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Wärme - und Stoffübertragung

, Volume 22, Issue 3–4, pp 185–188 | Cite as

Thermal wave propagation within a medium with the inert heat source

  • L. Malinowski
Article

Abstract

A heat conduction equation of a new type is derived which takes into account the finite velocity of heat flux propagation and the relaxation of heat source capacity. The equation is solved for a semi-infinite body and a step change in temperature at the surface. The analysis shows that as the time increases the obtained solution moves from the solution of the classical hyperbolic equation without energy generation towards the solution of the classical hyperbolic equation with energy generation.

Keywords

Heat Flux Wave Propagation Heat Conduction Heat Source Apply Physic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

a

thermal diffusivity,k/(ϱ cp

cp

specific heat at constant pressure

C

speed of heat propagation

C1,C2

constants

k

thermal conductivity

qv

steady capacity of internal heat source

qvd

transient capacity of internal heat source

r1,r2

roots of characterisitc equation

t

time

tk

relaxation time of heat flux

tq

relaxation time of internal heat source capacity

T

temperature

T0

surface temperature

u(τ)

unit step function

x, y, z

Cartesian coordinates

X

dimensionless coordinate

α, β

constant coefficients

Θ

dimensionless temperature

ϱ

density

τ

dimensionless time

τr-tqtk

ratio of relaxation times

Ψ

dimensionless steady capacity of internal heat source

Ψd

dimensionless transient capacity of internal heat source

Ausbreitung thermischer Wellen in einem Medium mit träger Wärmequelle

Zusammenfassung

Es wird eine neuartige Wärmeleitungsgleichung abgeleitet, welche die endliche Geschwindigkeit der Ausbreitung des Wärmestromes und die Relaxation der Kapazität der Wärmequelle berücksichtigt. Die Gleichung wird für einen halbunendlichen Körper und eine schrittweise Temperaturänderung an der Oberfläche gelöst. Die Analyse zeigt, daß mit zunehmender Zeit sich die Lösung der klassischen hyperbolischen Gleichung ohne Wärmeerzeugung in eine solche mit ebenfalls klassischer hyperbolischer Gleichung mit Wärmeerzeugung wandelt.

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References

  1. 1.
    Vernotte, M. P.: Les paradoxes de la théorie continue de l'equation de la chaleur. Comptes Rendus 246 (1958) 3154–3155Google Scholar
  2. 2.
    Baumeister, K. J.; Hamill, T. D.: Hyperbolic heat-conduction equation —A solution for the semi-infinite body problem. J. Heat Transfer 91 (1969) 543–548Google Scholar
  3. 3.
    Taitel, Y.: On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. Int. J. Heat Mass Transfer 15 (1972) 369–371Google Scholar
  4. 4.
    Vick, B.; Özisik, M. N.: Growth and decay of a thermal pulse predicted by the hyperbolic heat conduction equation. J. Heat Transfer 105 (1983) 902–907Google Scholar
  5. 5.
    Özisik, M. N.; Vick, B.: Propagation and reflection of thermal waves in a finite medium. Int. J. Heat Mass Transfer 27 (1984) 1845–1854Google Scholar
  6. 6.
    Malinowski, L.: Heat conduction equation of superfast processes(in Polish). Proceedings of the 25 th Symposium Modelling in Mechanics No. 53 (1986) 262–269Google Scholar
  7. 7.
    Doetsch, G.: Anleitung zum praktischen Gebrauch der Laplace Transformation. München: Oldenbourg 1961Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • L. Malinowski
    • 1
  1. 1.Department of Mechanical EngineeringTechnical University of SzczecinSzczecinPoland

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