Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 334

By Eq. (6.287), we can readily obtain that, in two-phase-system heat conduction

2
β

2
σ

γ k σ + γ k β − 2 γ β γ σ k βσ

τ T
τ

0 =

1 + . (6.288)

γ β γ σ ( k β + k σ + 2 k βσ )

2
β

2
σ

It can be large, equal or smaller than 1 depending on the sign of γ k σ + γ k β −

2 γ β γ σ k βσ . Therefore, by the condition for the existence of thermal waves that re-

quires 1 (Section 6.10, Xu and Wang 2002), we may have thermal waves

in two-phase-system heat conduction when

τ T / τ 0 <

2
β

2
σ

γ k σ + γ k β − 2 γ β γ σ k βσ < 0 .

Note also that for heat conduction in two-phase systems there is a time-dependent
source term F in the dual-phase-lagging heat conduction [Eqs. (6.286)

and (6.287)]. Therefore, the resonance can also occur. This agrees with the experi-
mental data of casting sand tests in Tzou (1997). Discarding the coupled conductive
terms in Eqs. (6.273) and (6.274) assumes k βσ =

( r , t )

τ T / τ 0 is always larger
than 1, which leads to the exclusion of thermal oscillation and resonance (Vadasz
2005a, 2005c, 2006a) and generates an inconsistency between theoretical and ex-
perimental results in the literature regarding the possibility of thermal waves and
resonance in two-phase-system heat conduction (Tzou 1997, Vadasz 2005a, 2005c,
2006a). The coupled conductive terms in Eqs. (6.273) and (6.274) are thus respon-
sible for the thermal waves and resonance in two-phase-system heat conduction.
These thermal waves and possibly resonance are believed to be the driving force
for the extraordinary conductivity enhancement reported in nanofluids (Assael et al.
2006, Choi et al. 2001, 2004, Das 2006, Das et al. 2006, Eastman et al. 2001, 2004
Jang and Choi 2004, Kumar et al. 2004, Murshed et al. 2006, Peterson and Li 2006,
Phelan 2005, Putnam et al. 2006, Strauss and Pober 2006, Wang and Mujumdar
2007, Yu and Choi 2006).
Although each

0sothat

τ 0 and τ T is ha υ -dependent, the ratio τ T / τ 0 is not; this makes its

evaluation much simpler.