Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 31

and the formula for calculating internal energy in thermodynamics, the first law of
thermodynamics yields

c ∂

T ( r , t )

ρ

t + ∇ ·

q − F ( r , t )= 0 , (1.72)

∂

where and c are the density and the specific heat of the material, and F is the rate

of internal energy generation per unit volume. Equation (1.72) is called the energy
equation , and it contains two unknowns T and q . By usinga constitutive relation of
heat flux density, we may eliminate q from Eq. (1.72) to obtain an equation of tem-
perature T . Since both the Fourier law (1.23) and the CV constitutive relation (1.25)
are the special cases of the dual-phase-lagging constitutive relation (1.35), we use
Eq. (1.35) to derive the heat-conduction equations.
Assuming constant material properties, the divergence of Eq. (1.35) yields

ρ

+ τ 0 ∂
∂

τ T ∂
∂

∇ · q

t [ ∇ ·

q ]= − k Δ T − k

t [ Δ

T ] . (1.73)

Substituting the expression of ∇ · q from Eq. (1.72)

c ∂

T
∂

∇ · q = F − ρ

t

into Eq. (1.73) and introducing the thermal diffusivity α = k / ( ρ c ) leads to

F

2 T
∂

1
α

∂ T

∂ t + τ 0

∂

T ∂
∂

1
k

0 ∂

F
∂ t 2 = Δ

T + τ

t ( Δ

T )+ + τ . (1.74)

α t

This is called the dual-phase-lagging heat-conduction equation .When τ T = 0, it

reduces to the hyperbolic heat-conduction equation

F

2 T
∂

1
α

∂ T

t + τ 0

∂ 1

k

+ τ 0 ∂

F
∂ t 2 = Δ

T + . (1.75)

∂ α t

In the absence of two phase lags, i. e. when τ 0 = τ T = 0 , it reduces to the classical

parabolic heat-conduction equation

1
α

∂ T

∂

1
k F t = Δ

T + . (1.76)

For steady-state heat conduction, both the first and the second derivatives of T with
respect to t are zero. All three kinds of heat-conduction equations reduce to potential
equations .