Prediction of dendrite arm spacings in unsteady-and steady-state heat flow of unidirectionally solidified binary alloys

Abstract

Various theoretical dendrite and cell spacing formulas have been tested against experimental data obtained in unsteady- and steady-state heat flow conditions. An iterative assessment strategy satisfactorily overcomes the circumstances that certain constitutive parameters are inadequately established and/or highly variable and that many of the data sets, in terms of gradients, velocities, and/or cooling rates, are unreliable. The accessed unsteady- and steady-state observations on near-terminal binary alloys for primary and secondary spacings were first examined within conventional power law representations, the deduced exponents and confidence limits for each alloy being tabularly recorded. Through this analysis, it became clear that to achieve predictive generality the many constitutive parameters must be included in a rational way, this being achievable only through extant or new theoretical formulations. However, in the case of primary spacings, all formulas, including our own, failed within the unsteady heat flow algorithm while performing adequately within their steady-state context. An earlier untested, heuristically derived steady-state formula after modification, $$\lambda _1 = 120\left( {\frac{{16X_0^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} G_0 (\varepsilon \sigma )T_M D}}{{(1 - k)m\Delta H G R}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ ultimately proved its utility in the unsteady regime, and so it is recommended for purposes of predictions for general terminal alloys. For secondary spacings, a Mullins and Sekerka type formula proved from the start to be adequate in both unsteady- and steady-state heat flows, and so it recommends itself in calibrated form, $$\lambda _2 = 12\pi \left( {\frac{{4\sigma }}{{X_0 (1 - k)^2 \Delta H}}\left( {\frac{D}{R}} \right)^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ for future predictions.