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On hyperbolic heat conduction in solids: minimum mean free path of energy carriers and a method of estimating thermal diffusivity and heat propagation characteristics

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Abstract

The paper gives the analytical solution to the one dimensional hyperbolic heat conduction equation in an insulated slab-shaped sample that is heated uniformly on the front face with δ or laser impulse. The solution results in a formula that enables to estimate the minimum mean free path of energy carriers in the sample to detect the second sound (i.e. the thermal wave) at the sample rear face. A method of experimental data evaluation at the second sound effect is proposed, which gives the thermal diffusivity of the sample and the parameters of heat propagation.

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Abbreviations

a :

thermal diffusivity

b :

dimensionless parameter

c :

specific heat capacity

c v :

specific heat capacity per volume unit

I0 (x), I1 (x):

modified Bessel functions of the first kind

L :

sample thickness

L c :

sample critical thickness

ℓ:

mean free path

ℓmin :

minimum mean free path

Q :

total delivered energy per square meter

Q w :

total energy of the wave per square meter

\({\bar{Q}}_{w}\) :

total relative energy of the wave per square meter

q :

density of energy flow rate

T :

temperature

\({\bar{T}}\) :

relative temperature

T ∞ :

resultant temperature

t :

time

t*:

wave transit time

\({\bar{t}}\) :

dimensionless time

t T :

duration of the impulse

\({\bar{t}}_{T}\) :

relative duration of the impulse

t c :

characteristic time of the impulse

\({\bar{t}}_{c}\) :

relative characteristic time of the impulse

v :

mean velocity of heat carriers in substance

x :

coordinate

\({\bar{x}}\) :

dimensionless coordinate

δ(x):

delta-function

μ (x):

unit-step function

λ:

thermal conductivity

ρ:

mass density

τ:

relaxation time

ω:

wave velocity

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  1. Faculty of Natural Sciences, Constantine the Philosopher University, Trieda A. Hlinku 1, 949 74, Nitra, Slovakia

    Ján Beňačka

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Beňačka, J. On hyperbolic heat conduction in solids: minimum mean free path of energy carriers and a method of estimating thermal diffusivity and heat propagation characteristics. Heat Mass Transfer 44, 873–887 (2008). https://doi.org/10.1007/s00231-007-0289-9

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  • DOI: https://doi.org/10.1007/s00231-007-0289-9

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