Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 22

other delay time τ 0 is interpreted as the relaxation time due to the fast-transient

effects of thermal inertia (or small-scale effects of heat transport in time) and is
called the phase-lag of the heat flux . Both of the phase-lags are treated as intrinsic
thermal or structural properties of the material. The corresponding heat-conduction
equation reads (Xu and Wang 2005)

t ) 1

α

∂ T ( r , 1

k F

t − τ )+ t ) , t =

= Δ T ( r , ( r , t + τ 0 , τ = τ 0 − τ T ,

∂ t

0and t > τ 0 , for τ 0 − τ T > (1.33)

or

t − τ )
∂

1
α

∂ T ( r , 1

k F

t )+ t − τ ) , t = = Δ T ( r , ( r , t + τ T , τ = τ T − τ 0 ,

t

0 nd t > τ for τ − τ < . (1.34)

0 T T

Unlike the relation (1.28) according to which the heat flux is the result of a tem-
perature gradient in a transient process, the relation (1.32) allows either the temper-
ature gradient or the heat flux to become the effect and the remaining one the cause.
For materials with τ 0 > τ T , the heat flux density vector is the result of a temperature

gradient. It is the other way around for materials with τ T > τ 0 . The relation (1.28)

corresponds to the particular case where τ 0 = τ T (not nec-

essarily equal to zero), the response between the temperature gradient and the heat
flux is instantaneous; in this case, the relation (1.32) is identical with the classical
Fourier law (1.23). It may also be noted that while the classical Fourier law (1.23) is
macroscopic in both space and time and the relation (1.28) is macroscopic in space
but microscopic in time, the relation (1.32) is microscopic in both space and time.
Also note that Eqs. (1.33) and (1.34) are of the delay and advance types, respec-
tively. While the former has a wave-like solution and possibly resonance, the latter
does not (Xu and Wang 2005). Both single-phase-lagging and dual-phase-lagging
heat conduction have been shown to be admissible by the second law of extended
irreversible thermodynamics (Tzou 1997) and by the Boltzmann transport equation
(Xu and Wang 2005).
Expanding both sides of Eq. (1.32 )by using the Taylor series and retaining only
the first-order terms of

τ 0 > 0and τ T = 0. If

τ

0 and

τ T , we obtain the following constitutive relation that

is valid at point r and time t ,

k )+ τ 0 ∂

q ( r , t )

)+ τ T ∂
∂

q ( r , t

t = −

∇ T ( r , t

t [ ∇

T ( r , t )] , (1.35)

∂

which is known as the Jeffreys-type constitutive equation of heat flux (Joseph
and Preziosi1989). In literature this relation is also called the dual-phase-lagging
constitutive relation .When τ 0 = τ T , this relation reduces to the classicalFourier

law (1.23), and it reduces to the CV constitutive relation (1.25) when 0.

Eliminating q from Eq. (1.35) and the classical energy equation leads to the dual-
phase-lagging heat conduction equation that reads, if all thermophysical material

τ T =