## 5 Numerical Example  
A box structure made of metal-foam was provided to verify the efficiency of the proposed method. In order to improve computational efficiency, the equivalent solid parameter model was adopted, where the actual performance of metal-foam structure can be simulated by adjusting the thermodynamic parameters of the equivalent model. The schematic diagram of the structure was shown in Figure 8, where the overall dimensions were 500 mm \(\times\) 500 mm \(\times\) 500 mm, and the thickness was 30 mm. Meanwhile, there is a \(100\times 100\times 100\) mm heating element on the middle of the bottom. The whole structure is made of the same material with emissivity \(\varepsilon\) and heat conductivity \(k\). To simulate the heat transfer performance of the box under low temperature and pressure environment at high altitudes, the ambient air temperature outside the box is set to -60 \({}^{\circ}\)C, and the initial ambient air temperature in  
Figure 7: Flowchart of the parameter interval updating strategy. the box is set to 20 \({}^{\circ}\)C. Besides, the combination of radiation and convection was considered both inside and outside the box, where the external convection heat transfer coefficient was set to 3 \(W/(\text{m}^{2}\cdot\text{K})\). Then, the finite element model of equivalent model was created in NX THERMAL/FLOW with 14,049 tetrahedron elements and 27,302 nodes, as shown in Figure 8b. Four nodes were selected as the feature response points, where point 1 and 2 were located at the center of the outer surface of the box, respectively, and point 3 and 4 were distributed along the centerline at the inner surface of the box, as shown in Figure 8a. And the temperature value of each feature point is the result of the steady state of the system. Furthermore, it should be noted that the thermodynamic parameters identified by this method are not parameters of metal-foam models, but the equivalent model. In other words, we can obtain a more accurate equivalent model that can simulate the original foam structure. Due to the uncertainty of the metal-foam material properties, as well as the uncertainty of the complicated external factors, various uncertainties unavoidably exist in the heat transfer system of the box structure. According to the description in Section 2, if the equivalent heat conductivity, emissivity and the power of heating element are known in advance to change in the intervals \(k\in[10,20]\)\(W/(\text{m}\cdot\text{K})\), \(\varepsilon\in[0.4,0.8]\), and \(\text{p}\in[10,15]\)\(W\), the accurate intervals of four feature points can be calculated by the optimization algorithm, where the temperature responses are changed in the ranges \([-56.85,\,-49.57]\), \([-49.80,\,-34.97]\), \([-56.56,\,-49.11]\), \([-55.36,\,-47.83]\) respectively. In order to simulate the situation lack of measurement datas in actual engineering, 8 samples of each temperature response intervals are randomly selected with the assumption of normal distribution, which are listed in Equation (28). \[ \begin{array}{l}T_{1}=\{-53.85,\quad-54.21,\quad-53.33,\quad-55.12,\quad-53.2 1,\quad-54.86,\quad-51.04,\quad-54.10\};\\ T_{2}=\{-44.20,\quad-42.96,\quad-41.46,\quad-38.40,\quad-41.33,\quad-45.14, \quad-39.02,\quad-43.78\};\\ T_{3}=\{-52.01,\quad-54.25,\quad-54.68,\quad-54.90,\quad-52.94,\quad-53.53,\quad -51.10,\quad-53.04\};\\ T_{4}=\{-52.39,\quad-50.32,\quad-50.97,\quad-50.62,\quad-53.11,\quad-51.30, \quad-51.70,\quad-53.31\};\\ \end{array}  \tag{28}\]  
By utilizing the traditional extremum description method, the intervals of above sample set can be described as  
\[ \begin{array}{l}T_{1}^{\varepsilon}=[-55.12,-51.04]\quad T_{2}^{\varepsilon} =[-45.14,-38.40]\\ T_{3}^{\varepsilon}=[-54.90,-51.10]\quad T_{4}^{\varepsilon}=[-53.31,-50.32] \end{array}  \tag{29}\]  
Similarly, the unbiased estimation method is applied to obtain the unbiased intervals of the above sample set, which shown in Equation (30). \[