Inverse problem in a porous medium: estimation of effective thermal properties

Figures & data

Figure 1. The hot wire technique.

Figure 2. Dimensionless numerical and analytical wire evolutions (Bii = 10, A = 1.5, Z = A/2, γ = 5 × 10⁻³).

Figure 3. Influence of the temperature sensor position (R) on the determinant, the condition number and the lowest eigenvalue.

Figure 4. Reduced sensitivities versus time.

Figure 5. Measured temperature versus time (σ = 0.01 K).

Table 1. Identification results.

	λe(Wm^−1 K^−1)	(ρCp)e × 10^−6(Jm^−3 K^−1)	hi(Wm^−2 K^−1)
Expected values	0.37048	1.004903	10	Relative error (%)(---581----5)	Itera. Nb	Least squares J
Initial values (+10%)	0.407528	1.105393	11
σ = 0 K	0.370479	1.004904	10	1.28 × 10^−6	7	4.39 × 10^−11
σ = 0.01 K (0.2% Tmax)	0.370577	1.004451	10	2.37 × 10^−4	10	7.09 × 10^−2
σ = 0.1 K (2% Tmax)	0.371461	1.000370	10	2.39 × 10^−3	21	7.097
σ = 0.2 K (4% Tmax)	0.372441	0.995756	10	4.80 × 10^−3	6	28.39
Initial values (+30%)	0.481624	1.306374	13
σ = 0 K	0.37048	1.004903	10	0	9	5.05 × 10^−13
σ = 0.01 K (0.2% Tmax)	0.370577	1.004719	10	1.48 × 10^−4	17	7.09 × 10^−2
σ = 0.1 K (2% Tmax)	0.371462	1.000357	10	2.39 × 10^−3	17	7.097
σ = 0.2 K (4% Tmax)	0.345918	1.048558	10	3.78 × 10^−2	7	28.38
Initial values (+50%)	0.55572	1.507355	15
σ = 0 K	0.37048	1.004903	10	1.70 × 10^−8	6	2.39 × 10^−11
σ = 0.01 K (0.2% Tmax)	0.370577	1.004451	10	2.37 × 10^−4	10	7.09 × 10^−2
σ = 0.1 K (2% Tmax)	0.371454	1.000395	10	2.37 × 10^−3	10	7.097
σ = 0.2 K (4% Tmax)	0.314602	0.900846	10	4.74 × 10^−3	8	28.38
Initial values (+70%)	0.629816	1.708335	17
σ = 0 K	0.370479	1.004904	10	1.37 × 10^−6	22	4.50 × 10^−11
σ = 0.01 K (0.2% Tmax)	0.370579	1.004409	10	2.53 × 10^−4	35	7.09 × 10^−2
σ = 0.1 K (2% Tmax)	0.371454	1.000398	10	2.37 × 10^−3	17	7.097
σ = 0.2 K (4% Tmax)	0.372433	0.995912	10	4.74 × 10^−3	9	28.38
Initial values (+90%)	0.703912	1.909316	19
σ = 0 K	0.370479	1.004904	10	1.37 × 10^−6	66	4.68 × 10^−11
σ = 0.01 K (0.2% Tmax)	0.370578	1.004445	10	2.40 × 10^−4	78	7.09 × 10^−2
σ = 0.1 K (2% Tmax)	0.371454	1.000395	10	2.37 × 10^−3	28	7.097
σ = 0.2 K (4% Tmax)	0.372432	0.995914	10	4.74 × 10^−3	15	28.38

Table

A	The cylinder aspect ratio, A = H/D
Bi	Biot number, (Bi = hD/λ)
Ci	The wire specific heat capacity (J kg^−1 K^−1)
Cpf	The fluid specific heat capacity (J kg^−1 K^−1)
Cps	The solid specific heat capacity (J kg^−1 K^−1)
Cp	Effective specific heat capacity (J kg^−1 K^−1)
D	The cylinder gap width, D = ro − ri (m)
g	Gravity acceleration (m s^−2)
H	Cylinder height (m)
h	Heat transfer coefficient (W m^−2 K^−1)
Ka	The medium volumetric heat capacity ratio
Ks	The hot wire volumetric heat capacity ratio
Mi	Linear wire mass (kg m^−1)
n	Number of measurements
q	Linear heat flux density (W m^−1)
Q	Reference heat flux density (W m^−2)
r, z	Space coordinates (m)
R, Z	Dimensionless space coordinates
t	Time (s)
T	Temperature (K)
p	Number of parameters

Table

ε	Average porosity
θ	Dimensionless temperature, θ = (T−T0)/Tref
γ	Dimensionless wire radius, γ = ri/D
κ	Radius ratio, κ = ro/ri
ρ	Fluid density (kg m^−3)
λe	Effective thermal conductivity (Wm^−1 K^−1)
τ	Dimensionless time

Table

f	Fluid phase
h	High cylinder base
i	Inner wall (wire)
l	Lower cylinder base
o	Outer wall (ambient and initial)
ref	Reference
s	Solid phase
e	Effective