A. J. (1967). Advances in Free Convection. Advances in Heat Transfer, 4(C), 1–64. Doi: https://doi.org/10.1016/S0065-2717(08)
-7.
Minkowycz W. J. & Sparrow E.M. (1974). Local nonsimilar solutions for natural convection on a vertical cylinder. Journal of Heat Transfer, 96, 178–183. Doi: https://doi.org/10.1115/1.3450161.
Sparrow E.M. & Stretton A J. (1985). Natural convection from variously oriented cubes and from other bodies of unity aspect ratio. International Journal of Heat and Mass Transfer, 28(4), 741–752. Doi: https://doi.org/10.1016/0017-9310(85)90224-8.
Fishenden M. & Saunders O.A. (1950). An Introduction to Heat Transfer. Oxford University Press, London.
Lloyd J.R. & Moran W R. (1974). Natural convection adjacent to horizontal surface of various planforms. Journal of Heat Transfer, 96, 443–447.Doi: https://doi.org/10.1115/1.3450224.
Goldstein R.J., Sparrow E.M. & Jones D.C. (1973). Natural convection mass transfer adjacent to horizontal plates. International Journal of Heat and Mass Transfer, 16(5), 1025–1035. Doi: https://doi.org/10.1016/0017-9310(73)90041-0.
Aung W. (1987). Mixed Convection in Internal Flows. In: . S. Kakaç, R.K. Shah & W. Aung (eds.), Handbook of Single-Phase Convective Heat Transfer, John Wiley & Sons, New York, pp. 15.1–15.51.
Gebhart B. (1971). Heat Transfer, 2nd ed., McGraw-Hill, New York.
Chen T.S., Armaly B.F. & Aung W. (1985). Mixed Convection in Laminar Boundary-layer Flow. In: , S. Kakaç, W. Aung & R. Viskanta, Natural Convection: Fundamentals and Applications, pp. 699–725.
Sparrow E.M., Eichhorn R., Gregg J.L. (1959). Combined forced and free convection in a boundary layer flow. Physics of Fluids, 2(3), 319–328. Doi: https://doi.org/10.1063/1.1705928.
Lloyd J.R. & Sparrow E.M. (1970). Combined forced and free convection flow on vertical surfaces. International Journal of Heat and Mass Transfer, 13(2), 434–438. https://doi.org/10.1016/0017-9310(70)90119-5.
Chen T.S., Mucoglu A. & Sparrow E.M. (1977). Mixed convection in boundary layer flow on a horizontal plate. Journal of Heat Transfer, 99(1), 66–71. Doi: https:// doi.org/10.1115/1.3450657.
Mori Y. & Futagami K. (1967). Forced convective heat transfer in uniformly heated horizontal tubes (2nd report, theoretical study). International Journal of Heat and Mass Transfer, 10(12), 1801–1813. Doi: https://doi.org/10.1016/0017-9310(67)90051-8.
Ramachandran N., Armaly B F. & Chen T.S. (1987). Measurements of laminar mixed convection flow adjacent to an inclined surface. Journal of Heat Transfer, 109(1), 146–150. Doi: https://doi.org/10.1115/1.3248035.
Churchill S.W. (1977). A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE Journal, 23(1), 10–16. Doi: https://doi.org/10.1002/aic.690230103.
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Copyright (c) 2024 Weston Olson, Michael Stemmler, Erik Fernandez, Jayanta Kapat
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