Approximated Analytical Solution for One-Dimensional Ablation Problem with Time-Variable Heat Flux

Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems

“…This choice is motivated through an analysis of the heating up stage before ablation commences. Their results are compared with an analysis of Braga et al [4,5] who use functions of the form (1) u = a 0 + a 1 (δ − x) n .…”

“…Other investigations with h s = 0 and Q constant are discussed [4,5,15], whereas time dependent boundary conditions are described in [4,18,45]. In the case of the convective boundary condition with h s = 0, the HBI method has only been analysed in [6,46], whilst consideration with both Q , h s = 0 are absent from the literature.…”

“…When studying the melting of a sub-cooled material subject to a Robin condition melting is not always immediate and this means there will be a pre-melting stage where a standard solution to the heat equation may be found and compared with the approximate solution. This approach was employed in [4,5,35] for example. In a series of papers, Braga & Mantelli [4,5] investigate ablation, using the HBIM, which has a pre-ablation stage.…”

“…This approach was employed in [4,5,35] for example. In a series of papers, Braga & Mantelli [4,5] investigate ablation, using the HBIM, which has a pre-ablation stage. They choose the power n for the approximating function of equation (1) from matching the time to ablation predicted by an exact solution and by the HBIM.…”

“…In general when we provide numerical solutions we will be using parameter values appropriate to ice and water, since this data is most readily available. Perhaps the most well studied example of the HBI method on ablation occurs in the context of heat shields on space vehicles, see [4,5] for example. For that example we will use parameter values appropriate to teflon.…”