Studies On Momentum, Heat And Mass Transfer In Binary Alloy Solidification Processes
Abstract
The primary focus of the present work is the development of macromodels for numerical simulation of binary alloy solidification processes, consistent with microscopic phasechange considerations, with a particular emphasis on capturing the effects of nonequilibrium species redistribution on overall macrosegregation behaviour. As a first step, a generalised macroscopic framework is developed for mathematical modelling of the process. The complete set of equivalent singlephase governing equations (mass, momentum, energy and species conservation) are solved following a pressurebased Finite Volume Method according to the SIMPLER algorithm. An algorithm is also developed for the prescription of the coupling between temperature and the meltfraction.
Based on the above unified approach of solidification modelling, a macroscopic numerical model is devised that is capable of capturing the interaction between the doublediffusive convective field and a localised fluid flow on account of solutal undercooling during nonequilibrium solidification of binary alloys. Numerical simulations are performed for the case of twodimensional transient solidification of PbSn alloys, and the simulation results are also compared with the corresponding experimental results quoted in the literature. It is observed that nonequilibrium effects on account of solutal undercooling result in an enhanced macrosegregation. Next, the model is extended to capture the effects of dendritic arm coarsening on the macroscopic transport phenomena occurring during a binary alloy solidification process. The numerical results are first tested against experimental results quoted in the literature, corresponding to the solidification of an AlCu alloy in a bottomcooled cavity. It is concluded that dendritic arm coarsening leads to an increased effective permeability of the mushy region as well as an enhanced eutectic fraction of the solidified ingot. Consequently, an enhanced macrosegregation can be predicted as compared to that dictated by shrinkageinduced fluid flow alone.
For an orderofmagnitude assessment of predictions from the numerical models, a systematic approach is subsequently developed for scaling analysis of momentum, heat and species conservation equations pertaining to the case of solidification of a binary mixture. A characteristic velocity scale inside the mushy region is derived, in terms of the morphological parameters of the twophase region. A subsequent analysis of the energy equation results in an estimation of the solid layer thickness. It is also shown from scaling principles that nonequilibrium effects result in an enhanced macrosegregation compared to the case of an equilibrium model For the sake of assessment of the scaling analysis, the predictions are validated against computational results corresponding to the simulation of a full set of governing equations, thus confirming the trends suggested by the scale analysis.
In order to analytically investigate certain limiting cases of unidirectional alloy solidification, a fully analytical solution technique is established for the solution of unidirectional, conductiondominated, alloy solidification problems. The results are tested for the problem of solidification of an ammonium chloridewater solution, and are compared with those from existing analytical models as well as with the corresponding results from a fully numerical simulation. The effects of different microscopic models on solidification behaviour are illustrated, and transients in temperature and heat flux distribution are also analysed. An excellent agreement between the present solutions and results from the computational simulation can be observed.
The generalised numerical model is subsequently utilised to investigate the effects of laminar doublediffusive RayleighBenard convection on directional solidification of binary fluids, when cooled and solidified from the top. A series of experiments is also performed with ammonium chloridewater solutions of hypoeutectic and hypereutectic composition, so as to facilitate comparisons with numerical predictions. While excellent agreements can be obtained for the first case, the second case results in a peculiar situation, where crystals nucleated on the inner roof of the cavity start descending through the bulk fluid, and finally settle down at the bottom of the cavity in the form of a sedimented solid layer. An eutectic solidification front subsequently progresses from the top surface vertically downwards, and eventually meets the heap of solid crystals collected on the floor of the cavity. However, comparison of experimental observations with corresponding numerical results from the present model is not possible under this situation, since the associated transport process involves a complex combination of a number of closely interconnected physical mechanisms, many of which are yet to be resolved.
Subsequent to the development of the mathematical model and experimental arrangements for macroscopic transport processes during an alloy solidification process, some of the important modes of doublediffusive instability are analytically investigated, as a binary alloy of any specified initial composition is directionally solidified from the top. By employing a closeformed solution technique, the critical liquid layer heights corresponding to the onset of direct mode of instability are identified, corresponding two a binary alloy with three different initial compositions.
In order to simulate turbulent transport during nonequilibrium solidification processes of binary alloys, a modified k8 model is subsequently developed. Particular emphasis is given for appropriate modelling of turbulence parameters, so that the model merges with singlephase turbulence closure equations in the pure liquid region in a smooth manner. Laboratory experiments are performed using an ammonium chloridewater solution that is solidified by cooling from the top of a rectangular cavity. A good agreement between numerical and experimental results is observed.
Finally, in order to study the effects of threedimensionality in fluid flow on overall macrosegregation behaviour, the interaction between doublediffusive convection and nonequilibrium solidification of a binary mixture in a cubic enclosure (cooled from a side) is numerically investigated using a threedimensional transient mathematical model. Investigations are carried out for two separate model systems, one corresponding to a typical metalally analogue system and other corresponding to an actual metalalloy system. As a result of threedimensional convective flowpatterns, a significant solute macrosegregation is observed in the transverse sections of the cavity, which cannot be captured by twodimensional simulations.
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