## 4 Results and Discussion
### Parameter Optimization of the PDE-Based Model  
The parameterization of the computational model is one of the significant challenges. It is also beneficial to find kinetic and transport parameters from the experiment charge/discharge data. Such techniques are typically formulated to minimize objective function (OF) given by the sum-of-squares differences between the model outputs and their experimentally measured values for each _ith_ cycle. The output of this particle-based model is measured in terms of the total charge in the flow channels, while the current measured from the FE charging in the EFC cell at constant voltage is used to estimate the total charge in charged FEs. The OF is given as,  
\[\text{OF}=min_{\theta i}\sum_{j=1}^{n_{i}}[y_{i}(t_{i})-y_{model,i}(t_{i};\theta _{i})]^{2} \tag{10}\]  
where \(y_{i}(t_{i})\) is the measured charge at the constant voltage cycle in time \(t_{i}\) for cycle \(i\), \(y_{model,i}(t_{i};\theta_{i})\) is the charge computed from the particle-based model at time \(t_{i}\) for cycle \(i\) for the vector of model parameters \(\theta_{i}\)(the parameters estimated from the experimental data), and \(n_{i}\) is the number of time points in cycle \(i\). Here, the goal is to evaluate the capability of COMSOL and its interfacing module to determine these parameters [57]. Since the  
Figure 6: Loop diagram of the numerical solution process. total charge in this model is dependent on the SP size \(D_{sp}\) from Equation (8), mass and concentration of SP are also connected by Equation (5); thus, the optimization requires the modeling parameters such as SP size \(D_{sp}\) and mass \(m_{sp}\), the actual number of SPs \(actual_{NSP}\), and the average number of positive ions per carbon particle \(\beta_{Na}\). The optimization step requires a set of global variables for gradient-based and gradient-free optimization on an existing COMSOL model. BOBYQA is a gradient-free algorithm used to optimize the model to determine input values to yield an output that matches the experimental data [58]. The optimization was performed individually for selective parameters, starting with the size and mass of an SP. The actual numbers of SPs and \(Na^{+}\) ions per carbon particle were estimated using an SP's optimized mass and size. The scales, upper bounds, and lower bounds for control variables and parameters are given in Table 2.  
To assess the suitability of the above modeling approach to the actual systems, the numerical results of the total charge in the model were compared with the corresponding potentiostatic EFC charging in the control voltage scenario. The total charge was obtained by integrating potentiostatic EFC current under the constant voltages of 0.75, 0.50, and 0.25 V, respectively. The optimization yielded values for the SP diameter, SP mass, number of SPs used in the flow channel, and average number of \(Na^{+}\) ions for a single carbon particle for the best match, as listed in Table 2 under the heading labeled \"Exact\". An optimization plot of charge in the experiment vs. time resulting from the potentiostatic charging of the EFC is shown in Figure 7. The error analysis is based on a comparison of the experimental measurements and the model's predictions--a comparison of them showed excellent agreement. The maximum difference between the charge obtained by the model and experiment was \(1.406\times 10^{-6}\) C. We define the error measure \(\Delta\)C as  
\[\Delta\text{C }=\frac{(\text{simulation point}-\text{experimental point})}{(\text{simulation point}+\text{experimental point})/2}\times 100\ \%\Delta\text{C}=8.3\ \%\]