**1. Introduction**

The rapid economic development has led to increasingly serious conflict between resources and the environment, which forced the world to search for sustainable and green energy to substitute traditional energy [1–3]. One promising and reliable opinion is to extract energy from the salinity difference between seawater and freshwater. It is based firmly on the fact that when two solutions with different salinity are mixed, the entropy of the system will increase, which can be captured and converted into electrical energy [4]. Theoretically, about 2.5 MJ of free energy could be generated by controlled mixing of 1 m3 river water with a large amount of seawater. The global potential for energy extraction from the world coast would, then, reach 2 TW of power, which satisfies around 20% of the world's energy demand [5,6].

By considering the tremendous amount of energy available from salinity difference, some techniques were proposed to harvest this energy. The pressure-retarded osmosis (PRO) [7,8] and reverse electrodialysis (RED) [9] are two advanced techniques and have been demonstrated at a pilot scale [7,10]. In PRO, the seawater and freshwater are separated by a semi-permeable membrane, which drives water from the freshwater to permeate

**Citation:** Zou, Z.; Liu, L.; Meng, S.; Bian, X.; Li, Y. Applicability of Different Double-Layer Models for the Performance Assessment of the Capacitive Energy Extraction Based on Double Layer Expansion (CDLE) Technique. *Energies* **2021**, *14*, 5828. https://doi.org/10.3390/en14185828

Academic Editor: Marcin Kami ´nski

Received: 16 August 2021 Accepted: 12 September 2021 Published: 15 September 2021

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into the more concentrated seawater due to the difference in osmotic pressure. The expanding volume of the seawater can be depressurized through a hydroturbine to generate electricity. In RED, the solutions with different salt concentrations flow alternately in compartments, which are separated by a stack of positively and negatively charged ionexchange membrane. The ion diffusion across the membranes generates a current that can be extracted [10,11]. Up to now, the highest reported power density for PRO and RED are around 10 and 1 W/m2, respectively [12]. Although significant progress has been made in both PRO and RED, the drawbacks of these techniques are also obvious, including the high cost and short lifetime of membranes, as well as the use of additional converters (hydroturbine in PRO) for effectively producing electricity. These drawbacks hampered the commercialization of both PRO and RED techniques and drive researchers to develop new technologies, such as Capacitive mixing (CapMix).

The so-called CapMix is an innovative technology that was recently introduced for extracting energy from salinity differences. It is the common name for several electrodebased technologies [13–16], including Capacitive energy extraction based on Double Layer Expansion (CDLE), Capacitive energy extraction based on Donnan Potential (CDP) and Soft Electrode technique (SE). In CDLE, the porous electrodes are first charged in salty water by an external power source and then discharged in fresh water; this process leads to an electrical double layer (EDL) expansion and results in an increase in electric potential. In CDP, the porous electrodes are covered by ion-exchange membranes, which only allow anion or cation to pass through and thus generates a Donnan potential difference across the membrane. In the SE technique, instead of using membranes in CDP, the electrodes are made of an activated carbon core together with a polyelectrolyte layer, either cationic or anionic. The major disadvantage of CapMix is the intermittent power production as well as the low power density, and the largest reported power density for CDLE, CDP and SE are 35, 105 and 50 mW/m2, respectively.

Among the techniques of the CapMix family, CDLE is the earliest technique that was first proposed and implemented experimentally by Brogioli [13] in 2009. It is the simplest one in terms of structure, composed of two electrodes that are parallel to each other and a spacer serving as a channel for water flowing through the cell without the use of ion-exchange membranes. The performance of the CDLE technique is dependent on the properties of the electrodes, cell structure as well as operation method. In the study of material properties, Iglesias et al. [17] investigated the effect of carbon wettability and pore size distribution on the performance of CDLE and found that electrodes with hydrophilized material improve the energy production of CDLE. They also concluded that activated carbon with a predominant pore population in the 1 nm region gives an optimum result. In another study about the effects of pore sizes of the porous electrode, Nasir et al. [18] also suggested that the optimum average pore diameter of electrodes for CDLE is about 1 nm. Furthermore, Iglesias et al. [19] investigated the possibility of stacking individual CDLE cells in series to increase energy production. They found that multiple cells in series might increase the potential rise and that such an increase is limited and cannot compensate for the increase in internal resistance. The influence of the operation conditions, such as the flow velocity and the solution temperature, on the performance of CDLE, was also investigated. It was reported that a higher flow rate might lead to an improvement in the power production of CDLE [18] and that by controlled mixing solutions with different temperatures, the potential rise can be maximized, and thus the energy production can be increased [20].

By following experimental works, theoretical studies were also conducted over the years to provide a platform for identifying the influences of different parameters on the CDLE process. The theoretical models in CDLE focus mainly on the description of thermodynamic properties of EDL as well as the transport of ions inside the porous electrode. Among different equilibrium models, the Gouy–Chapman–Stern (GCS) model has been widely used to simulate the thermodynamic CDLE cycle [18,21–23]. This model is simple but does not account for the effects of EDL overlap and the finite size of the ions. To remedy this problem, Jiménez et al. [24] developed a modified Possion–Boltzmann–Stern (MPBS) model and applied it to predict the maximum energy production of CDLE. It was also extended to consider the influences of multi-ionic solutions and cylindrical pores, suggesting that the presence of multivalent ions would reduce the net energy gain in a CDLE cycle [25].

The model for the description of EDL at equilibrium alone cannot be applied to describe the dynamic behavior of CDLE cells. It should, in principle, be coupled with ionic and current transport models to give a fully quantitative description of the complex mechanisms affecting the performance of the CDLE as performed by Rica et al. [26,27]. The dynamic model originally proposed by Rica et al. [27] was based on a 1D theory that was developed by Biesheuvel and Bazant [28] with the use of the GCS model. It was later found that by using a modified Donnan (mD) model instead, the kinetics of ionic transport and adsorption in the CDLE could be better described in the cases where the EDLs are overlapped within the micropores of the electrodes.

For whatever purpose, an accurate description of the structural and the thermodynamic properties of the EDL at equilibrium is essential in understanding the behavior of CDLE cells. In the literature, however, the use of different EDL models in the study of CDLE is somehow arbitrary without giving a detailed discussion on the applicability of the models, especially the rationality and the physical interpretation of the relevant parameters contained in the models, even though these models have been successfully applied in many works [19,22,23]. However, the physical meaning behind the parameters is important for better understand the thermodynamic properties of the EDLs. Mainly, for this reason, we strive to highlight the physical differences between some commonly used EDL models and then evaluate the applicability of different EDL models for the performance assessment of the CDLE technique.

The remainder of this contribution is organized as follows. In the next section, a brief description is provided of the experiment setup and the operation scheme. Then, a detailed elaboration of different EDL models is made in Section 3. The comparison between the experimental results with the simulations of different EDL models is then presented in Section 4, followed by physical interpretations of the parameters used in the models and a discussion on the models' applicability. The contribution ends with concluding remarks.
