Computational Investigation of Liquid Holdup and Wetting Efficiency Inside Trickle Bed Reactors with Different Catalyst Particle Shapes

Abstract

Liquid holdup and wetting efficiency are essential parameters for design of trickle bed reactors. Both parameters play an important role in reactor performance including pressure drop, conversion, and heat transfer. Empirical formulas are usually employed to calculate liquid holdup and wetting efficiency. However, factors such as particle shape and the wetting ability of liquid on the particle surface are not described clearly in traditional formulas. In this paper, actual random packing was built by DEM and CFD simulations were performed to investigate the factors affecting liquid holdup and wetting efficiency in trickle bed reactors, including particle shape, surface tension, contact angle, liquid viscosity, liquid density, liquid, and gas superficial velocity. Detailed fluid flow and liquid-solid interaction were described by VOF model. Four different particle shapes were investigated. It showed the particle shape has great effect and the 4-hole cylinder packing gained both highest liquid holdup and wetting efficiency. The overall simulations gave a detailed description of phase interactions and fluid flow in the voids between catalyst particles and these results could give further guidance for the design and operation of trickle bed reactors.

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 ε_L measures the quantity of liquid inside TBRs:

$${\epsilon }_L=\frac{\mathrm{volume}\mathrm{of}\mathrm{liquid}}{\mathrm{volume}\mathrm{of}\mathrm{TBR}}$$

(1)

Another description of liquid quantity inside TBRs is the liquid saturation:

$${\beta }_L=\frac{\mathrm{volume}\mathrm{of}\mathrm{liquid}}{\mathrm{volume}\mathrm{of}\mathrm{voids}\mathrm{between}\mathrm{particles}\mathrm{inside}\mathrm{TBR}}$$

(2)

And the wetting efficiency is the percentage of wetting area of catalyst surface:

$${\eta }_L=\frac{\mathrm{wetting}\mathrm{area}\mathrm{of}\mathrm{particles}}{\mathrm{particles}\mathrm{surface}\mathrm{area}}$$

(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 [5,9,10,11]. 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]. This drawback is more significant when we perform simulations with random packings, since a random packing has lots of tiny voids where surface tension has a great effect. Volume of fluid (VOF) model is one of the best implicit free-surface reconstruction methods. The VOF model is a surface-tracking technique applied to a fixed Eulerian mesh when the locus of the interface between two or more immiscible fluids is of interest. The main hurdles of using VOF rest on preserving a sharp boundary between the immiscible fluids and the computation of surface tension [14]. The velocity field and bubble profile in a vertical gas-liquid slug flow inside the capillaries has been modeled with a VOF technique, and the result was found to be in good agreement with published experimental measurements [15]. In the past two decades, the VOF model has been predominately practiced in the field of CFD simulation of TBRs [6,13,16,17,18,19,20].

Many researchers are devoted to conducting simulation research about TBRs. Jindal et al. performed Eulerian multi-fluid simulations to study the effect of bed characteristics of local liquid spreading in a TBR [10]. Lu et al. developed a new porous media model for CFD simulations of the gas-liquid two-phase flow in rotating packed beds [11]. However, these papers either modeled solid particles as a continuum phase or regarded the whole simulation region as a porous media. They only calculated the fluid-solid interactions through specific methods, without considering the influence of the actual particle shape on the fluid flow. Several authors made efforts to perform particle-resolved TBR simulations. Lopes et al. proposed a high-resolution CFD approach using discrete particles and designed a TBR with regular positioned catalyst particles for the Euler-Euler [21] and Eulerian VOF simulations [13], and studied the catalytic wet oxidation of phenolic wastewaters in their later studies [22,23]. Du et al. performed CFD simulations on regular particles arrangement within a rectangular domain to investigate the local wetting behavior in a TBR [6,16]. Simulations performed by both groups modeled particles with regular configuration and fixed distance between each other, which is difficult to achieve in a real reactor where catalyst particles are packed randomly, and the particle’s shape can be complex such as a cylinder or hollow cylinder. Besides, their grid resolution are not high enough to give clear descriptions of the liquid flow configuration on the discrete particle surfaces.

In this paper, detailed packing geometries with different particle shapes by discrete element method (DEM). In addition, the effects of factors such as surface tension and contact angle are discussed, which are hardly shown in empirical formulas. The CFD simulation was employed with the volume of fluid (VOF) method to investigate the factors affecting liquid holdup and wetting efficiency in a trickle bed reactor. This research can give guidance in trickle bed reactor design and operation in the future.

2. Models and Methods

2.1. Governing Equations

2.1.1. DEM Model

To construct the actual geometry of a random packed bed, discrete element method (DEM) was employed to generate random packing recently. The DEM and CFD methods were developed and show better performance in many applications, especially in fixed bed reactor simulations. Eppinger et al. introduced the DEM-CFD meshing method for fixed beds consisting of monodisperse spherical particles [24]. Wehinger et al. combined particle packing structure with detailed reaction mechanisms of catalytic dry reforming of methane over rhodium and studied the interaction between chemical kinetics and transport of momentum, heat, and mass [25]. Zhang et al. studied the flow and heat transfer in fixed beds considering the packing structures of cylindrical particles and low tube to particle diameter ratios [26].

In this research, the DEM method was used to generate actual geometry of random packings with different particle shapes. The distinct characteristic of DEM is that inter-particle contact forces are included in the equations of motion. The size, shape, and orientation of each particle are taken into account. The governing equation describing particle motion is:

$$m_p\frac{d{v}_p}{dt}=F_s+F_b$$

(4)

where m_p mass of particle, v_p is particle velocity, t is time, F_s and F_b is surface and volume force respectively. F_b could be calculated by:

$$F_b=F_g+F_c$$

(5)

where F_g is gravity, F_c is the interaction force between particles and between particles and wall:

$$F_g=m_{p}g$$

(6)

$$F_c={\displaystyle \sum _{contacts}{F}_{cm}}$$

(7)

F_cm is contact force which could be calculated as:

$$F_{cm}=F_n+F_t$$

(8)

where F_n is the normal and F_t is the tangential force component:

$$F_n=-K_n{d}_n-N_n{v}_n$$

(9)

If $\left|K_t{d}_t\right|<\left|K_n{d}_n\right|C_{fs}$, F_t is defined as:

$$F_t=-K_t{d}_t-N_t{v}_t$$

(10)

Otherwise:

$$F_t=\frac{\left|K_n{d}_n\right|C_{fs}{d}_t}{\left|d_t\right|}$$

(11)

where K_n and K_t are the normal and tangential spring constant, N_n and N_t are the normal and tangential damping, d_n and d_t are overlaps in the normal and tangential directions at the contact point respectively, C_fs is the static friction coefficient. N_n and N_t could be calculated as follows:

$$N_n=2N_n{}_{{}_{}damp}\sqrt{K_n{M}_{eq}}$$

(12)

$$N_t=2N_t{}_{{}_{}damp}\sqrt{K_t{M}_{eq}}$$

(13)

N_{n damp} and N_{t damp} are normal and tangential damping coefficient respectively, M_eq is the equivalent particle mass. These two damping coefficient could be defined as follows:

$$N_n{}_{{}_{}damp}=\frac{-\ln \left(C_n{{}_{}}_{rest}\right)}{\sqrt{{\pi }^2+\ln {\left(C_n{{}_{}}_{rest}\right)}^2}}$$

(14)

$$N_t{}_{{}_{}damp}=\frac{-\ln \left(C_t{{}_{}}_{rest}\right)}{\sqrt{{\pi }^2+\ln {\left(C_t{{}_{}}_{rest}\right)}^2}}$$

(15)

where C_{n rest} and C_{t rest} are the normal and tangential coefficients of restitution respectively. The rolling of particles (especially non-sphere particles) has great impact on the motion of particles. The rotational momentum equation is given as:

$$I_p\frac{d{\omega }_p}{dt}=M_b+M_c$$

(16)

where I_p is the particle moment of inertia, ω_p is particle angular velocity, M_b is the moment that acts on the particle (rotational drag), M_c is the moment that acts on an individual particle due to contact force, which in turn acts on the particle at a point other than the particle center of gravity:

$$M_c={\displaystyle \sum _{contacts}\left(r_c\times F_{cm}+M_{cm}\right)}$$

(17)

where r_c is the vector from the particle center of gravity to the contact point and M_cm is the moment that act on the particle from contact models (rolling resistance).

2.1.2. VOF Model

The VOF model assumes that all immiscible fluid phases present in a control volume share their velocity, pressure, and temperature fields. Therefore, the same set of basic governing equations describing momentum, mass, and energy transport in a single-phase flow is solved. It is necessary to know the volume fraction α_i of phase i (liquid or gas), and the conservation equation that describes the transport of α_i is:

$$\frac{\partial ({\alpha }_i{\rho }_{\mathrm{i}})}{\partial t}+\mathsf{\nabla }\cdot \left({\alpha }_i{\rho }_{\mathrm{i}}v\right)=0$$

(18)

where ρ_i is the density, t is time and v is the velocity. The momentum equation is given as:

$$\frac{\partial (\rho v)}{\partial t}+\mathsf{\nabla }\cdot \left(\rho v\otimes v\right)=-\mathsf{\nabla }p+\mathsf{\nabla }\cdot \left[\mu \left(\mathsf{\nabla }v+\mathsf{\nabla }{v}^{\mathrm{T}}\right)\right]+\rho g+F$$

(19)

where p is pressure, g is gravity and F is the resultant of other body forces. The physical properties are calculated as functions of the physical properties of its constituent phases and their volume fractions:

$$\rho ={\displaystyle {\sum }_i{\rho }_i}{\alpha }_i$$

(20)

$$\mu ={\displaystyle {\sum }_i{\mu }_i}{\alpha }_i$$

(21)

The surface tension force is a tensile force tangential to the interface separating the two fluids. It works to keep the fluid molecules at the free surface in contact with the rest of the fluid and shows remarkable effects on liquid behaviors in TBRs. The surface tension force f_σ calculation method named continuum surface force (CSF) model was proposed by Brackbill et al. as the following [27]:

$$f_{\sigma }=\sigma \kappa n$$

(22)

σ is the surface tension coefficient and the vector normal to the interface n is calculated using the smooth field of the phase volume fraction α_i:

$$n=\mathsf{\nabla }{\alpha }_i$$

(23)

The curvature of the interface κ can therefore be expressed in terms of the divergence of the unit normal vector as follows:

$$\kappa =-\mathsf{\nabla }\cdot \frac{\mathsf{\nabla }{\alpha }_i}{\left|\mathsf{\nabla }{\alpha }_i\right|}$$

(24)

Wall adhesion was considered by adjusting normal vector with contact angle:

$$n=n_{wall}\cos \theta +n_t\sin \theta$$

(25)

where θ is the contact angle, n_wall is the unit normal vector of solid wall, and n_t is the unit vector inside the wall normal to the solid-liquid-gas contact line.

2.2. Numerical Setup

Two parts of simulations were performed in this research. To verify the accuracy of the simulation framework, a simulation of a droplet falling on a sphere particle was carried out according to Du et al.’s research [26], and the outcome was compared with their results of both experimental and numerical simulation. Then, a packed bed structure was generated by DEM method and factors affecting the liquid distribution were investigated in the complex region. The properties of gas and liquid employed in verify simulations and CFD simulations referred to air and water under standard conditions and are given in

Table 1. Gas and liquid properties employed in both verify and DEM-CFD simulations.

Gas and Liquid Properties	Value
Liquid viscosity μL (Pa·s)	8.8871 × 10−4
Gas viscosity μG (Pa·s)	1.85508 × 10−5
Liquid density ρL (kg/m3)	1000
Gas density ρG (kg/m3)	1.225
Surface tension coefficient σ (N/m)	0.072
Contact angle θ	45°

.

All simulations mentioned in this research were conducted with polyhedral mesher. The reason to choose it is that polyhedral cells are relatively easy and efficient to build, requiring no more surface preparation than the equivalent tetrahedral mesh. Besides, they also contain approximately five times fewer cells than a tetrahedral mesh for a given starting surface. It provides great advantages to the simulation regions, since it contains extremely complex geometry and requires large number of cells to describe the shape of the random packing. It will help us to build cells with high quality and speed up the burdensome simulations. All these simulations were implemented with the commercial software STAR-CCM+ 12.02 [28].

2.2.1. Validation Model

Du conducted the experiment and performed the 2D VOF simulation which received satisfactory flow patterns comparable with the experiment results [19]. However, the 2D simulation was closer to a situation where a cylindrical drop of infinite length falls on an infinite-high cylinder, which is far from reality. Therefore, to better verify the accuracy of this method, the 3D simulations were performed.

The mesh plot of 3D verification model and measuring positions are shown in Figure 1. The simulation scales are the same as those in Du’s experiments [19]. The validation experiment was conducted under atmospheric pressure and at a temperature of 273 K. A particle of 10 mm in diameter was positioned in a cylinder region whose diameter and length are 20 mm and 25 mm respectively. A water droplet of 4 mm in diameter was patched 5 mm above the particle and slightly offset from the central axis (about 2.82 mm). For the region within 17 mm of the particle and droplet surface, meshes with the highest resolution were used (0.07 mm for 3D cases). For the other region, the largest-scale nonstructural meshes were adopted. The top entrance was defined as the velocity inlet for the gas phase and the exit was defined as pressure outlet conditions. The time step used in the simulation was 1 × 10⁻⁵ s, and the initial velocity for the liquid droplet was set to 0 so that the droplet fell down only by gravity. The simulated results were recorded every 5 ms, and the liquid film thickness was measured at different positions of the particle surface at different time intervals.

2.2.2. CFD Simulation

The mesh of the simulated TBR model is shown in Figure 2. The 6 mm × 6 mm × 9 mm TBR model was defined as the simulation region. Four different particle shapes were considered in this research, including sphere, cylinder, single-hole cylinder, and 4-hole cylinder.

Table 2. Geometric size of four kinds of particles.

Shape	Characteristic Size (mm)		Shape	Characteristic Size (mm)
Sphere	Diameter	2	Cylinder	Height	2.27
4-hole cylinder	Height	2.27		Diameter	1.36
	Diameter	1.36	Single-hole cylinder	Height	2.27
	Inner holes diameter	0.34		Diameter	1.36
	Inner holes center distance	0.68		Inner hole diameter	0.68

shows the geometric size data of four kinds of particles.

The diameter of the sphere was 2 mm. The surface area of the cylinder particle was the same as the sphere, and the ratio of diameter to height was 3/5. The diameters of the inner holes of the single-hole cylinder and 4-hole cylinder are half and a quarter of the cylinder diameter, respectively.

About 56 random spheres and 77 cylindrical particles inside rectangular domains were generated through DEM method. The liquid was introduced into the TBR from the top through the four tubes with diameter D_in = 0.5 mm and length L_in = 1 mm, which were located on the top forming a 3 mm × 3 mm square. Different numbers of inlets were tested, including 1, 4, and 9 inlets on the top. We found that there was no significant difference in simulation results between different inlet numbers. Therefore, the middle number of 4 inlets was selected. The residual area of the top was the gas inlet. The gas and liquid inlets were defined as velocity inlets and the region bottom was a pressure outlet. Four cross sections were set as symmetrical planes. The VOF model suffer from the generation of spurious currents near the interface. However, to get convergent simulation results, there is no treatment about spurious currents in this research. The time step was 2 × 10⁻⁴ s and the mesh numbers were 269,516, 1,112,159, 1,395,718, and 1,604,862, respectively. Two high-performance compute cluster with 8 CPU-cores were used to perform the simulations. The calculation time of four different models are 9.0, 48.0, 57.3, and 65.9 h respectively.

The independence of time step and mesh number was confirmed before the decisions were made. Except for the investigation of particle shape’s influence, the investigations of the factors were implemented on a random 4-hole cylinder packing. All simulations were conducted in the theoretical trickle flow regime. The volume of liquid and the wetted area of each case are monitored. At first both two quantities increased fast, while after some time they stopped significant increase and began to fluctuate. After that the result data was calculated by time-averaging. In the following chapters, only the investigated factors were changed, and the rest of fluid properties were kept constant as shown in Table 1.

3. Results and Discussion

3.1. Method Validation

The comparison of snapshots is shown in Figure 3. It can be seen that 3D simulations gave more accurate results for they described the extreme thin liquid film on the top of the particle at 20 ms and 25 ms better. This is due to the fact that the 3D region allowed liquid to flow along the particle surface in various directions. These results proved that the simulation framework was capable of describing the behavior of liquid on solid particles. However, it could be found that some differences exist between the reported experiment and 3D simulation results.

Table 3. Thickness of liquid film on particle surface of different positions.

t (ms)		Liquid Film Thickness (mm)
		0°	20°	40°	60°	80°	100°
10	Exp.	1.30	3.79	1.82	1.33
	Sim.	0.23	3.35	2.87
15	Exp.	1.25	0.85	1.22	1.56	2.05
	Sim.	0.49	2.94	2.14	0.95
20	Exp.	0.09			0.90	1.58	1.89
	Sim.	0.62	0.47	0.78	0.85	1.11
25	Exp.				1.21	2.44	2.56
	Sim.	0.81				0.52	1.22

shows the thickness of liquid film on particle surface of different positions. Dry surface and unmeasurable thin film are ignored in the table. It can be seen there are still several significant difference between reported experiment and 3D simulation. Most of thicknesses in the experiment are larger than those in 3D simulations, and the wetting of particle surface is faster in the experiment. All these differences could be concluded from two reasons. First the droplet in simulation is perfect sphere while in experiment is nearly ellipsoidal, which is larger in volume. Therefore the thickness of liquid film and flowing velocity are larger in experiment. Second, it can be seen that in the experiment there is a droplet hanging on the bottom of the particle at t = 0 ms, which means the particle was wetted. However, the particle surface in the simulation was totally dry. This could lead to the differences between the experiment and simulation.

3.2. Factors Investigation

3.2.1. Catalyst Particle Shape

In TBRs, the voids formed between particles affect the fluid flow structure significantly hence affect the liquid holdup and wetting efficiency. The scope is limited when altering the local characteristics of packing/voids, but the manipulation of particle size and shape may allow some degree of control [1]. Packings formed by particles of different shape have different porosities and specific surface areas, and the different flow channels of various complexity have a great influence on the distribution of the flow field. Here four particle shapes (sphere, cylinder, single-hole cylinder, and 4-hole cylinder) were investigated for their characteristics of liquid holdup and wetting efficiency.

Characteristic data of three kinds of random packings was given in

Table 4. Characteristic data of four kinds of random packing structures.

Shape	Porosity	Liquid Holdup	Liquid Saturation	Surface Area (m2)	Wetting Efficiency	Pressure Drop (Pa)
Sphere	0.55	0.1784	0.3246	4.49 × 10−4	0.5883	236.50
Cylinder	0.51	0.1711	0.3329	6.20 × 10−4	0.6322	401.60
Single-hole cylinder	0.63	0.2272	0.3600	8.29 × 10−4	0.6289	344.19
4-hole cylinder	0.66	0.2633	0.3955	1.02 × 10−3	0.6566	351.00

, and the liquid distributions inside four packings are shown in Figure 4. It can be clearly seen that the liquid inside four packings tended to accumulate in the inner holes and voids between particles. The cylinder packing had the lowest porosity liquid holdup among all. The difference in liquid holdup between spheres and cylinders was not significant since the contact form between cylinder particles was more complex than spheres, including point contact, line contact, and surface contact. This caused a higher number and more complex shapes of voids between particles, from which the cylinder packing got a stronger ability of keeping liquid by capillary effect. However, cylinders got higher liquid saturation than spheres due to their lower porosity. Single-hole cylinders showed much higher liquid holdup and saturation than both spheres and cylinders. In addition, 4-hole cylinders had both the highest liquid holdup and saturation. It can be found that the inner holes in the 4-hole cylinder packing were easier to be filled with liquid than those in a single-hole cylinder packing, due to the smaller inner hole diameter. The larger number of holes made the 4-hole cylinder packing gain the highest porosity and improved the ability of trapping liquid significantly.

The particle surface area showed a remarkable difference between four kinds of particles. Sphere packing had the lowest surface area and wetting efficiency. For cylinders, single-hole cylinders, and 4-hole cylinders, the surface area increased with the rise of the inner hole number. However, their wetting efficiencies were comparable, as the small inner holes of hollow cylinders got wet easily due to the capillary effect. It can be concluded that sphere particles were suitable in situations which prefer low liquid holdup and wetting efficiency, while single-hole and 4-hole cylinders showed the opposite characteristics. Cylinder particles give lower liquid holdup, but higher wetting efficiency.

As for the pressure drop, the cylinder packing got the highest one due to the lowest porosity. Comparing cylinder and sphere packing, their liquid holdup and saturation were similar, but the lower porosity gave less free space for gas flow between cylinders, making a remarkable higher pressure drop. While for the single-hole and 4-hole cylinders, whose liquid holdups were much higher than sphere and cylinder packings, the high porosity provided a lot of space for gas flow even though most of them were filled with liquid. Therefore, their pressure drops were lower than the cylinder packing. However, the pressure drop in 4-hole cylinders packing was slightly higher than that in the single-hole cylinder.

Comparing these four kinds of particles, the 4-hole cylinder has both the highest liquid holdup and wetting efficiency, which are preferred by most of the gas-liquid-solid catalyzed systems. Therefore, the 4-hole cylinder was decided as the main investigation object in the following chapters.

3.2.2. Surface Tension Coefficient

Surface tension coefficient gives the measurement on the contractility of the liquid-gas interface. Surface tension is caused by cohesive forces between similar molecules and it shows distinct effect when liquid flows through tiny regions like voids between particles and inner holes of particles. The effect of surface tension coefficient on liquid holdup and wetting efficiency was investigated and the results are shown in

Table 5. Liquid holdup and wetting efficiency of three surface tension coefficient.

Surface Tension Coefficient (N/m)	Liquid Holdup	Wetting Efficiency
0.036	0.2401	0.6008
0.072	0.2633	0.6566
0.108	0.2701	0.6687

and Figure 5.

It was clearly shown that both liquid holdup and wetting efficiency increased with the rise of surface tension coefficient. Higher coefficient brought higher surface tension, which gave an opposing effect on the inertia force and made liquid hard to leave the sphere packing. Figure 5 shows that with the increase of surface tension coefficient, more voids between particles and inner holes were filled with liquid, contributing to the increase of liquid holdup and wetting efficiency. Difference between 0.072 N/m and 0.108 N/m was less significant because most of the inner holes were filled with liquid when the coefficient rose to 0.072 N/m. The rise of wetting efficiency can also be explained by the opinion of Du [6], which declared that the under lower surface tension conditions rivulet flow patterns are found on the particle surface.

3.2.3. Contact Angle

The contact angle describes the wetting ability of a liquid on a specific solid surface, which is affected by multiple factors including liquid component, solid component, and surface roughness. A higher contact angle means lower wetting ability, while a lower contact angle means that the liquid can spread easier on solid surfaces. The simulation results regarding the effect of contact angle are shown in

Table 6. Liquid holdup and wetting efficiency of three contact angles.

Contact Angle (°)	Liquid Holdup	Wetting Efficiency
20	0.2503	0.6919
45	0.2633	0.6566
70	0.2488	0.5691

and Figure 6.

It can be seen that the wetting efficiency decreased significantly with the rise of contact angle because a higher contact angle resulted in accumulation rather than the spread of the liquid on particle surfaces. The same conclusion was made in other investigations on sphere particles. However, there was a noticeable phenomenon that 45° got the highest liquid holdup. Figure 6 shows that liquid in the 20° simulation tended to stay close to the particle surface. Most of the voids were free from liquids and some inner holes were not full of liquid. In the 45° simulation most of the space between particles was filled with liquid, while particles in the 70° were less surrounded by liquid for the weaker wetting ability. It can be seen that some liquid in the 70° simulation tended to gather as a liquid drop (the circled position in Figure 6). This discovery of liquid holdup variation might be explained by the fact that liquid with a low contact angle tends to flow on particle surface as a thin film, making it harder to fill the voids and inner holes inside the packing. In this case, the gas gains more flow channels and it becomes easier for the liquid to leave the random packing, while liquid with a too high contact angle is less hydrophilic, which tends to gather into a liquid drop on surface and slips down. So it can be concluded that a moderate contact angle can result in high liquid holdup due to its ability of silting channels and cohere with the particle surface.

3.2.4. Liquid Viscosity

Liquid viscosity is another factor affecting liquid distribution. It affects fluid turbulence directly and thus affects the flow form of the liquid inside TBRs. Investigation results of the variation of liquid viscosity are given in

Table 7. Liquid holdup and wetting efficiency of three liquid viscosities.

Liquid Viscosity (Pa·s)	Liquid Holdup	Wetting Efficiency
4.4435 × 10−4	0.2392	0.6237
8.8871 × 10−4	0.2633	0.6566
13.3305 × 10−4	0.2654	0.6518

, and the liquid distribution inside the packing is shown in Figure 7.

It can be clearly seen from Figure 7 that the quantity of liquid inside the packing increased with the rise of liquid viscosity. This is because the higher viscosity lead to a stronger liquid-solid shear, making it difficult for turbulence to develop, and resulted in a more uniform liquid flow on the particle surface. In high liquid viscosity cases the liquid-solid shear plays a greater role than the gas-liquid interactions [1], making the liquid harder to leave the particle surface and the whole packing. Film flow was easier to be found under high viscosity while rivulet flow can be observed under low viscosity [16], which is responsible for the increase in wetting efficiency. Differences between the first and the second was greater than that between the second and the third. For the wetting efficiency it increased by about 3% when the viscosity increased from 4.4435 × 10⁻⁴ Pa·s to 8.8871 × 10⁻⁴ Pa·s. However, a further increase of viscosity caused a slight drop in wetting efficiency, which did not vary distinctly. It shows the same tendency as liquid holdup, which means that the increase of liquid viscosity is more effective when the viscosity is low.

3.2.5. Liquid Density

The investigation result of influence on liquid holdup and wetting efficiency by liquid density is shown in

Table 8. Liquid holdup and wetting efficiency of three liquid densities.

Liquid Density (kg/m3)	Liquid Holdup	Wetting Efficiency
500	0.2694	0.6625
1000	0.2633	0.6566
1500	0.2543	0.6313

, and the liquid distribution is shown in Figure 8. It can be found that the liquid holdup and wetting efficiency decreased with the rise of liquid density. In Figure 8, the quantity of liquid in the voids between particles decreased noticeably, for more liquid trapped by surface tension and shear force was drawn out from the voids due to the increased gravity force. The increase of density strengthened the gravity effect on the liquid flow, leading to a faster downward acceleration. Thus, the liquid attained a higher speed and was able to leave the packing and the particle surface faster, resulting in a lower liquid holdup and wetting efficiency. Besides, it was reported that a higher liquid density makes rivulet flow easier than film flow [16], which was another reason for the lower wetting efficiency.

3.2.6. Liquid and Gas Velocity

The liquid and gas velocities inside TBRs have a great effect on the liquid distribution by affecting the inertia force. The simulation results of different liquid and gas superficial velocities are shown in

Table 9. Liquid holdup and wetting efficiency at different liquid and gas velocities.

Liquid Superficial Velocity (m/s)	Gas Superficial Velocity (m/s)	Liquid Holdup	Wetting Efficiency
0.002	0.245	0.2096	0.5686
0.005		0.2565	0.6339
0.008		0.2633	0.6566
0.008	0.082	0.3086	0.7163
	0.163	0.2898	0.6881
	0.245	0.2633	0.6566

, and the liquid distribution inside the spherical packing is shown in Figure 9 and Figure 10. The increase of liquid velocity lead to a rise in both liquid holdup and wetting efficiency, while the increasing of gas velocity had the opposite effect.

In Figure 9 the accumulation of liquid inside the 4-hole cylinder packing is clearly shown. With increasing velocity, liquid tends to flow through more voids, then fills all possible space inside the TBR. It can be found that the quantity of liquid in v_L = 0.002 m/s was much less than v_L = 0.005 m/s and 0.008 m/s. Although the increase of liquid velocity gave rise to the liquid inertia force, it still cannot force liquid to leave the packing faster since the input quantity increased too. It is worth noticing that the increased liquid holdup from v_L = 0.005 m/s to 0.008 m/s was much less remarkable that that from 0.002 m/s to 0.005 m/s. It can be explained by the fact that most of the attainable free space was filled with liquid when v_L = 0.005 m/s and the increase of v_L from 0.005 m/s to 0.008 m/s may have happened in the pulse flow regime, where the increase of liquid holdup with v_L was less significant than which in the trickle flow regime.

The rise of the gas velocity brought a higher liquid-gas shear stress, making it easier for the liquid to leave the particle surface and the packing. The difference of liquid quantities shown in Figure 10 was remarkable, as the liquid quantity under v_G = 0.245 m/s was less than the other situations.

4. Conclusions

This paper employed CFD with a volume of fluid (VOF) method to implement high-precision simulations of the co-currently downward gas and liquid flow between catalyst particles inside TBRs. Traditional experimental methods have intrinsic limitations on liquid holdup and wetting efficiency measurement and most empirical equations which were concluded from these methods do not consider all effective factors such as the contact angle and surface tension coefficient. The detailed effective factors were investigated in this paper and remarkable findings are presented below.

(1)

Four different catalyst particle shapes including sphere, cylinder, single-hole cylinder, and 4-hole cylinder were simulated to study their characteristics about liquid holdup and wetting efficiency. The 4-hole cylinder obtained both the highest liquid holdup and wetting efficiency. Spherical and cylindrical particles of the same volume have similar liquid holdups, but the wetting efficiency of cylindrical particles is higher than that of spherical particles. In addition, the liquid holdup of cylindrical particles increases with the number of inner holes (surface area), while the wetting efficiency does not change significantly.

(2)

The variation of the contact angle and surface tension coefficient directly affects the accumulation and flow of liquid in voids between the particles inside the 4-hole cylinder packing, which further affect the liquid holdup and wetting efficiency. Both liquid holdup and wetting efficiency increase with the surface tension coefficient. The wetting efficiency decreases with the increase of contact angle, but a moderate contact angle can result in a high liquid holdup.

(3)

Liquid viscosity and density affect the liquid holdup and wetting efficiency by affecting the liquid-solid shear force and gravity force, respectively. A higher viscosity or lower density of liquid can result in an increase of both liquid holdup and wetting efficiency.

(4)

Gas superficial velocity affects the liquid holdup and wetting efficiency by affecting the inertial force and has the opposite effect on the surface tension and liquid-solid shear force. While liquid superficial velocity affects those two factors mainly by varying the liquid input quantity. The increase in liquid velocity gives a rise in both liquid holdup and wetting efficiency, while the increase in gas velocity can reduce them.