## 2 Model Development
### Latent Heat Loss due to Sweating during Walking/logging  
The evaporation of sweat is an effective cooling mechanism for thermal regulation. Neither the sweating rate, nor the evaporation rate is uniform over the skin surface. It will be shown later that each part of foot sole has a different sweating evaporation. Cooling due to sweating is more considerable in the arc zone and is negligible in the ball of the foot zone. In the absence of more precise equations, the following equation is used to simulate the sweating-cooling effect:  
\[q^{{}^{\prime\prime}}_{sw}=h_{fg}h_{e}\times 4.7\times 10^{-5}\times(P_{sat}-P) \Big{(}T_{f}-T_{f0}\Big{)} \tag{16}\]  
Equation (16) was adopted from [35]. \(h_{fg}=2.43\times 10^{6}\) J/kg is the water latent heat [36]. According to Equation (16), the sweating rate increases by increasing the temperature elevation \(\Big{(}T_{f}-T_{f0}\Big{)}\). In the above equation, \(P\) is the vapour pressure in vicinity of the ball  
Figure 4: Thermal conditions for 10 min rest in the experiment. zone (the area that sweating is considerable). \(T_{f}\) is the foot sole skin temperature and \(P_{sat}\) is the saturate vapour pressure at \(T_{f}\). It was assumed that the vapour pressure at the skin is at the saturated pressure [37; 38]. So, when the vapor pressure reaches the saturated pressure, evaporation stops. \(P_{sat}\) can be estimated by the Antoine equation [36]:  
\[P_{sat}=100\ \text{exp}\left(18.956-\frac{4030.18}{235+T_{f}}\right) \tag{17}\]  
However, the vapor pressure can be defined based on relative humidity, \(RH\):  
\[P=RH\cdot P_{sat} \tag{18}\]  
and Equation (16) can be represented as:  
\[q_{sw}^{n}=h_{fg}h_{e}\times 4.7\times 10^{-5}\times P_{sat}(1-RH)\Big{(}T_{f}- T_{f0}\Big{)} \tag{19}\]  
\(h_{e}\) is the evaporative heat transfer coefficient, which is proportional to the square root of velocity [39]:  
\[h_{e}=0.000192\times LR\times V^{0.5} \tag{20}\]  
where \(LR\) is the Lewis ratio and approximately equals \(16.5\times 10^{-3}\) K/Pa. **3. Heat Transfer Modeling of a Shod Foot**  
The results of the current simulation were compared against the results from the literature [3]. The experiment was comprised of two stages [3]. In that experiment, first, participants wore the shoe with no socks, and after 10 min of rest, they did a jogging exercise for 30 min [3]. Foot skin temperatures were measured at various points every minute by thermocouples [3]. Participants were seven healthy adult males (1.72 \(\pm\) 0.07 m height, 61.8 \(\pm\) 4.7 kg weight, and 23.8 \(\pm\) 4.3 years old) [3]. The footwear was a running shoe with a woven upper (Figure 5) [3]. The footsole inside length is \(L\) = 25 cm (1 cm less than the shoe outside length due to shoe thickness) [3]. The experiments were performed in a climate chamber with a constant air temperature of 25 \({}^{\circ}\)C, 58.5% \(RH\), no solar radiation, and wind for three different speeds, i.e., 3, 6, and 9 Km/h [3]. More details can be followed in [3]. The shoe sole was made-up of EVA08, and its properties are presented in Table 1: \begin{table}
\begin{tabular}{c c c c c c} \hline \(c_{ps}\) & \(k_{s}\) & \(p_{s}\) & \(E^{n}\) & \(E\) & \(E\) \\ (J\(\cdot\)kg\({}^{-1}\).K\({}^{-1}\)) & (W\(\cdot\)m\({}^{-1}\).K\({}^{-1}\)) & (kg\(\cdot\)m\({}^{-3}\)) & (Pa) & (Pa) & **Hardness** \\ \hline
2585 & 0.11 & 240 & \(1.55\times 10^{6}\) & \(12.6\times 10^{6}\) & A45 \\ \hline \end{tabular}
\end{table} 
Table 1: EVA08 thermal–mechanical properties based on what was reported in [23]. Figure 5: The running shoe used for the experiment (adapted from [3]). The equations were solved numerically using the Crank-Nicolson scheme. Since the set of equations is nonlinear, they had to be solved iteratively. To validate the numerical method, the problem described in [28] was solved numerically and compared with the proposed analytical solution in [28]. The maximum considered difference was less than 0.05%.