## 1 Introduction  
The gas-liquid-solid three-phase catalytic reaction is a common reaction type in chemical engineering. Many kinds of gas-liquid-solid three-phase reactors are employed in the industry, among which trickle bed reactors (TBRs) are widely used in fields such as petroleum, petrochemical, fine chemicals, and biochemical industries. They offer many advantages such as easy operation, high solid loading, and operation pressure [1]. In a catalyst packed bed, gas and liquid reactants can flow either concurrently downward or in opposite directions. The liquid phase on particles surface appears as a film or droplet. The phase interaction between gas, liquid, and solid in TBR makes the hydrodynamic phenomena quite complicated, which is usually investigated by experimental method [2; 3; 4]. When gas and liquid flow concurrently downward through a packed bed, four flow regimes were found. They are trickle flow, pulse flow, spray flow, and bubbly flow [5]. The characteristics of these flow regimes are different, and they affect the hydrodynamic parameters of TBRs significantly. Among these parameters,liquid holdup and wetting efficiency are of great importance, which give descriptions of the liquid distribution. Liquid holdup \(\epsilon_{L}\) measures the quantity of liquid inside TBRs:  
\[\epsilon_{L}=\frac{\text{volume of liquid}}{\text{volume of TBR}} \tag{1}\]  
Another description of liquid quantity inside TBRs is the liquid saturation:  
\[\beta_{L}=\frac{\text{volume of liquid}}{\text{volume of voids between particles inside TBR}} \tag{2}\]  
And the wetting efficiency is the percentage of wetting area of catalyst surface:  
\[\eta_{L}=\frac{\text{wetting area of particles}}{\text{particles surface area}} \tag{3}\]  
The overall liquid holdup and wetting efficiency are usually calculated by empirical formulas. However, local liquid distribution phenomenon on the particle scale such as flow maldistribution, channeling and catalyst wetting can also significantly affect the overall reactor performance (e.g., pressure drop, catalyst use, and reactant conversion). The rigorous mathematical models for calculating them are still tough [6]. The liquid flow textures in a bed consist of several features: liquid film flow, rivulets over the particles, pendular structures, liquid-filled channels, and liquid-filled pockets [7]. They have various particle surface liquid coverages and perform differently with different gas-liquid catalytic reactions (liquid-limited or gas-limited). Also, the maldistribution of liquid caused by variation of voidage can lead to the local hot spot formation, causing temperature runaway and operation safety problems. The flow phenomenon mentioned above is significantly affected by factors such as particle shape, wetting ability of liquid on the particle surface, which are not described clearly in traditional formulas. Since the local liquid distribution has a significant impact on reactor performance, better models are needed for its accurate prediction. They will significantly aid the predictions of local temperatures and local concentration distributions, facilitating better reactor design [8]. Therefore, it is very important to accurately estimate the local liquid distribution on the particle scale in TBRs for further design and operation. In recent years, computational fluid dynamics (CFD) has become a powerful method for investigating fluid behaviors inside TBRs and has attracted a lot of attention [9; 10; 5; 11]. To simulate multiphase flow inside TBRs with CFD, many multiphase models have been employed. The Eulerian-Eulerian model for gas-liquid-solid flows is the most commonly used CFD model to predict the dynamic behavior of TBRs [12]. However, the Eulerian-Eulerian approach does not accomplish interface tracking by solving a continuity equation of the volume fraction of one (or more) phases, and for this reason it is unable to capture the wetting characteristics of the gas-liquid interface in the TBRs [13].