Stagnant film model of the effect of natural convection on the dendrite operating state

Abstract

We develop a simple model of the influence of natural convection on the selection of the operating state (dendrite tip velocity, V, and tip radius, ϱ) for dendritic growth of a pure material. We hypothesize that the important aspects of natural convection can be accounted for by considering the global convection that would occur in the vicinity of a sphere of radius R that characterizes the size of a dendritic array that is growing from a point source. We estimate the thickness, δ, of a stagnant boundary layer surrounding this sphere by matching the value of the Nusselt number obtained from the heat transfer literature. We solve the steady-state problem of a paraboloidal dendrite at temperature T M growing toward a confocal paraboloid at temperature T ∞ located at a distance δ from the tip. This results in a new transcendental equation, that depends on the gravitational acceleration, g, and transport properties of the melt, for the dimensionless supercooling, S, in terms of the Peclet number, Pe. By assuming that the selection parameter σ is not directly affected by natural convection, we are able to compute values of V and ϱ as functions of the supercooling, Δ T, for various values of g. Results are quite insensitive to the precise choice of R and in remarkably good agreement with the data of Glicksman et al. for growth of succinonitrile on Earth and in microgravity. The value of g at which our model indicates that convective effects become significant has a much stronger dependence (∼ Δ T 9.5) on supercooling than that (∼ Δ T 5.5) calculated by using the analogy of Gill et al. between natural convection and forced convection.