Non-local effects and size-dependent properties in Stefan problems with Newton cooling

• Published in: Applied Mathematical Modelling Vol. 76; pp. 513 - 525
• Main Author: Calvo-Schwarzwälder, Marc
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.12.2019; Elsevier BV
• Subjects: Asymptotic methods; Biot number; Cooling effects; Dimensionless numbers; Fourier law; Nanoscale; Newton cooling; Non-local effects; Phase change; Size-dependent thermal conductivity; Solidification; Stefan problem; Thermal conductivity; Thermal energy; Nanoscale; Non-local effects; Stefan problem; Size-dependent thermal conductivity; Newton cooling; Phase change
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2019.06.008

•A solidification process is modeled using a modified Fourier law with a sizedependent thermal conductivity.•It is shown that the Biot number is expected to be small.•Solutions are obtained using numerical and asymptotic methods.•The study shows that non-local effects become less important for small Stefan or Biot numbers. We model the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment. In the limit of a large Biot number, this condition becomes the commonly used fixed-temperature condition. It is shown that in practice the size of this non-dimensional number is very small. We study the effect of a small Biot number on the solidification process with numerical and asymptotic solution methods. The study indicates that non-local effects become less important as the Biot number decreases.