Simultaneous estimation of temperature-dependent thermal conductivity and heat capacity based on modified genetic algorithm

Figures & data

Figure 1. One-dimensional conductive model.

Figure 2. Flowchart of the proposed method.

Figure 3. Function f6 vs. x with y set to 0.

Figure 4. Average and the best fitness evolution of function f6.

Figure 5. Simulated measurements of temperature at the heated surface and some internal locations for .

Figure 6. Filtered temperatures with σ = 0.01T_max at the heated surface for .

Figure 7. Scaled dimensionless sensitivity coefficients of parabolic TDTPs for .

Figure 8. Effect of the heating duration on the dimensionless D-optimality criterion for estimating parabolic TDTPs.

Table 1. Estimation of linear TDTPs using MEGA and L–M method with genetic parameters: n_s = 100, n_g = 1000, P_c = 0.99, P_m = 0.05, P_r = 0.9, and r_c = 0.9 for σ = 0 and σ = 0.01T_max

Run			c1	c2	k1	k2	RMS error
σ = 0	MEGA	1	448.17	0.295	14.12	0.0165	0.033
		2	448.55	0.299	14.14	0.0164	0.059
		3	446.60	0.285	14.03	0.0169	0.109
		4	448.78	0.306	14.16	0.0163	0.082
		5	447.21	0.280	14.09	0.0168	0.097
		Mean	447.86 ± 1.15	0.293 ± 0.013	14.11 ± 0.06	0.0166 ± 0.0005	0.076 ± 0.038
	L–M		448.00	0.291	14.10	0.0166	1e−8
σ = 0.01Tmax	MEGA	1	450.41	0.275	14.28	0.0160	2.10
		2	446.87	0.311	13.88	0.0173	2.08
		3	449.86	0.273	14.19	0.0169	2.11
		4	448.63	0.285	13.93	0.0175	2.09
		5	446.25	0.314	14.25	0.0165	2.10
		Mean	448.40 ± 2.25	0.292 ± 0.024	14.11 ± 0.232	0.0168 ± 0.0008	2.10 ± 0.01
	L–M		447.22	0.304	14.14	0.0167	2.08
	L–M*		447.69	0.296	14.12	0.0166	1.85
*Estimation with filtered data.

Table 2. Estimation of parabolic TDTPs using MEGA and L–M method with genetic parameters: n_s = 100, n_g = 1000, P_c = 0.99, P_m = 0.05, P_r = 0.9, and r_c = 0.9 for σ = 0.01T_max

Run			c1	c2	c3	k1	k2	k3	RMS error
σ = 0.01Tmax	MEGA	1	446.92	0.5237	−4.9e−4	13.70	0.0197	−2.31e−5	2.83
		2	445.64	0.5121	−5.2e−4	13.97	0.0209	−1.92e−5	2.91
		3	449.78	0.5386	−4.3e−4	14.28	0.0205	−1.84e−5	2.78
		4	443.21	0.5069	−5.3e−4	13.56	0.0208	−2.03e−5	2.85
		5	445.11	0.5425	−4.5e−4	14.13	0.0196	−2.12e−5	2.94
		Mean	446.10 ± 3.03	0.5248 ± 0.020	−4.8e−4 ± 5e−5	13.93 ± 0.37	0.0203 ± 0.001	−2.03e−5 ± 7e−7	2.86  ± 0.08
	L–M		446.42	0.5296	−5.0e−4	13.94	0.0205	−2.05e−5	2.51
	L–M*		447.35	0.5387	−5.2e−4	13.98	0.0204	−2.12e−5	2.18
*Estimation with filtered data.

Table 3. Estimation of linear and parabolic TDTPs with respect to the different functional forms of the estimates

Simulation	Estimation	c1	c2	c3	k1	k2	k3	RMS error
Linear TDTPs	Constant	566.13	–	–	17.69	–	–	6.37
	Linear	447.69	0.296	–	14.12	0.0166	–	1.85
	Parabolic	444.67	0.311	−7.2e−5	13.91	0.0187	−4.6e−6	4.37
Parabolic TDTPs	Constant	548.35	–	–	17.23	–	–	5.23
	Linear	451.89	0.247	–	14.31	0.0147	–	3.39
	Parabolic	447.35	0.5387	−5.2e−4	13.98	0.0204	−2.12e−5	2.18

Figure 9. Average and the best fitness evolutions of function f(β) from the MEGA for parabolic TDTPs with σ = 0.01T_max.

Figure 10. Estimated heat capacity parameter (c₁) from the MEGA for linear TDTPs with σ = 0.01T_max.

Figure 11. Estimated heat capacity parameter (c₂) from the MEGA for the linear TDTPs with σ = 0.01T_max.

Figure 12. Estimated heat capacity for the linear TDTPs with σ = 0.01T_max.

Figure 13. Estimated thermal conductivity for the linear TDTPs with σ = 0.01T_max.

Figure 14. Estimated heat capacity for the parabolic TDTPs with σ = 0.01T_max.

Figure 15. Estimated thermal conductivity for the parabolic TDTPs with σ = 0.01T_max.

Figure 16. Residual distribution for the parabolic TDTPs with σ = 0.01T_max and different number of nodes.