## 2 Capacitive Deionization (CDI)
### Standard Analysis Metrics  
The desalination performance of CDI is often determined by standard analysis metrics such as salt adsorption capacity (SAC), salt adsorption rate (SAR), charge efficiency, and energy consumption (Table 1) [1; 2]. SAC represents the amount of removed ions divided by the total mass of electrode materials during the charging process (Equation (1)). In particular, the maximum salt adsorption capacity (mSAC, SAC obtained at the equilibrium state) is useful to explore the sorption capacity without the influence of system parameters. Note that the SAC is affected by the operational conditions [1]. SAR expresses the ratio of removed ions to the operation time (Equation (2)) to show the rate capability of the system. Since the SAR is influenced by various operational parameters, including cell architecture, electrode thickness, porosity, and pore structures, it is a useful parameter to optimize the system [38]. Then, the correlation between SAC and SAR can be presented in the CDI Ragone, allowing systematic understandings and comparisons among several types of CDI systems [39].  
\[\text{SAC (mg/g) }=\ \frac{M_{NaCl}\cdot\int\ (C_{0}\ -\ C_{t})\cdot\
u\cdot dt}{m} \tag{1}\]  
\[\text{SAR (mg/g/s) }=\ \frac{M_{NaCl}\cdot\int\ (C_{0}\ -\ C_{t})\cdot\
u \cdot dt}{m_{wk}\cdot t} \tag{2}\]  
where \(C_{0}\) and \(C_{t}\) are the initial and effluent concentrations (mM) which were converted from the measured conductivities, \(\
u\) is the volumetric flow rate (L/min), \(M_{NaCl}\) is the molecular weight of _NaCl_ (g/mol), m is the total mass of the carbon electrodes (g), \(m_{wk}\) is the total mass of the working electrodes (g), and \(t\) is the operation time for the charging step or the total desalination cycle(s).  
Charge efficiency and energy consumption are the parameters to evaluate how efficiently CDI is operated (Equations (3) and (4)). Charge efficiency is derived by the ratio of the removed ions  
Figure 1: Schematic illustration of basic cell configurations and operations in capacitive deionization (CDI).  
to the invested electric charge, while the energy consumption represents the energy required to remove a single ion [1,2]. Since the charge efficiency shows the degree of electric charge utilized to remove the ions, it is a major index to determine the presence of parasite reactions as well as energy consumption [40]. The energy consumption in kT is the thermodynamic energy consumption based on the specific Gibbs free energy of separation, offering the scientific evidence to compare the energy efficiency of CDI systems with that of other desalination technologies [41].  
\[\text{Charge efficiency}\ =\ \frac{SAC\cdot\text{m}}{M_{Nucl}}/(\frac{\int\ Idt}{F}) \tag{3}\]  
\[\text{Energy consumption}\ (kT)\ =\ \frac{V}{n}\int\ Idt\ (\text{at CV mode}),\ \frac{I}{n}\int\ Vidt\ (\text{at CC mode}) \tag{4}\]  
where \(F\) is the Faradaic constant (96,485 C/mol), \(I\) is the current (A) during the charging step, \(V\) is the voltage (V), and \(n\) is the total number of ions desalinated during the operation. 1 kT is equivalent to \(2.48\times 10^{3}\) J/mol for the case of removing NaCl. Note that CV stands for the constant voltage operation, while CC stands for the constant current operation.