SOLUTIONS   Heat Conduction


Fourier's law relating heat flux and temperature gradients is widely adopted in heat conduction analysis and associated finite element modeling software. While accurate in most contexts, significant departures from Fourier's law are observed in a number of applications, particularly ones involving very small time and/or space scales as well as configurations using heterogeneous materials such as composite structures or granular materials. To model the heat conduction processes in these cases, a generalization of Fourier's law can be developed through the introduction of additional phase lag parameters. For micro-scale thermal models, these lags can expressed in terms of well-established physical processes. For macroscopic applications, these lags are viewed as correction parameters whose values can be extracted from experimental measurement.

As part of an SBIR effort, CDI developed finite element software to model the 3D non-Fourier thermoelastic response in solid structures. The model employs an extension of the Dual Phase Lag (DPL) model for the heat conduction description together with the traditional stress-strain relationships. The DPL model contains two phase lags and subsumes both Fourier's law and the hyperbolic heat conduction models previously used to model non-Fourier materials. These laws were reduced to discrete form using the finite element method. The resulting software is configured to accommodate a variety of boundary conditions including unsteady and nonlinear heating conditions. Validation efforts included the recovery of difficult solutions involving the propagation of temperature and stress waves resulting from step heat flux inputs. A 1D version of the DPL-based finite element software was also used in conjunction with least squares estimation techniques to extract phase lag estimates from flash diffusivity experiments involving selected materials in rocket nozzles. The software was successfully applied to intense transient heating scenarios including rocket nozzles and heat shields. It can also be used for MEMS, nanotechnology, thin films, laser technology, cryogenics and composite structures.