Multi-model method for solving nonlinear transient inverse heat conduction problems

Figures & data

Figure 1. (a) One-dimensional transient heat conduction system. (b) Grid division of time and space.

Figure 2. Block scheme of the multi-model inversion system.

Table 1. Thermal properties of Armco (99.75% pure) [20].

Temperature (K)	Thermal conductivity (W m^−1 K^−1)	Specific heat (J kg^−1 K^−1)	Density (kg m^−3)
200	80.6	384	7870
400	65.7	490	7870
600	53.1	574	7870
800	42.2	680	7870
1000	32.3	975	7870
1200	28.7	609	7870
1500	31.4	654	7870

Figure 3. Comparison of temperature evolution calculated by the heat balance method in this work and the finite element software Comsol.

Table 2. Effect of the number of local models on the inversion results.

Number of local models	Set of balance points /K	Computation time/s	Sq/(W m^−2)
N = 1	{600}	3.04	1.00 × 10^6
N = 2	{500, 900}	3.27	1.30 × 10^5
N = 4	{300, 500, 700, 900}	3.35	1.23 × 10^5
N = 8	{200, 300, 400, 500, 600, 700, 800, 900}	3.53	1.22 × 10^5

Figure 4. Inversion results of the multi-model method with different numbers of local models.

Table 3. Comparison of the single model, adaptive model and multi-model method.

Inverse method	Set of balance points/K	Computation time/s	Sq/(W m^−2)
Single model	{600}	3.04	1.00 × 10^6
Multi-model	{300, 500, 700, 900}	3.35	1.23 × 10^5
Adaptive model	–	6.42	1.20 × 10^5

Figure 5. Comparison of the single model, adaptive model and multi-model method.

Figure 6. DMC digital filter coefficients at different balance point temperatures ($\mathit{\alpha }=1\times {10}^{-10}$ K² m⁴ W⁻², k_p = 30, k_f = 10, t_k = 18 s).

Figure 7. The DMC digital filter coefficients for α = 5 × 10⁻¹⁰, 1 × 10⁻¹⁰ and 5 × 10⁻¹¹ K² m⁴ W⁻² (k_p = 30, k_f = 10, t_k = 18 s, $v_{\text{b,}n}$ = 600 K).

Figure 8. Effect of the initial guess value of the surface heat flux on the inversion results by DMC filter. ($\mathit{\alpha }=1\times {10}^{-10}$ K² m⁴ W⁻²)

Figure 9. Comparison of the inversion results of multi-model and single model method associated with DMC digital filter with noisy-free measured temperature. ($\mathit{\alpha }=1\times {10}^{-10}$ K² m⁴ W⁻², k_p = 20, k_f = r = 10).

Figure 10. Comparison of the inversion results of multi-model and single model method associated with DMC digital filter with $\mathit{\sigma }=0.5$ K. ($\mathit{\alpha }=1\times {10}^{-10}$ K² m⁴ W⁻², k_p = 20, k_f = r = 10).

Incropera FP, Dewitt DP, Bergles TL, et al. Fundamentals of heat and mass transfer. 6th ed. Hoboken (NJ): Wiley; 2007.

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