Abstract
Ingots of metals generally consist of three zones of different shapes of grains (Fig. 1). The boundaries of the three zones can be determined, if we compare the inward propagation velocities of melting point υm=(_??_x/_??_t)θ=θm with Tamman's “Kristallisationsgeschwindigkeit” KG at each point of ingots. A zone will be constituted with columnar grains, if therein KG_??_υm; and with granular grains, if therein KG<υm. The behaviours of vcurves in ingots are shown in Figs. 3 and 4. A simplified ideal case was calculated by solving the differential equation of thermal conduct ivity, and therefrom actual case was deduced. The reason why the values of υm excel both at surface and center of ingots is as fallows: at the surface as the cooling velocity _??_θ/_??_t is large, υm becomes prominent, and at the center as the thermal gradient _??_θ/_??_x is small, υm becomes again prominent. The relative positions of υm curve and KG curves are illustrated in Fig. 5. Four typycal cases are distinguishable.
(1) KGI: Ingot will be constituted entirely with columnar grains.
(2) KGII: Ingot will consist of two zones, the outer one (from surface to intersection of two curves) being constituted with columnar grains and the inner one (from intersection to center) with granular grains.
(3) KGIII: Ingot will consist of three zones, two intersections of two curves representing the boundaries of zones (Fig. 6).
(4) KGIV: Ingot will be constituted entirely with granular grains.
Inversely utilising this consideration, we shall be able to determine Tammann's KG curve, which is not yet determined numerically. The method is as follows: When the calculated υm curve cuts the boundaries of zones of cast ingot, then the intersections will furnish the points on KG curve. Repeating this process at various conditions of casting, we shall be able to gain the whole KG curve.
