A study of the convective cooling of large industrial billets

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The thermodynamic heat-transfer mechanisms, which occur as a heated billet cools in an air environment, are of clear importance in determining the rate at which a heated billet cools. However, in finite element modelling simulations, the convective heat transfer term of the heat transfer mechanisms is often reduced to simplified or guessed constants, whereas thermal conductivity and radiative emissivity are entered as detailed temperature dependent functions. As such, in both natural and forced convection environments, the fundamental physical relationships for the Nusselt number, Reynolds number, Raleigh parameter, and Grashof parameter were consulted and combined to form a fundamental relationship for the natural convective heat transfer as a temperature-dependent function. This function was calculated using values for air as found in the literature. These functions were then applied within an FE framework for a simple billet cooling model, compared against FE predictions with constant convective coefficient, and further compared with experimental data for a real steel billet cooling. The modified, temperature-dependent convective transfer coefficient displayed an improved prediction of the cooling curves in the majority of experiments, although on occasion a constant value model also produced very similar predicted cooling curves. Finally, a grain growth kinetics numerical model was implemented in order to predict how different convective models influence grain size and, as such, mechanical properties. The resulting findings could offer improved cooling rate predictions for all types of FE models for metal forming and heat treatment operations
Original languageEnglish
Article number137
Number of pages15
JournalJournal of Manufacturing and Materials Processing
Issue number4
Publication statusPublished - 16 Dec 2021

Bibliographical note

Funding Information:
Many thanks to colleagues in the School of Metallurgy and Materials at the University of Birmingham for the support and advice offered during the work. Thanks to Wilde Analysis (Stockport, UK) and to Wilde Analysis staff James Farrar and Tomas Brownsmith for their technical support with the Deform FE software.

Publisher Copyright:
© 2021 by the author. Licensee MDPI, Basel, Switzerland.


  • Nusselt
  • Grashof
  • Reynolds
  • Rayleigh
  • finite element
  • steel
  • grain size

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering


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