Simplified equations for transient heat transfer problems at low Fourier numbers

Martin Gram Christensen, Jens Adler-Nissen

Research output: Contribution to journalJournal articleResearchpeer-review


This paper proposes an analytical solution to transient heat transfer, which also applies for the initial heating/cooling period (Fo <0.2) of solids with simple geometries subjected to convective boundary conditions, with negligible mass transfer and phase-change. The new equation is presented and validated for infinite slabs, infinite cylinders and spheres and by an industrial application example, covering the center temperature and the volume average temperature. The approach takes ground in the residual difference between a 1 term series solution and a 100 term solution to the Fourier equation of the thermal response for solids subjected to convective heat transfer. By representing the residual thermal response as a function of the Biot number and the first eigenvalue, the new approach enables the description of the thermal response in the whole Fourier regime. The presented equation is simple and analytical in form, which allows an easy implementation into spreadsheets and thus serves as a transparent and fast tool for crude process calculations in e.g. process planning or introduction of new products to existing lines. The prediction error of the new equation is low (RMSD <0.015) for 0 <Fo <0.2 and 0.1 <Bi <100 for infinite slabs, infinite cylinders, spheres and typical examples of finite bodies.
Original languageEnglish
JournalApplied Thermal Engineering
Pages (from-to)382-390
Number of pages9
Publication statusPublished - 2015


  • Analytical solutions
  • Fourier-numbers
  • Heat-transfer
  • Low
  • Modeling
  • Cylinders (shapes)
  • Eigenvalues and eigenfunctions
  • Fourier series
  • Mass transfer
  • Models
  • Convective boundary conditions
  • Convective heat transfer
  • Fourier
  • Infinite cylinders
  • Process calculation
  • Simplified equations
  • Transient heat transfer
  • Heat convection


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