## 4 Results and Discussion
### Capacity Estimation Models  
Based on the storage experiments, it can be known that capacity attenuates significantly during storage at high temperatures. Hence, it is necessary to estimate the capacity loss caused by self-discharge to accurately obtain the remaining capacity of underwater vehicles in the operating stage. For primary batteries, the electricity is generated due to the chemical reaction of lithium metal and the consumption of Li decays the battery capacity. Hence, the rate of capacity decay is closely related to the chemical reaction rate of lithium metal, as given below:  
\[Kt_{n}=Q_{0}-Q_{n,25} \tag{2}\]  
where \(K\) refers to the reaction rate of lithium metal, \(t_{n}\) represents the storage time (days), \(Q_{0}\) denotes the maximum available capacity of the new battery, and \(Q_{n,25}\) corresponds to the maximum available capacity of the battery stored for \(t_{n}\) days and discharged at 25 \({}^{\circ}\)C. According to the experimental results, the reaction rate of lithium metal at different temperatures can be derived, as listed in Table 3. Herein, the mean values of three samples are taken as the final experimental results in different cases. It can be known that the reaction rate remains constant in the temperature range of 0 \({}^{\circ}\)C to 25 \({}^{\circ}\)C, whereas the reaction rate exhibits a 20-fold increase from 25 \({}^{\circ}\)C to 54 \({}^{\circ}\)C. Finally, it is determined that the capacity decay model is developed only at \(>\)25 \({}^{\circ}\)C.  
It is generally considered that the effect of temperature on the rate of chemical reaction conforms to the Arrhenius equation. From the Section 2, we know that temperature renders a significant influence on the rate of the chemical reaction, and the Arrhenius equation further defines the relationship between temperature and chemical reaction rate. Hence, taking logarithms of both sides of Equation (1), the linear form of Arrhenius equation can be obtained:  
\[\ln K=\ln A-\frac{E_{\rm a}}{R}\cdot\frac{1}{T} \tag{3}\]  
Figure 8 can be drawn by incorporating the reaction rate constant \(K\) and corresponding temperature \(T\) into Equation (3). After fitting analysis, the relationship between \(\ln(K)\) and  
Figure 7: The capacity variation at different temperatures and current rates. (**a**) capacity vs. temperature at different current rates; (**b**) capacity vs. current rate at different temperatures. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(K_{0}\) **(Ah/d)** & \(K_{25}\) **(Ah/d)** & \(K_{45}\) **(Ah/d)** & \(K_{54}\) **(Ah/d)** \\ \hline \(2.7\times 10^{-3}\) & \(3.4\times 10^{-3}\) & \(44.4\times 10^{-3}\) & \(74.0\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The reaction rate of lithium metal at different temperatures. \(1/T\) is established, as given by Equation (4). Meanwhile, parameters ln\(A\) and \(E_{\text{a}}/R\) are found to be 32.81 and 11,515.24, respectively. \[\ln(K)=-11515.24\cdot\frac{1}{T}+32.81 \tag{4}\]  
By changing the form of Equation (3), the chemical reaction rate \(K\) can be expressed as:  
\[K=\exp(\frac{-11515.24}{T}+32.81) \tag{5}\]  
By substituting Equation (5) into Equation (1), the capacity decay model after \(t_{n}\) days of storage at temperature \(T\) can be developed as:  
\[Q_{n,25}=Q_{0}-t_{n}K=Q_{0}-t_{n}\exp(\frac{-11515.24}{T}+32.81) \tag{6}\]  
Furthermore, the error analysis is performed to assess the accuracy of abovementioned model. The mean square error (\(MSE\)) is the most important and common index to measure the deviation between predicted and actual values. The larger \(MSE\) corresponds to the lower model accuracy. The \(MSE\) can be given as:  
\[MSE=\frac{1}{n}\sum_{i=1}^{n}{(\hat{y}_{i}-y_{i})^{2}} \tag{7}\]  
where \(n\) refers to the number of samples, \(y_{i}\) represents the true value of the sample \(i\), and \(\hat{y_{i}}\) corresponds to the predicted value of sample \(i\).