Why solidification has an S-shaped history

A. Bejan*, S. Lorente, B. S. Yilbas, A. Z. Sahin

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

31 Scopus citations


Here we show theoretically that the history of solid growth during "rapid" solidification must be S-shaped, in accord with the constructal law of design in nature. In the beginning the rate of solidification increases and after reaching a maximum it decreases monotonically as the volume of solid tends toward a plateau. The S-history is a consequence of four configurations for the flow of heat from the solidification front to the subcooled surroundings, in this chronological order: solid spheres centered at nucleation sites, needles that invade longitudinally, radial growth by conduction, and finally radial lateral conduction to interstices that are warming up. The solid volume (Bs) vs time (t) is an S-curve because it is a power law of type Bs ∼ tn where the exponent n first increases and then decreases in time (n = 3/2, 2, 1, ⋯). The initial portion of the S curve is not an exponential.

Original languageEnglish
Article number1711
JournalScientific Reports
StatePublished - 26 Apr 2013

Bibliographical note

Funding Information:
The authors acknowledge the Deanship of Scientific Research, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for funding the project IN111042 during the course of this work.

ASJC Scopus subject areas

  • General


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