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Heat and Mass Transfer

, Volume 31, Issue 6, pp 443–450 | Cite as

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

  • A. Barletta
  • E. Zanchini
Originals

Abstract

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.

Keywords

Fourier Heat Flux Heat Conduction Apply Physic Constitutive Equation 
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

thermodynamic system

A1,A2

thermodynamic states ofA

B

function ofη, ω and Λ defined by Eq. (28)

c

specific heat

F

function oft employed in Eq. (6)

i

imaginary unit

k

thermal conductivity

L

1/2 of the slab thickness

n

non-negative integer number

M

function ofη, ω and Λ defined by Eq. (34)

p

Laplace transformed time variable

q

heat flux density component in thex direction

q

heat flux density vector

q0

constant heat flux employed in Eq. (6)

Q(t)

heat supplied to the slab in the time interval [0,t]

Q

limit ofQ(t) whent tends to + ∞

Re

real part of a complex number

Res (;)

residue of a complex function at a pole

s

entropy per unit boundary area

S

entropy

t

time

tp

time constant employed in Eq. (45)

T

temperature

T0

uniform value of temperature att=0

x

spatial coordinate

equal by definition

Greek symbols

α

=k/(ρc), thermal diffusivity

γ

real number employed in Eq. (18)

Γ

dimensionless entropy change defined by Eq. (55)

δQ

infinitesimal quantity of heat

η

dimensionless time defined in Eq. (8)

ϑ

dimensionless temperature defined in Eq. (8)

ϑ0

function ofη andξ defined in Eq. (16)

λn

dimensionless parameters expressed by Eq. (19)

Λ

dimensionless parameter defined in Eq. (8)

μn

dimensionless parameters expressed by Eq. (20)

ξ

dimensionless spatial coordinate defined in Eq. (8)

Ξ

=2Lq0/(kT0), dimensionless parameter

Π

dimensionless parameter defined in Eq. (46)

ρ

mass density

σ

entropy production rate per unit volume

τ

thermal relaxation time

Φ

function ofη and Λ defined in Eq. (11)

χ

dimensionless heat flux density defined in Eq. (8)

ω

dummy variable employed in Eqs. (28) and (34)

Superscripts

Laplace transformed function dummy integration variable

Nicht-Fouriersche Wärmeleitung in einer ebenen Platte mit aufgeprägtem Wärmefluß am Rande

Zusammenfassung

Es wird die nichtstationäre Wärmeleitung in einer unendlich ausgedehnten, ebenen Platte mit aufgeprägtem Wärmefluß am Rande untersucht. Letzterer kann einem beliebigen Zeitgesetz folgen; ferner sei eine endliche thermische Relaxationszeit unterstellt. Das Temperatur- und das Wärmeflußfeld werden analytisch, unter Verwendung der Beziehung für die Wärmestromdichte nach Cattaneo-Vernotte ermittelt. Es läßt sich nachweisen, daß zwischen Temperatur- und Wärmeflußverteilungen einerseits und der Hypothese vom lokalen thermischen Gleichgewicht Unvereinbarkeiten bestehen können. Durch Grenzübergang zu verschwindend kurzer thermischer Relaxationszeit läßt sich ein Vergleich mit der auf dem Fourierschen Gesetz basierenden Lösung herstellen.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • A. Barletta
    • 1
  • E. Zanchini
    • 1
  1. 1.Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA)Universitä di BolognaBolognaItaly

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