Transient Heat Conduction in a Thin Layer Between Semi-Infinite Media in Polymer Shaping


Heat conduction lasting for only a short time is quite common in thin layers embedded between two semi-infinite media. Basically, the temperature in such conditions has singularities at zero time and depths when initial temperatures differ. Consequently, these singularities have been seen to hamper accurate numerical solution and worse off, complicate analytical solutions. For a purely resistive layer between semi-infinite media, general solutions for the case of composite slabs or multilayers have been derived by separation of variables, using Laplace transform, Sturm–Liouville integral transform and by natural Eigen function expansion. However, they still tend to be complex and often require the evaluation of integrals, eigenvalues, or infinite series with an uncertain truncation error.

Recently, Leendert van der Tempel (currently at Signify Research), Willem Potze and Jeroen Lammers at Philips Research in the Netherlands developed a novel series expansion and a new approximation with accurate singularity treatment for the temperature in the case of transient heat conduction in a single thin layer with a surface heat flux between two semi-infinite media at different uniform initial temperatures. Their work is currently published in Journal of Heat Transfer.

The research method employed entailed the derivation of the approximation and expansion series for the temperature for the stated cases. Next, Laplace transformation of the model was used to simplify the partial differential equations to ordinary differential equations. Lastly, the temperature accuracy of the derived series was evaluated for two test cases in the field of thermoplastic shaping of polymers.

The authors observed that the series converged rapidly in the injection molding and fused deposition modeling test cases below 1 μK truncation error within two terms fitting in one spreadsheet cell and involving only 8–12 function evaluations. In addition, the researchers noted that the relative accuracy at the singularity in the origin was excellent. Moreover, they found out that the approximation fitted in one spreadsheet cell and involved 14 function evaluations. For the injection molding test case, it was seen to have 56 K and in the fused deposition modeling test case 3.5 mK approximation error. The asymptotic behavior of the relative accuracy for short and long times was also seen to be correct.

In a nutshell, Leendert van der Tempel and colleagues successfully presented the derivation of a series expansion and an approximation for temperature in the case of transient heat conduction in a thin layer between two semi-infinite media at different uniform initial temperatures. The series expansion and the approximation were evaluated as 3 orders of magnitude more accurate in the fused deposition modeling test case than the previously published convective approximation. Altogether, the developed series expansion enables quick yet accurate thermal analysis of compression molding, injection molding, and fused deposition modeling and in particular the depth distribution of the vitrification rate. The logarithmically weighted depth average of the vitrification rate explains the measured post-molding volume shrinkage.

Transient Heat Conduction in a Thin Layer Between Semi-Infinite Media in Polymer Shaping-Advances in Engineering

About the author

Ir. Leendert van der Tempel is a scientist at Signify Research and formerly at Philips Research. He graduated cum laude in Mechanical Engineering at University of Twente in the Netherlands in 1984. He is author of a book chapter on Deformation of polycarbonate optical disks by water sorption and ageing.

He holds 6 granted US patents, including a patent with Herman Borg on optical disk warpage reduction which is applied in all 2½ billion Blu-ray Disks, saving worldwide over 350 M€ material cost.

This publication is about his life’s work on analytical multilayer transient conduction models, successfully applied to compression molding of glass, injection molding of polycarbonate, thermal effusivity measurement, ironing, initialization of rewritable optical disks and fused deposition modelling (FDM). His thermomechanical research interests include :

  • analytical multilayer transient conduction models
  • ophthalmic microscope illumination with improved red reflex and retinal safety
  • direct bonding, injection molding and Fused Deposition Modelling
  • automatic ironing
  • OLED defectivity (using lock-in thermography) and stochastic degradation & lifetime modelling
  • moisture sorption, physical ageing, creep and consequent warpage of polycarbonate optical disks
  • thermal effusivity and conductivity measurement methods
  • thermal radiation and conduction, especially in glass and polycarbonate processing


Leendert van der Tempel, Willem Potze, Jeroen H. Lammers (2018). Transient Heat Conduction in a Thin Layer Between Semi-Infinite Media in Polymer Shaping. Journal of Heat Transfer, volume 140 page 041301-8.

Go To Journal of Heat Transfer

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