Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 25

While the former is caused only by the fast transient effects of thermal inertia, the
latter comes from these effects as well as the delayed response due to the microstruc-
tural interaction. Tzou (1997) refers to the former wave as the CV-wave and thelatter
wave as the T-wave. By Eqs. (1.27) and (1.39), we have

2

τ T
τ

V T = V CV . (1.40)

0

Therefore, the T-wave is always slower than the CV-wave because Eqs. (1.37)
and (1.38) are valid only for

τ 0 >> τ T . This has been shown by the heat propa-
gation in superfluid helium at extremely low temperatures (Tzou 1997). It is inter-
esting to note that Eq. (1.37) is the simplest constitutive relation that accounts for
the dual-phase-lagging effects and yields a heat-conduction equation of hyperbolic
type. If the second order term in τ T is also retained, the resulting heat-conduction

equation will no longer be hyperbolic (Tzou 1997). It is also of interest to note that
Eq. (1.38) closely resembles the energy equation describing the ballistic behavior of
heat transport in an electron gas (Qiu and Tien 1993, Tzou 1997).
In this section, we have presented a brief review of some valid models for heat
conduction from macro- to micro-scales: the Fourier model based on Eqs. (1.23)
and (1.24),the CV model based on Eqs. (1.25) and (1.26), the wave model based on
Eqs. (1.30) and (1.31), the single-phase-lagging model based onEq. (1.28),
and the three dual-phase-lagging models the [dual-phase-lagging model based
on Eqs. (1.32), (1.33) and (1.34); the first-order dual-phase-lagging model based
on Eqs. (1.35) and (1.36); the second-order dual-phase-lagging model based on
Eqs. (1.37) and (1.38)]. In literature, the dual-phase-lagging model usually refers
to the first-order dual-phase-lagging model [Eqs. (1.35) and (1.36)], which is a gen-
eralized and unified model for heat conduction from macro- to micro-scales with
the Fourier, wave and CV models as its special cases. A relatively comprehensive
list of literature on these models can be found in the References.

1.3.2 The Boltzmann Transport Equation
and Dual-Phase-Lagging Heat Conduction

The Boltzmann Transport Equation

Consider a classical system of N particles. Each particle has s degrees of freedom
so that the number of coordinates needed to specify positions of all N particles is
l Ns . The classical mechanical state of the system can be completely described

by l spatial coordinates q i and l corresponding velocity coordinates

=

i . Introduce
a conceptual Euclidean hyperspace of 2 l dimensions, with a coordinate axis for
each of the 2 l spatial coordinates and velocities. This conceptual space is usually
termed as the phase space for the system. The state of the classical N -particle or
N -body system at any time t is completely specified by the location of one point in

υ