Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 19

The Fourier Law

In deriving the classical heat-conduction equation (1.21), we have used the Fourier
law of heat conduction. It was the first constitutive relation of heat flux and was
proposed by the French mathematical physicist Joseph Fourier in 1807 based on
experimentation and investigation (Wang 1994). For heat conduction in a homoge-
neous and isotropic medium, the Fourier law of heat conduction reads

q ( r , t )= − k ∇ T ( r , t ) , (1.23)

where r stands for the material point, t the time, T the temperature and the gradi-

ent operator. k is the thermal conductivity of the material, which is a thermodynamic
state property. By the state theorem of thermodynamics, k should be a function of
two independent and intensive dynamic properties (normally pressure and temper-
ature; Cengel and Boles 2006). The second law of thermodynamics requires that k
is positive-definite (Wang 1994, 1995, 2001). In engineering applications, we of-
ten take k as a material constant because variations in pressure and temperature are
normally sufficiently small. The value of k is material-dependent. If the material is
not homogeneous or isotropic, k becomes a second order tensor (Wang 1994, 1995,
1996, 2001). Along with the first law of thermodynamics, this equation gives the
classical parabolic heat-conduction equation (See Section 1.3.3 for the derivation)

∇

1
α

∂ T

∂

1
k F

t = Δ

T + . (1.24)

Here is the thermal diffusivity of the material, F is the rate of internal energy

generation per unit volume, and

α

is the Laplacian.
The Fourier law of heat conduction is an early empirical law. It assumes that
q and

Δ

T appear at the same time instant t and consequently implies that thermal
signals propagate with an infinite speed. If the material is subjected to a thermal
disturbance, the effects of the disturbance will be felt instantaneously at distances
infinitely far from its source. Although this result is physically unrealistic, it has
been confirmed by many experiments that the Fourier law of heat conduction holds
for many media in the usual range of heat flux q and temperature gradient (Wang
1994).

∇

The CV Constitutive Relation

With the development of science and technology such as the application of ultra-
fast pulse-laser heating on metal films, heat conduction appears in the range of high
heat flux and high unsteadiness. The drawback of infinite heat propagation speed
in the Fourier law becomes unacceptable. This has inspired the work of searching
for new constitutive relations. Among many proposed relations (Wang 1994), the