A criterion for determining when convection needs to be considered in calculations of solidification/melting in thermal cavities

In general, modeling the solidification/melting in a rectangular cavity requires an accounting of the buoyancy driven fluid motion and associated convective heat transport. This study asks the question—Under what conditions can the contribution of the convective heat transport be ignored? To answer this question, we consider the problem of the solidification of a pure material in a rectangular cavity. Dimensionless governing equations for this problem are obtained. A numerical parametric study is performed, varying the Prandtl number, the Grashof number, thermal diffusivity ratio, scaled wall temperatures, and cavity aspect ratio. For 55 separate cases, steady state predictions of the fraction of solid in the cavity are obtained. These values are compared to steady state solid fraction predictions from a heat conduction alone model, i.e., a model that neglects convection. This analysis leads to a proposal of dimensionless group $V$ which provides a criterion for when convection is important. We find that when the value of $V\le 4000$, calculations based solely on conduction are sufficient.