Advertisement

Heat and Mass Transfer

, Volume 44, Issue 4, pp 381–392 | Cite as

Heat wave phenomena in solids subjected to time dependent surface heat flux

  • G. HeidarinejadEmail author
  • R. Shirmohammadi
  • M. Maerefat
Original

Abstract

Hyperbolic heat conduction in a plane slab, infinitely long solid cylinder and solid sphere with a time dependent boundary heat flux is analytically studied. The solution is based on the separation of variables method and Duhamel’s principle. The temperature distribution, the propagation and reflection of the temperature wave and the effect of geometry on the shape of the wave front are studied for the case of a rectangular pulsed boundary heat flux. Comparisons with the solution obtained for Fourier heat conduction are performed by considering the limit of a vanishing thermal relaxation time.

Keywords

Wave Front Solid Sphere Solid Cylinder Temperature Wave Jump Point 
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.

List of symbols

a

thermal diffusivity

c1, c2

constant coefficient

C,Cn

constant coefficient

f(Fo)

function defined by Eq. 10

Fo

Fourier number, at/x o 2

g(Fo)

function employed in Eq. 21

h(X)

function employed in Eq. 21

k

constant employed in Eq. 12

n

non-negative integer number

N

value of n that for n > N n has an imaginary value

p

constant employed in Eq. 3

q(t)

function employed in Eq. 4

qo

reference value of the heat flux

q

heat flux density

RnX)

eigen function

t

time

t

dummy integration variable employed in Eq. 32

T

temperature field

\({\hat{T}}\)

dimensionless temperature field defined by Eq. 6

To

initial temperature

Tn (Fo),

function defined by Eq. 34

Ve

Vernotte number, aτ o /x o 2

Wn (Fo)

function defined by Eq. 31

x

spatial variable

xo

the thickness of slab or the radius of cylinder and sphere

X

dimensionless spatial variable, x/x o

Greek symbols

βn

variable defined by Eq. 26

β1n

 = |β n |, real variable

φ (X)

function employed in Eq. 11

Φ (X,Fo)

function defined by Eq. 30

ηn

variable defined by Eq. 28

λ

thermal conductivity

θ (x,t)

function employed in Eq. 11

τo

thermal relaxation time

−ω2

separation constant

ωn

eigen values

ψ (Fo)

function employed in Eq. 11

References

  1. 1.
    Xu YS, Guo ZY (1995) Heat wave phenomena in IC chips. Int J Heat Mass Transf 38:2919CrossRefGoogle Scholar
  2. 2.
    Tang DW, Araki N (1996) Analytical solution of non-Fourier temperature response in a finite medium under laser pulse heating. Heat Mass Transf 31:359Google Scholar
  3. 3.
    Cattaneo C (1958) Sur une forme de l’equation de la chaleur elinant le paradox d’une propagation instantance. C R Acad Sci 247:431MathSciNetGoogle Scholar
  4. 4.
    Vernotte P (1958) Les paradoxes de la theorie continue de l’equation de la chaleur. C R Acad Sci 246:3154MathSciNetGoogle Scholar
  5. 5.
    Peshkov V (1944) Second sound in helium II. J Phys VIII, USSR, 381Google Scholar
  6. 6.
    Jackson HE, Walker CT (1971) Thermal conductivity, Second sound, and Phonon-Phonon interactions in NaF. Phys Rev B3:1428CrossRefGoogle Scholar
  7. 7.
    Narayanamurti V, Dynes RC (1972) Observation of second sound in Bismuth. Phys Rev Lett 28:1561CrossRefGoogle Scholar
  8. 8.
    Kaminiski W (1990) Hyperbolic heat conduction for materials with nonhomogeneous inner structure. ASME J Heat Transf 112:555CrossRefGoogle Scholar
  9. 9.
    Mitra K, Kumar S, Vedavarz A, Moallemi MK (1995) Experimental evidence of hyperbolic heat conduction in processed meat. ASME J Heat Transf 117:568Google Scholar
  10. 10.
    Kao TT (1977) Non-Fourier heat conduction in thin surface layers. ASME J Heat Transf 99:343Google Scholar
  11. 11.
    Frankel JI, Vick B, Ozisik MN (1985) Flux formulation of hyperbolic heat conduction. J Appl Phys 58:3340CrossRefGoogle Scholar
  12. 12.
    Hector LG, Kim WS, Ozisik MN (1992) Propagation and reflection of thermal waves in a finite medium due to axisymmetric surface sources. Int J Heat Mass Transf 35:897zbMATHCrossRefGoogle Scholar
  13. 13.
    Tang DW, Araki N (1996) Non-Fourier heat conduction in a finite medium under periodic surface thermal disturbance. Int J Heat Mass Transf 39:1585zbMATHCrossRefGoogle Scholar
  14. 14.
    Barletta A, Zanchini E (1996) Non-Fourier heat conduction in a plane slab with prescribed boundary heat flux. Heat Mass Transf 31:443Google Scholar
  15. 15.
    Barletta A, Zanchini E (1996) Hyperbolic heat conduction and thermal resonance in a cylindrical solid carrying a steady-periodic electric field. Int J Heat Mass Transf 39:1307CrossRefGoogle Scholar
  16. 16.
    Barletta A, Pulvirenti B (1998) Hyperbolic thermal waves in a solid cylinder with a non-stationary boundary heat flux. Int J Heat Mass Transf 41:107zbMATHCrossRefGoogle Scholar
  17. 17.
    Barletta A, Pulvirenti B (1998) Periodic heat conduction with relaxation time in cylindrical geometry. Heat Mass Transf 33:319CrossRefGoogle Scholar
  18. 18.
    Tan ZM, Yang WJ (1999) Propagation of thermal waves in transient heat conduction in a thin film. J Franklin Inst 336B:185CrossRefGoogle Scholar
  19. 19.
    Liu KC, Chen HT (2004) Numerical analysis for the hyperbolic heat conduction problem under a pulsed surface disturbances. Appl Math Comput 159:887zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Carslaw HS, Jaeger JC (1948) Conduction of heat in solids, 2nd edn. Clarendon Press, OxfordGoogle Scholar
  21. 21.
    Barletta A, Zanchini E (1997) Thermal wave heat conduction in a solid cylinder which undergoes a change of boundary temperature. Heat Mass Transf 32:285CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • G. Heidarinejad
    • 1
    Email author
  • R. Shirmohammadi
    • 1
  • M. Maerefat
    • 1
  1. 1.Department of Mechanical EngineeringTarbiat Modares UniversityTehranIran

Personalised recommendations