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International Journal of Thermophysics

, Volume 34, Issue 1, pp 122–138 | Cite as

Thermal Analysis of Non-linear Convective–Radiative Hyperbolic Lumped Systems with Simultaneous Variation of Temperature-Dependent Specific Heat and Surface Emissivity by MsDTM and BPES

  • Mohsen Torabi
  • Hessameddin Yaghoobi
  • Karem Boubaker
Article

Abstract

With the advent of temperatures near absolute zero, it is often claimed that at very low temperatures the effect of thermal wave propagation must be included by the hyperbolic heat conduction equation (HHCE). In this paper the non-linear convective–radiative HHCE is investigated. Opposite to common numerical analyses, analytical expressions are obtained for the temperature variations by the multi-step differential transformation method. Some conclusions about alteration of the specific heat of the material, temperature steeping, and Vernotte number have been formulated.

Keywords

Convective–radiative heat transfer Hyperbolic heat conduction MsDTM Temperatures near absolute zero Variable-specific heat coefficient 

List of Symbols

Variables

\(A\)

Dimensionless parameter describing variation of the thermal conductivity

\(B\)

Dimensionless parameter describing variation of the surface emissivity

\(c\)

Specific heat (J\(\,\cdot \,\)kg\(^{-1}\cdot \, \)K\(^{-1}\))

\(c_\mathrm{a} \)

Specific heat at the temperature \(T_\mathrm{a}\) (J\(\,\cdot \, \)kg\(^{-1}\cdot \, \)K\(^{-1}\))

\(D\)

Domain

\(E\)

Surface emissivity

\(E_\mathrm{s} \)

Surface emissivity at the temperature \(T_\mathrm{s} \)

\(Fo\)

Fourier number

\(H\)

Constant

\(h\)

Heat transfer coefficient (W\(\cdot \, \)m\(^{-2}\cdot \, \)K\(^{-1}\))

\(Nr\)

Dimensionless radiation–conduction parameter

\(S\)

Surface area (m\(^{2}\))

\(T\)

Temperature (K)

\(t\)

Temporal coordinate (s)

\(T_\mathrm{a} \)

Convection sink temperature (K)

\(T_\mathrm{i} \)

Initial temperature (K)

\(T_\mathrm{s} \)

Radiation sink temperature (K)

\(V\)

Volume (m\(^{3}\))

\(Ve\)

Vernotte number

\(X(k)\)

Transformed analytical function

\(x(t)\)

Original analytical function

Greek Symbols

\(\alpha \)

Measure of thermal conductivity variation with temperature (K\(^{-1}\))

\(\beta \)

Measure of surface emissivity variation with temperature (K\(^{-1}\))

\(\rho \)

Mass density (kg\(\cdot \, \)m\(^{-3}\))

\(\sigma \)

Stefan–Boltzmann constant (W\(\cdot \, \)m\(^{-2}\cdot \, \)K\(^{-4}\))

\(\tau \)

Thermal relaxation time (s)

\(\theta \)

Dimensionless temperature

\(\theta _\mathrm{a} \)

Dimensionless convection sink temperature

\(\theta _\mathrm{s} \)

Dimensionless radiation sink temperature

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Mohsen Torabi
    • 1
  • Hessameddin Yaghoobi
    • 1
  • Karem Boubaker
    • 2
  1. 1.Young Researchers Club, Central Tehran BranchIslamic Azad UniversityTehranIran
  2. 2.École Supérieure de Sciences et Techniques de Tunis (ESSTT)Université de TunisMahdiaTunisia

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