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Journal of Thermal Analysis and Calorimetry

, Volume 132, Issue 2, pp 1389–1397 | Cite as

Notes on using temperature-dependent thermal diffusivity—forgotten rules

  • Gyula Gróf
Article

Abstract

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.

Keywords

Nonlinear heat conduction Thermal diffusivity Temperature-dependent properties Integral transformation methods 

List of symbols

a1

Constant in diffusivity function (m2 s−1)

a2

Constant in diffusivity function (s K−1 m−2)

b

Constant in conductivity function (K−1)

cp

Specific heat, pressure constant (J kg−1 K−1)

cv

Specific heat, volume constant (J kg−1 K−1)

\(\overline{\overline{f}}\)

Stress tensor (N m−2)

\(\bar{g}\)

Force field (m s−2)

k

Thermal conductivity (W m−1 K−1)

k0

Base value in conductivity function (W m−1 K−1)

km

Mean value of conductivity (W m−1 K−1)

p

Pressure (N m−2)

\(\dot{\bar{q}}_{\text{F}}\)

Conductive heat flux (W m−2)

\(\dot{q}_{\text{V}}\)

Volume heat generation (W m−3)

t

Time (s)

u

Internal energy specific (J kg−1)

v

Specific volume (m3 kg−1)

w

Velocity (m s−1)

x

Space coordinate (m)

A

Constant in diffusivity function

B

Constant in conductivity function

C

Constant in conductivity function

D

Constant in conductivity function

K

Constant in volume heat capacity function

T

Temperature (K)

TH

Lower limit of temperature (K)

TL

Upper limit of temperature (K)

V

Goodman temperature (K)

Greek scripts

α

Thermal diffusivity (m2 s−1)

β

Vol. thermal expansion (K−1)

ρ

Density (kg m−3)

ρc

Volume heat capacity (J m−3 K−1)

σ

Stress (normal) (N m−2)

τ

Stress (shear) (N m−2)

F

Dissipation (W m−3)

θ

Kirchhoff temperature (K)

θ0

Lower limit of Kirchhoff temperature (K)

ϕ

General function of temperature

ϑ

Auxiliary variable

Subscripts/superscripts

H

Higher

L

Lower

0

Optional or starting

P

Constant pressure

v

Constant volume

j

Superscript in conductivity function

m

Superscript in diffusivity function

n

Superscript in conductivity function

r

Superscript in volume heat capacity function

Notes

Acknowledgements

The work was supported by the grant OTKA K116375.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Energy Engineering, Faculty of Mechanical EngineeringBudapest University of Technology and EconomicsBudapestHungary

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