## 3 Result and Discussion  
Firstly, we assumed the initial condition under which a molten liquid layer of 1 mm thickness is formed at \(t=0\) s around the ETDP by referring to previous literature [11]. Figure 3a-c shows the temperature for the case of \(T_{tip}=368\) K over time. The heat is diffused from the heated probe tip toward the ice region. The highest temperature can be found near the ETDP. The enlarged temperature and enthalpy fields are highlighted by a green box and are illustrated in Figure 3d,e, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Heating Power** & **Rate of Penetration (Experiment Result)** & **Rate of Penetration (Prediction Model Result)** & **Error** \\ \hline
240 W & 0.380 mm/s & 0.329 mm/s & 13.4\% \\
300 W & 0.458 mm/s & 0.411 mm/s & 10.2\% \\
360 W & 0.508 mm/s & 0.494 mm/s & 2.7\% \\
480 W & 0.55 mm/s & 0.65 mm/s & 18\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the predicted results of rate of penetration (ROP) with the experimental results [13]. Figure 3: The temperature distribution around the electro-thermal drilling probe (ETDP) for the case of heated probe tip temperature of \(T_{\text{tip}}=368\) K at time (**a**) \(t=0\) s, (**b**) \(t=30\) s, and (**c**) \(t=60\) s. The enlarged fields of (**d**) temperature and (**e**) enthalpy within the green dotted box. Interestingly, the ice that contacted with the heated probe tip remained at 273.15 K since it is the mushy zone (highlighted by a blue dotted box) where the phase change occurs. Although the mushy zone has a constant temperature of 273.15 K, as shown in Figure 3d, it is worth noting that enthalpy within the mushy zone presents a gradient toward the vertical direction, as shown in Figure 3e. This means that when the temperature of ice reaches 273.15 K (melting temperature), enthalpy begins to increase with a fixed temperature (\(T\) = 273.15 K) and the corresponding liquid fraction from Equation (8) also increases during the melting process. Figure 4a-c shows the liquid fraction fields for the case of a heated probe tip temperature, \(T_{tip}\), of 368 K over time. As mentioned above, the ice region, where the temperature reaches 273.15 K, becomes a mushy zone where the liquid fraction is between 0 and 1. The liquid fraction inside the green dotted box in Figure 4d. Not surprisingly, it shows very similar distribution with the enthalpy distribution shown in Figure 3e. The thin liquid layer is formed near the heated probe tip. This implies that ETDP can move in a downward direction by pushing and squeezing the liquid layer, as shown in Figure 4e. The ETDP slowly moves downward. The vertical positions of the heated probe tip, \(z\), are 0, 14.7 and 29.8 mm at \(t\) = 0, 30 and 60 s, respectively. We can calculate the ROP of ETDP from the expression of \(\text{ROP}=dz/dt\). As a result, the ROP of ETDP, \(dz/dt\), has a roughly constant value of 0.49 mm/s for the case of \(T_{tip}\) = 368 K.  
Figure 5 represents the penetration depth of ETDP, \(\Delta z\), for the different temperatures of the heated probe tip. The present study determines the penetration depth of ETDP, \(\Delta z(t)\), from the following expression, \(z(t)\)-\(z(0)\), as shown in the subset image of Figure 5. The evolution of the penetration depth of ETDP shows a linear tendency with time for all cases of \(T_{tip}\). The ROP are 0.38, 0.43 and 0.49 mm/s for the different \(T_{tip}\) of 348, 358 and 368 K, respectively. According to Equation (10), the ROP of ETDP, \(v\), is proportional with the heating power, \(P\), where \(P\) is the \(\int_{0}^{R}q^{\prime}(r)\pi r^{2}dr\) and \(q^{\prime}(r)\) indicates the local heat flux at the heated probe tip. Therefore, the fact that ROP has a constant value over time can be interpreted as that the heating power of ETDP also maintains a constant value during the drilling process. The ROP, \(v\), increases with the temperature of the heated probe tip, \(T_{tip}\), due to the increase of heating power, \(P\).