Notes on using temperature-dependent thermal diffusivity—forgotten rules

New materials and structures are applied in order to maintain the engineering products more and more efficient. Different types of thermal analysis/modelling of such materials and structures are also needed for the proper planning or design. The thermal analysis/modelling processes are supported by many commercial codes but in some cases, the experts might have applied a self-developed method also. The success of the analysis/modelling depends on many factors, but the basis is the correct application of the heat conduction theory which is more important in case of transient thermal processes. As the used tools are more and more powerful, the users pay less attention to the theory background, according to the author experience. It seems that the long tradition of the inconsistent application of the temperature-dependent thermal diffusivity is continuing, even strengthened by the increased availability of data of the temperature-dependent thermal properties. Regarding the origination of the thermal diffusivity, this might lead to the misinterpretation or misunderstanding of the nonlinear heat conduction problem solution procedures. This paper calls attention to avoid this, shows what kind of mistakes might occur related to this widely observable practice, summarizes the proper application and interprets where and how one could find room for the temperature-dependent thermal diffusivity.

[1]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[2]  W. Schützenhöfer,et al.  Thermal Diffusivity and Thermal Conductivity of Five Different Steel Alloys in the Solid and Liquid Phases , 2015 .

[3]  Richard L. Phillips,et al.  Theory of the non-stationary arc column , 1967 .

[4]  Derek B. Ingham,et al.  A note on the determination of the thermal properties of a material in a transient nonlinear heat conduction problem , 1995 .

[5]  E. Astm.  Standard test method for thermal diffusivity of solids by the flash method , 1992 .

[6]  Carlos Alberto Brebbia,et al.  The dual reciprocity boundary element formulation for diffusion problems , 1987 .

[7]  James V. Beck,et al.  Inverse Heat Conduction , 2023 .

[8]  Sin Kim,et al.  A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation , 2001 .

[9]  Yongsheng Zhou,et al.  Temperature dependence of thermal diffusivity and conductivity for sandstone and carbonate rocks , 2018, Journal of Thermal Analysis and Calorimetry.

[10]  Zhang Guan-ren.  Solution of nonlinear heat conduction equation and image method , 1984 .

[11]  Xiao-Wei Gao,et al.  A Modified Conjugate Gradient Method for Transient Nonlinear Inverse Heat Conduction Problems: A Case Study for Identifying Temperature-Dependent Thermal Conductivities , 2014 .

[12]  E. Kaschnitz,et al.  Thermal Diffusivity of the Aluminum Alloy Al–17Si–4Cu (A390) in the Solid and Liquid States , 2007 .

[13]  C. Snowden,et al.  Electro-thermal device and circuit simulation with thermal nonlinearity due to temperature dependent diffusivity , 2000 .

[14]  W. K. Yeung,et al.  Second-order finite difference approximation for inverse determination of thermal conductivity , 1996 .

[15]  Yu. V. Zhernovyi,et al.  Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem , 2000 .

[16]  Zhifei Zhang,et al.  Global regularity for the initial–boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity , 2013 .

[17]  Kamil Dychtoń,et al.  Microstructure and thermal characteristics of SiC–Al2O3–Ni composite for high-temperature application , 2016, Journal of Thermal Analysis and Calorimetry.

[18]  Hisham A. Abdel-Aal.  A remark on the flash temperature theory , 1997 .

[19]  Werner Theisen,et al.  Temperature-dependent thermal conductivities of non-alloyed and high-alloyed heat-treatable steels in the temperature range between 20 and 500 °C , 2014, Journal of Materials Science.

[20]  Antonio Fasano,et al.  On the estimation of thermophysical properties in nonlinear heat-conduction problems , 1974 .

[21]  Qiang Sun,et al.  Olivine thermal diffusivity influencing factors , 2018, Journal of Thermal Analysis and Calorimetry.

[22]  Anne M. Hofmeister,et al.  Corrigendum to ‘The influence of temperature-dependent thermal diffusivity on the conductive cooling rates of plutons and temperature-time paths in contact aureoles’ [Earth Planet. Sci. Lett. 317–318 (2012) 157–164] , 2015 .

[23]  A. Hofmeister.  Thermal diffusivity of garnets at high temperature , 2006 .

[24]  Anne M. Hofmeister,et al.  Temperature-dependent thermal diffusivity of the Earth's crust and implications for magmatism , 2009, Nature.

[25]  David A. Yuen,et al.  Effects of temperature-dependent thermal diffusivity on shear instability in a viscoelastic zone: implications for faster ductile faulting and earthquakes in the spinel stability field , 2000 .

[26]  Guoping Chen,et al.  Inverse determination of temperature-dependent thermophysical parameters using multiobjective optimization methods , 2015 .

[27]  John M. Mihaljan,et al.  A Rigorous Exposition of the Boussinesq Approximations Applicable to a Thin Layer of Fluid. , 1962 .

[28]  Gyula Gróf,et al.  Simultaneous Identification of Temperature-Dependent Thermal Properties via Enhanced Genetic Algorithm , 2012 .

[29]  Manuel G. Velarde,et al.  On the /non linear/ foundations of Boussinesq approximation applicable to a thin layer of fluid , 1975 .

[30]  Damien Meresse,et al.  Thermal diffusivity identification by 2nd derivative analysis of transient temperature profile , 2016, Journal of Thermal Analysis and Calorimetry.

[31]  R. J. Jenkins,et al.  Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity , 1961 .

[32]  C. Snowden,et al.  Reply to Comment on 'Electro-thermal device and circuit simulation with thermal nonlinearity due to temperature dependent diffusivity' , 2001 .

[33]  Q. T. Pham.  A fast, unconditionally stable finite-difference scheme for heat conduction with phase change , 1985 .

[34]  A. Göbel,et al.  Determination of the Thermal Diffusivity of Electrically Non-Conductive Solids in the Temperature Range from 80 K to 300 K by Laser-Flash Measurement , 2010 .

[35]  M. N. Özişik,et al.  Inverse Heat Transfer: Fundamentals and Applications , 2000 .

[36]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[37]  Dean-Mo Liu,et al.  Effect of temperature on measuring the thermal diffusivity of ceramic of different specimen thicknesses by using a laser-pulse method , 1997 .

[38]  Leandro dos Santos Coelho,et al.  Apparent thermal diffusivity estimation of the banana during drying using inverse method , 2008 .

[39]  Anne M. Hofmeister,et al.  Thermal diffusivity and thermal conductivity of single-crystal MgO and Al2O3 and related compounds as a function of temperature , 2014, Physics and Chemistry of Minerals.

[40]  Anne M. Hofmeister,et al.  The influence of temperature-dependent thermal diffusivity on the conductive cooling rates of plutons and temperature-time paths in contact aureoles , 2012 .

[41]  L. Damkilde,et al.  Comments on 'Electro-thermal device and circuit simulation with thermal nonlinearity due to temperature dependent diffusivity' , 2001 .

[42]  M. Maeda,et al.  [Heat conduction]. , 1972, Kango kyoshitsu. [Nursing classroom].

[43]  Alexei I. Zhurov,et al.  Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time , 2013 .

[44]  R. P. Tye,et al.  Reference materials for thermal transport properties measurements , 2017, Journal of Thermal Analysis and Calorimetry.

[45]  Juan Pou,et al.  Thermal properties measurement of slate using laser flash method , 2012 .

[46]  Hisham A. Abdel-Aal,et al.  On friction-induced temperatures of rubbing metallic pairs with temperature-dependent thermal properties , 1998 .

[47]  Cheng-Hung Huang,et al.  An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity , 1995 .

[48]  Antonio Fasano,et al.  Numerical solution of phase-change problems , 1973 .

[49]  Derek B. Ingham,et al.  Identification of the Thermal Conductivity and Heat Capacity in Unsteady Nonlinear Heat Conduction Problems Using the Boundary Element Method , 1996 .

[50]  K. D. Maglić,et al.  Compendium of Thermophysical Property Measurement Methods , 2012 .

[51]  Kozo Katayama,et al.  Numerical Method of Transient Heat Conduction with Temperature Dependent Thermal Properties , 1972 .

[52]  L. E. Goodrich.  A one-dimensional numerical model for geothermal problems , 1974 .

[53]  Roland E. Lee.  HEAT TRANSFER TO THE THROAT REGION OF A SOLID PROPELLANT ROCKET NOZZLE , 1963 .

[54]  G. Comini,et al.  On the solution of the nonlinear heat conduction equations by numerical methods , 1973 .

[55]  Werner Theisen,et al.  Thermo-physical properties of heat-treatable steels in the temperature range relevant for hot-stamping applications , 2015, Journal of Materials Science.

[56]  John O. Mingle.  The method of differential quadrature for transient nonlinear diffusion , 1977 .

[57]  Ching-yu Yang,et al.  Estimation of the temperature-dependent thermal conductivity in inverse heat conduction problems , 1999 .

[58]  Anne M. Hofmeister,et al.  Inference of high thermal transport in the lower mantle from laser-flash experiments and the damped harmonic oscillator model , 2008 .

[59]  T. Goodman.  Application of Integral Methods to Transient Nonlinear Heat Transfer , 1964 .

[60]  Jesús Ildefonso Díaz Díaz,et al.  On the Boussinesq system with nonlinear thermal diffusion , 1997 .

[61]  Jean-Luc Bodnar,et al.  Artificial intelligence tools and inverse methods for estimating the thermal diffusivity of building , 2011 .