Finite volume solution of 2-D hyperbolic conduction in a heterogeneous medium

ABSTRACT

Finite volume method that has been shown to be an effective tool for the study of non-Fourier conduction in one-dimensional heterogeneous material is extended to two-dimensional (2D) heterogeneous materials. The accuracy of 2D numerical formulation is examined by comparing the numerical results with the exact solutions for a homogeneous material with constant thermal properties. The effects of temperature-dependent properties due to both heating and cooling of the medium on the wave propagation are investigated. Transient energy balance shows significant effects of finite thermal wave speed on the transient behavior of internal and wave energy changes in the system. The effects of finite thermal wave speed on the entropy production for the cooling case are examined by invoking the second law of thermodynamics. The 2D transient temperature distributions in a heterogeneous medium show the expected effects of flux lagging constant on the thermal wave propagations through heterogeneous material.

Nomenclature

C	=	= specific heat (J kg−3 K−1)
EG, H, I, τ	=	= energy contents defined by Eqs. (22c), (22b), (22a), and (22d), respectively
H	=	= thickness in y-dimension (10 nm)
L	=	= thickness in x-dimension (10 nm)
M, N	=	= number of grid points in y- and x-direction, respectively
qx,y	=	= heat flux in x- and y-direction, respectively (Wm−2)
	=	= energy generation rate (Wm−3)
	=	= entropy generation rate (J K−1 m−3 s−1)
S	=	= total entropy generation (J K−1)
t	=	= time (s)
tL	=	= characteristic time (L/vs = 6 × 10−13 s)
T	=	= temperature (K)
vs	=	= thermal wave speed (ms−1)
x	=	= spatial coordinate in x-direction (m)
y	=	= spatial coordinate in y-direction (m)
Greek symbols	=
α	=	= thermal diffusivity (m2s−1)
(δx)e,w; (δy)n,s	=	= distances between control volumes as shown in Figure 1 (m)
Δt	=	= time step (s)
Δx, Δy	=	= control volume size in x- and in y-dimension, respectively (m)
ρ	=	= density (kg m−3)
κ	=	= thermal conductivity (W m−1 K−1)
∅, φ	=	= defined by Eqs. (29a) and (29b), respectively, for exact solution
τ	=	= heat flux lagging constant (s)
Subscripts	=
0	=	= initial condition
e, w, n, s	=	= east, west, north, and south side control volume surfaces
E, W, N, S	=	= east, west, north, and south side control volumes
P	=	= control volume under consideration
Superscripts	=
1, 0, and −1	=	= time level at t − Δt, t, and t − Δt, respectively
1*	=	= predicted value at t + Δt

Nomenclature

C	=	= specific heat (J kg−3 K−1)
EG, H, I, τ	=	= energy contents defined by Eqs. (22c), (22b), (22a), and (22d), respectively
H	=	= thickness in y-dimension (10 nm)
L	=	= thickness in x-dimension (10 nm)
M, N	=	= number of grid points in y- and x-direction, respectively
qx,y	=	= heat flux in x- and y-direction, respectively (Wm−2)
	=	= energy generation rate (Wm−3)
	=	= entropy generation rate (J K−1 m−3 s−1)
S	=	= total entropy generation (J K−1)
t	=	= time (s)
tL	=	= characteristic time (L/vs = 6 × 10−13 s)
T	=	= temperature (K)
vs	=	= thermal wave speed (ms−1)
x	=	= spatial coordinate in x-direction (m)
y	=	= spatial coordinate in y-direction (m)
Greek symbols	=
α	=	= thermal diffusivity (m2s−1)
(δx)e,w; (δy)n,s	=	= distances between control volumes as shown in Figure 1 (m)
Δt	=	= time step (s)
Δx, Δy	=	= control volume size in x- and in y-dimension, respectively (m)
ρ	=	= density (kg m−3)
κ	=	= thermal conductivity (W m−1 K−1)
∅, φ	=	= defined by Eqs. (29a) and (29b), respectively, for exact solution
τ	=	= heat flux lagging constant (s)
Subscripts	=
0	=	= initial condition
e, w, n, s	=	= east, west, north, and south side control volume surfaces
E, W, N, S	=	= east, west, north, and south side control volumes
P	=	= control volume under consideration
Superscripts	=
1, 0, and −1	=	= time level at t − Δt, t, and t − Δt, respectively
1*	=	= predicted value at t + Δt