## 4 Results and Discussion
### Influence of Vacuum Level on Heat Transfer Characteristics
At 1.0 atm, the convection heat transfer and radiation heat transfer each takes approximately 50% of the overall heat flux. As the environmental pressure gets smaller, the radiation heat transfer becomes the dominant heat transfer approach. Under 0.1 atm, although the magnitude of radiation heat flux decreases compared with higher pressure cases, the weight of radiation heat flux over the total heat flux increases significantly. The radiation heat flux takes more than 80% of the overall heat flux under 0.1 atm. Based on the results in this study, at low-vacuum conditions, the radiation heat flux serves as the dominant heat transfer approach. As the environmental pressure gets lower, the difference between convection heat flux and radiation heat flux gradually increases. Figure 10 shows the influence of vacuum level on accumulated total and convection heat through the magnet surface over different vacuum levels under selected time frame. As the vacuum level increases from 1.0 atm to 0.1 atm, the total heat dissipating from levitation electromagnet is decreased by 49% (with more than 80% reduction of convection heat). A regression analysis is conducted to better capture the trend. It is found that both the total heat and convection heat can be well represented by a quadratic equation which contains the vacuum level term. In Figure 10a, the fitted curve has an \(R^{2}\) value greater than 0.999, while the fitted curve has an \(R^{2}\) value greater than 0.996 in Figure 10b. Therefore, it can be concluded that the total heat and the convection heat are approximately proportional to the square of the vacuum level. Figure 10: Relationship between accumulated heat through the magnet surface and vacuum level: (**a**) Total heat (**b**) Convection heat. Figure 9: Weight of convection heat flux over the total heat flux on the fluid–solid interface during the heating process. The convection heat transfer coefficient reflects the magnitude of convection heat transfer through the solid-fluid interface. This coefficient is a simplification and approximation of rigorous mathematical derivation. In numerical study and industrial research on conjugate heat transfer problems, the application of convection heat flux can effectively reduce the computation time and cost. The convection heat transfer coefficient can be affected by many factors such as external flow type (laminar flow and turbulent flow) and temperature of the solid surface. Both factors make it very challenging to predict [43]. Experiments usually measure convection heat transfer coefficients based on calibrated samples under typical conditions. The derived convection heat transfer coefficients are then applied to the entire system subsequent research [44]. In this study, the effective convection heat transfer coefficient is derived based on the numerical results according to Equation (8). The results show that during the heating process, under each vacuum level, the convection heat transfer coefficient has a variance of less than 10%. Therefore, a weighted average value of convection heat transfer coefficient is taken, this averaged convection heat transfer coefficient is referred as effective heat transfer coefficient (h\({}_{\text{e}}\)). This effective convection heat transfer coefficient decreases with lowered environment pressure. Table 7 shows the effective convection heat transfer coefficient of the EMU under different vacuum levels. Figure 11 shows the regression analysis on effective convection heat transfer coefficient and the vacuum level. The results from regression analysis indicate that the effective convection heat transfer coefficient of EMUs is linearly proportional to the vacuum level. When the vacuum is reduced to 0.1 atm, the effective convection heat transfer coefficient is less than 0.5 W/(m\({}^{2}\cdot\)K).