## 1 Introduction  
In recent years, the energy shortage has become a serious problem due to the depletion of fossil fuels [1]. The use of renewable energy is one effective way to address the energy shortage [2]. The development and utilization of renewable energy sources such as wind and solar power require efficient energy storage techniques. Battery energy storage systems are a crucial part of the new power system architecture of the future, allowing the power system to remain resilient and stable under a variety of control methods [3, 4]. Many new challenges continue to emerge regarding battery chemistry [5]. Metal-air batteries have gained increasing attention due to their high energy density [6, 7, 8]. Metal-air batteries mainly include magnesium-air batteries, lithium-air batteries, sodium-air batteries, and aluminum-air batteries [9, 10]. Among them, magnesium-air and lithium-air batteries have theoretical energy densities of up to 6800 Wh/kg [11] and 11,400 Wh/kg [12], respectively. Magnesium-air batteries use magnesium alloys as anode materials and rely on the reduction reaction of oxygen at the gas diffusion electrode to provide a cathodic current. The battery reaction principles are as follows:  
\[\mathrm{Mg}\rightarrow\mathrm{Mg}^{2+}+2\mathrm{e}^{-} \tag{1}\]  
\[\mathrm{O}_{2}+2\mathrm{H}_{2}\mathrm{O}+4\mathrm{e}^{-}\to 4\mathrm{OH}^{-} \tag{2}\]  
In the research of magnesium-air batteries, there is a type called the magnesium-dissolved oxygen seawater battery [13, 14]. This type of battery can directly use seawater as the electrolyte, and therefore, it has several advantages: (i) the battery has a simple and open structure; (ii) the flowing seawater as the electrolyte can alleviate electrode polarization,improve electrode efficiency and battery stability; (iii) the absence of traditional electrolyte reduces the weight of seawater batteries and avoids a series of safety issues caused by carrying liquid electrolytes. Previous research found that the mass transfer in the cathode largely affected the battery performance [15; 16; 17; 18; 19; 20]. For magnesium-dissolved oxygen seawater batteries, the source of oxygen in the cathode reaction is different from that of conventional metal-air batteries. In conventional metal-air batteries, the required oxygen comes from the air, while in seawater batteries, the majority of the required oxygen comes from dissolved oxygen in seawater. Therefore, the oxygen mass transfer on the cathode significantly affects the performance of seawater batteries. This explains why the cathodes of seawater batteries generally adopt a porous structure to obtain as much oxygen as possible. Moreover, the cathode porous structure affects oxygen mass transfer and further affects the battery's performance. Therefore, researchers have studied the influence of porous cathodes on the performance of metal-air batteries through experiments and simulations. Those studies mainly focus on parameters such as pore size [21; 22; 23; 24; 25], surface area [26; 27; 28; 29], pore volume [30; 31; 32; 33], and cathode wettability [34; 35; 36; 37]. Usually, the porous structure of the electrode is determined by the carbon materials and the binder utilized in the electrode. [PERSON] et al. [24] conducted research on carbon electrode materials in lithium-oxygen batteries and found that the discharge capacity of the battery was not directly related to the surface area and pore volume of the carbon material. Instead, the discharge capacity is proportional to the pore size within a certain range. To verify this conclusion, the authors used phenolic resin as the carbon precursor and prepared mesoporous and macroporous carbon materials with pore sizes ranging from 20 to 100 nm using silica gel particles with different particle sizes as templates. The results showed that the discharge capacity of the electrodes made from these self-made porous carbons increased with increasing pore size within the range of 20-80 nm, reaching the maximum discharge capacity at a pore size of 80 nm.