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Wärme - und Stoffübertragung

, Volume 17, Issue 1, pp 1–9 | Cite as

Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties

  • A. Campo
Article

Abstract

A method of analysis is described which yields quasianalytical solutions for one and multidimensional unsteady heat conduction problems with linearly dependent thermal properties, such as thermal conductivity and volumetric specific heat. The method accomodates rather general thermal boundary conditions including arbitrary variations in surface temperature or in surface heat flux or a convective exchange with a fluid having even varying temperature. Once the solution for the identical problem but with constant properties has been developed, its practical realization is rather direct, being facilitated by a reduced number of iterations. The four applied examples given in this work show that a wide variety of nonlinear heat conduction problems can be tackled by this procedure without much difficulty. These simple solutions compare favorably with more laborious results reported in the archival heat transfer literature.

Keywords

Heat Flux Surface Heat Flux Heat Conduction Problem Thermal Boundary Condition Constant Property 
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

side of square bar

Bi0

reference Biot number,hR/k0

Bi0T

transformed Biot number, equation (16)

c

geometric parameter, equation (8)

h

convective coefficient

k

thermal conductivity

k0

value ofk atT0

K

dimensionless thermal conductivity,k/k0

Ki

value ofK at Φi

Ki+1

value ofK at Φi+1

mk

slope of theK-Φ line, equation (3)

ms

slope of theS-Φ line, equation (4)

R

characteristic length

s

volumetric specific heat

s0

value of s at T0

S

dimensionless volumetric specific heat, s/s0

Si

value ofS at Φi

Si+1

value of S at Φi+1

t

time

T

temperature

T0

reference temperature

x, y

cartesian coordinates

X, Y

dimensionless cartesian coordinates,x/a andy/a

α

thermal diffusivity

θk

transformed time, equation (11)

θs

transformed time, equation (37)

τk

dimensionless time for variable conductivity, equation (8)

τs

dimensionless time for variable specific heat, equation (34)

Φ

dimensionless temperature,T/T0

η

dimensionless coordinate,r/R

α0

value of α at T0

Φi

lower value of the interval (Φi, Φi+1)

Φi+1

upper value of the interval (Φi, Φi+1

Berechnung nichtstationärer Wärmeleitvorgänge mit linear temperaturabhängigen Stoffwerten aus der Lösung für konstante Stoffwerte

Zusammenfassung

Es werden quasi-analytische Lösungen für ein- und mehrdimensionale nichtstationäre Wärmeleitprobleme mit linear temperaturabhängigen Stoffwerten, wie Wärmeleitfähigkeit und volumetrische Wärmekapazität, mitgeteilt. Die Methode gilt für recht allgemeine Randbedingungen wie beliebige Veränderungen der Oberflächentemperatur, der Wärmestromdichte oder auch konvektiven Wärmeaustausch mit veränderlicher Fluidtemperatur. Ist die Lösung für das identische Problem mit konstanten Stoffwerten bekannt, kann die Methode direkt mit einer begrenzten Zahl von Iterationen angewandt werden. Die vier hier mitgeteilten Beispiele zeigen, daß eine große Zahl nichtlinearer Wärmeleitprobleme auf diese Weise ohne Schwierigkeit angepackt werden können. Die einfachen Lösungen stimmen befriedigend mit komplizierteren Ergebnissen aus der Literatur überein.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • A. Campo
    • 1
  1. 1.Departamento de TermodinámicaUniversidad Simón BolívarCaracasVenezuela

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