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


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.


Wave Front Solid Sphere Solid Cylinder Temperature Wave Jump Point 
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List of symbols


thermal diffusivity

c1, c2

constant coefficient


constant coefficient


function defined by Eq. 10


Fourier number, at/x o 2


function employed in Eq. 21


function employed in Eq. 21


constant employed in Eq. 12


non-negative integer number


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


constant employed in Eq. 3


function employed in Eq. 4


reference value of the heat flux


heat flux density


eigen function




dummy integration variable employed in Eq. 32


temperature field


dimensionless temperature field defined by Eq. 6


initial temperature

Tn (Fo),

function defined by Eq. 34


Vernotte number, aτ o /x o 2

Wn (Fo)

function defined by Eq. 31


spatial variable


the thickness of slab or the radius of cylinder and sphere


dimensionless spatial variable, x/x o

Greek symbols


variable defined by Eq. 26


 = |β n |, real variable

φ (X)

function employed in Eq. 11

Φ (X,Fo)

function defined by Eq. 30


variable defined by Eq. 28


thermal conductivity

θ (x,t)

function employed in Eq. 11


thermal relaxation time


separation constant


eigen values

ψ (Fo)

function employed in Eq. 11


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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

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