# Evaluation of Present Accelerated Temperature Testing and Modeling of Batteries
## 3 Case Studies
### Case Study 1  
In portable electronic devices that drain relatively low power and energy, primary battery is the first option in consideration of cost and safety. A device company, as one of our clients, has extensively used a type of primary button-shaped battery in their product. The cathode, anode and electrolyte are manganese dioxide (MnO\({}_{2}\)) mixed with graphite, zinc (Zn), and Potassium hydroxide (KOH), respectively. The nominal capacity and minimum open-circuit voltage are 150 mAh and 1.5 V, respectively.  
For this type of battery, the company wants to determine its storage life at normal temperatures. Several these batteries were stored at 23 \({}^{\circ}\)C and 60 \({}^{\circ}\)C, respectively. In the testing process, a certain proportion of batteries tested at 23 \({}^{\circ}\)C were taken out and discharged using a standard profile after every 365 days, whereas the same proportion of batteries tested at 60 \({}^{\circ}\)C were discharged using the same profile after every 20 days.  
The testing data is plotted in Figure 2, where every 20 days' storage time on the degradation curve at 60 \({}^{\circ}\)C is drawn as 1 year. The normalized discharge capacity (NDC) is calculated based on the specified nominal capacity. It is observed that at both temperatures, the slopes of discharge capacity curves are becoming less negative over time, suggesting a decreasing instantaneous degradation rate.  
This company then conducted data analysis using the Arrhenius model. The samples stored at 60 \({}^{\circ}\)C for 40 days showed 7% capacity loss, whereas the samples stored at 23 \({}^{\circ}\)C for 730 days showed 6% capacity loss. The degradation rates at 23 \({}^{\circ}\)C and 60 \({}^{\circ}\)C were then calculated, respectively, assuming a linear capacity decrease over time as in Equations (4)-(6). Accordingly, the AF of 21.3, and activation energy of 0.703 eV can be derived.  
\[k_{1}=6\%/730\ \mathrm{days}=0.0082\%/\mathrm{day} \tag{4}\]  
\[k_{2}=7\%/40\ \mathrm{days}=0.175\%/\mathrm{day} \tag{5}\]  
Figure 2: Discharge capacity over storage time at 23 \({}^{\circ}\)C and 60 \({}^{\circ}\)C.  
\[AF=\frac{k_{2}}{k_{1}}=e^{E_{a}/R\cdot(\frac{1}{T_{1}}-\frac{1}{T_{2}})}=21.3 \tag{6}\]  
This modeling process is found to be incorrect. Only the capacity losses after the first 2 years' storage at 23 \({}^{\circ}\)C and 40 days' storage at 60 \({}^{\circ}\)C were picked, respectively, to calculate the degradation rates, then extrapolated the activation energy. The remaining data is not used in this case, and similar ways of validating the Arrhenius model and extracting the model parameters are seen in [33]. However, when the capacity losses at 23 \({}^{\circ}\)C from the 3 rd year to the 5 th year, and 60 \({}^{\circ}\)C from the 60 th day to the 100 th day are chosen for calculation, the AF and activation energy are changed to 15.6 and 0.632 eV, respectively. Therefore, the activation energy can change over storage time. If the reliability testing is carried out at other temperatures, it is expected that the activation energy may differ from the previous calculated value.