Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 20

constitutive relation proposed by Cattaneo (1958) and Vernotte (1958, 1961),

)+ τ 0 ∂

q ( r , t )

q ( r , t

t = −

k ∇ T ( r , t ) (1.25)

∂

is the most widely accepted. This relation is named the CV constitutive relation after
the names of the proposers. Here

0 is a material property and is called the relax-
ation time . The corresponding heat-conduction equation is thus (See Section 1.3.3
for the derivation)

τ 0 >

F

2 T
∂

1
α

∂ T

∂ t + τ 0

∂ 1

k

+ τ 0 ∂

F
∂ t 2 = Δ

T + . (1.26)

α t

Unlike its classical counterpart Eq. (1.24), this equation is of hyperbolic type,
characterizes the combined diffusion and wave-like behavior of heat conduction,
and predicts a finite speed

k
ρ

V CV = (1.27)

c τ 0

for heat propagation (Wang and Zhou 2000, 2001, Wang et al. 2007a).
Note that the CV constitutive relation is actually a first-order approximation of
a more general constitutive relation (single-phase-lagging model; Tzou 1992a),

q ( r , t + τ 0 )= − k ∇ T ( r , t ) . (1.28)

according to which the temperature gradient established at a point r at time t gives
rise to a heat flux vector at r at a later time t + τ 0 . There is a finite built-up time

τ 0 for the onset of heat flux at r after a temperature gradient is imposed there.
Thus the τ 0 represents the time lag needed to establish the heat flux (the result)

when a temperature gradient (the cause) is suddenly imposed. The higher t

corresponds to a larger derivation of the CV constitutive relation from the classical
Fourier law.
The value of

∂ q / ∂

τ 0 is material-dependent (Chandrasekharaiah 1986, 1998, Tzou
1997). For most solid materials, τ 0 varies from 10 − 10 sto10 − 14 s. For gases, τ 0

is normally in the range of 10 − 8 10 − 10 s. The value of τ 0 for some biological

materials and materials with non-homogeneous inner structures can be up to 10 2 s
(Beckert 2000, Kaminski 1990, Mitra et al. 1995, Peters 1999, Roetzel et al. 2003,
Vedavarz et al. 1992). Therefore, the thermal relaxation effects can be of relevance
even in common engineering applications where the time scales of interest are of
the order of a fraction of a minute.
Three factors contribute to the significance of the second term in the hyperbolic
heat-conduction equation (1.26): the value of

∼

τ 0 , the rate of change of temperature,
and the time scale involved. The wave nature of thermal signals will be over the
diffusive behavior through this term when (Tzou 1992a)

∂ T

∂

T r
2

>> ( / τ )

τ 0 exp

t (1.29) 0

t