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Wärme - und Stoffübertragung

, Volume 16, Issue 2, pp 113–118 | Cite as

Temperature in heat generating solids with memory

  • E. Lorenzini
  • M. Spiga
Article

Abstract

The present paper is concerned with time dependent heat transport by wave propagation in an homogeneous isotropic elastic solid with memory. The energy generated in the material for electrical heating or chemical or nuclear reactions is propagated with a finite speed.

The effect of this thermal wave speed is noticeable in many practical applications involving short time and high heat flux situations. The one dimensional, time dependent temperature distribution in a heat generating solid is analytically determined resorting to the Maxwell-Cattaneo-Vernotte equation, following the theory of complex functions of complex variables. Some results are reported and shortly discussed; a comparison with the classical Fourier theory is made.

Keywords

Heat Flux Heat Generate Heat Transport Nuclear Reaction Wave Speed 
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

heat flux relaxation function

bn

constant defined in (25)

b′n

constant defined in (30)

c

specific heat at constant pressure

E

elasticity modulus

Fo

Fourier number defined in (8)

i

imaginary unity

K

thermal conductivity

L

slab half thickness

N

integer number defined in (28)

p

integration variable for the inverse Laplace transform

q

heat flux vector

Q′

power produced in the unit volume

Q

temperature defined in (8)

Rm

residue associated to them-th pole

s

time

S

sound speed

t

time

T

temperature defined in (8)

T+

dimensionless temperature defined in (33)

u

function defined in (11)

ũ

inverse Laplace transform of the functionu

U

unit Heaviside step function

x

dimensionless coordinate defined in (8)

α

thermal diffusivity

vector differential operator

2

Laplace operator

γ

constant defined in (26)

ξ

cartesian coordinate

ϱ

mass density

gq

temperature

τ

dimensionless material thermal relaxation time defined in (8)

τR

material thermal relaxation time

Temperaturverteilung in einem festen Körper mit Wärmeerzeugung und thermischer Nachwirkung

Zusammenfassung

Die Arbeit behandelt den zeitabhängigen Wärmetransport durch eine fortschreitende Welle in einem homogenen isotropen elastischen Festkörper mit Gedächtnis. Die im Material durch elektrische Heizung oder durch chemische oder nukleare Reaktionen erzeugte Wärme wird mit endlicher Geschwindigkeit fortgeleitet. Die Wirkung dieser thermischen Wellengeschwindigkeit ist für manche praktische Anwendungen von Bedeutung, soweit kurze Zeiten und hohe Wärmeflüsse eine Rolle spielen. Die eindimensionale zeitabhängige Temperaturverteilung in einem wärmeerzeugenden Körper ist analytisch bestimmt unter Benutzung der Maxwell-Cattaneo-Vernotte-Gleichung nach der Theorie komplexer Funktionen komplexer Variablen. Einige Ergebnisse werden mitgeteilt, kurz diskutiert und mit der klassischen Fourier-Theorie verglichen.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • E. Lorenzini
    • 1
  • M. Spiga
    • 1
  1. 1.Istituto di Fisica Tecnia Termotecnica del ReattoreFacoltà di Ingeneria Università di BolognaBolognaItaly

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