## 3 Thermal Heat Modeling
Under the assumption that the heat generated by the skin metabolism is negligible compared to the effect of blood perfusion on the skin temperature distribution [52], the term \(Q_{m}\) in Equation (6) can be ignored. The corresponding initial and boundary conditions for Equation (6) are set as \(T(x,t)\mid_{t=0}=T_{0},\;T(x,t)\mid_{x=I_{e}}=T_{c}\), and \(-k\partial T/\partial x|_{x=0}=h(T-T_{e})\) with \(T_{0}\), \(T_{c}\), \(T_{e}\), \(L_{s}\), and \(h\) representing the initial skin temperature, the core skin temperature, the ambient temperature, the skin thickness, and the total heat transfer coefficient between the skin tissue and the environment, respectively. The core skin temperature can also be called deep tissue temperature, which refers to the subcutaneous tissue temperature. Our work referred to the skin temperature range in [53; 54], so 28.85 \({}^{\circ}\)C was selected as the initial skin temperature. Some of the values of these parameters used in the following calculations can be found in Table 1.  
In order to check the efficiency of the model by Equation (6), we estimate the temporal evolution of temperature due to the illumination of a 33.5 GHz channel onto a layer of skin, with the power density changing from 284 mW/cm\({}^{2}\) to 664 mW/cm\({}^{2}\). In Figure 3a, the variation of skin temperature versus illumination time is plotted. The temperature increases significantly for longer illumination time and higher power density and then comes to be a constant (equilibrium). The calculation agrees well with the measured data obtained by [PERSON] [47], who employed a layer of in vitro rat skin in the laboratory under the illumination of a continuous wave at 33.5 GHz and launched out from a lens antenna. The agreement between our calculations and the measured data indicates the efficacy of [PERSON] BHTE model in analyzing the temperature evolution of rat skin under electromagnetic radiation, provided that the total heat transfer coefficient (h) is determined as shown in Figure 3b. These results support the transferability of our findings to discussions on the thermal heating effect of human skin, despite its thickness being 5-6 times greater than that of rat skin. It is important to consider that as body temperature rises, mammals regulate their temperature through physiological activities, including increased heat dissipation, which results in an augmented heat transfer between the skin and the surrounding environment. Consequently, the heat transfer coefficient increases in response. Based on the curves shown in Figure 3b, which are calculated by fitting the measured data in Figure 3a, we can see that the coefficient \(h\) evolves with \(\Delta\)T along the same trend before stabilizing (\(h_{max}\)). This means there should be a correlation between the heat transfer coefficient and the temperature difference \(\Delta\)T (between the skin and ambient temperature), which has been demonstrated theoretically in reference [57]. To see the influence of a rainwater layer on human skin on a rainy day, we predicted the variation of thermal heating with/without a planar water layer by employing Equation (6). In the case of water layer coverage, we adopted the dielectric properties of wet skin. The heat transfer coefficient of rats in Figure 3b was used here because it has been proven to be close to that of human skin [58]. Calculations were completed using the parameters in Table 1 and setting the thickness of the planar water layer to 0.2 mm due to the relatively stable transmissivity over the whole frequency range as indicated in Figure 1d. The results are shown in Figure 3c. It is obvious that the temperature increases faster when there is a water layer. We attribute this to the insulation effect of the water layer, which has a much higher specific heat capacity than air [59]. It should also be noted that the temperature increases with respect to the operating frequency.