Revisiting the quasi-steady state approximation for modeling heat transport during directional crystal growth. The growth rate can and should be calculated!

Abstract

It is often feasible to use a quasi-steady-state (QSS) approach when analyzing heat transport in directional crystal growth systems. In such cases, it is common practice to approximate the growth velocity to be equal to the pull rate (the induced growth rate). The a priori unknown growth rate is needed for calculation of the rate of latent heat release during solidification. In this manuscript we propose and demonstrate the use of a modified QSS (MQSS) approach which allows for QSS analysis while taking into account, in a self-consistent manner, the calculated growth rate. The new method is shown to yield (for the model vertical gradient freeze system studied here) results which are in excellent agreement with those obtained from a fully transient model. This is achieved for all system parameter values considered in this manuscript, including those for which the standard QSS (SQSS) approach fails due to large deviations between the induced growth rate (the pull rate) and the actual, calculated growth rate. As expected, the SQSS approach is shown to yield satisfactory results when latent heat release is unimportant as well as in situations where the calculated growth rate is similar in value to the pull rate.