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2. DIRECTIONAL SOLIDIFICATION OF A SUBCOOLED LIQUID. Controlled solidification of liquids is of great importance in many applications. It is possible to cool a very pure liquid to below its melting temperature to a metastable state, and then trigger (nucleate) a controlled crystallization from a well defined sulid surface. We model this process in one direction to see the effect of the directional constraint. Employ a frameworkotation similar to that used by Deen in prablem 4-2 [the Stefan problem, from HW +i3) but with a few differences. Firstly assume the solid-liquid boundary always moves at a constant speed V presumably controlled by sn interfacial rate procnss occuring in the immediate vicinity of the boundary, transforming the liquid to a solid. Since we assume the solidification begins with a pure, subcooled liquid, the boundary propagates away from a well defincd nucleating surface into the liquid to reate the solid crystalline phase at the expense of the liquid. The moving boundary liberates the latent heat at the solid-liguid boundary at the constant rate V[=]L2TE This heat warms the liquid in the boundary's immediate vicinity and is wonductaxl aw:y contimously. As in Deen problem 4-2 assume that the temperature at the solid-liquid boundary is (piecewise) continuous and at the solid melting point. TP. Finally assume the entire solid portion becween the boundary and nucleating surface stavs essentially at TF. else the solid would melt if warmed to dbove TF by the latent heat release. (a) Consider a 1d solidification prones in Cartasian coordinates. Initially we have sub-cooled liquid at temperature T0. Write down a 1d mathematical model for the temperature distribution T(x,t), including a full set of auxiliary conditions, assuming rigid media with constant properties, and conforming to the assumptions given shove, Scale the model using the characteristic values x=V:t^=V2a:T=TxTF