Investigation of Liquid Spreading Processes Enhanced by Textured Structures on Hydrophilic Surfaces

Abstract

The liquid spreading on structured packings plays an essential role in affecting gas–liquid mass transfer in separation columns, yet the synergistic mechanism of surface wettability and textured geometries remains insufficiently understood. This study integrates experimental and computational methods to systematically investigate the liquid spreading characteristics on textured surfaces. The synergistic combination of hydrophilic modification and surface textures markedly enhances liquid spreading performance. Compared with the hydrophilic plane surface, the spherical cap texture increases the interface area and wetted area by 25.2% and 49.6%, respectively, while the pyramid-shaped texture leads to improvements of 24.5% and 48.9%, respectively. Based on Weber number analysis, it is identified that the competition between inertial force and surface tension governs the evolution of liquid spreading regimes. In addition, the results suggest that variations in liquid viscosity and density may further influence spreading behavior by modifying the balance among inertial, viscous, and surface tension forces. The geometric parameters of spherical cap textures are systematically examined, and it is revealed that a spherical cap with a non-uniform staggered configuration (Mode III) enables the efficient liquid spreading. A new non-uniform spherical cap texture is designed to enhance liquid spreading, which enhances spreading performance compared with the original plate, increasing the interface area by 27.3% and the wetted area by 47.4%. Although the liquid film thickness increases slightly, the wetted area ratio is significantly improved, indicating enhanced effective surface coverage. Both simulations and experiments confirm that the new textured structure further enhances liquid spreading performance on the textured surface. This research unveils a strategy to improve liquid spreading through tailored surface textures, opening up new possibilities for the design of efficient packings.

1. Introduction

Structured packings, characterized by their uniform geometry and well-defined flow channels, are widely used in gas–liquid separation processes and thus offer some advantages including low pressure drop, high throughput, superior separation efficiency, and excellent scalability. Structured packings are typically fabricated from corrugated sheets or wire gauze assemblies. Compared with wire gauze counterparts, corrugated sheet packings are frequently justified by superior mechanical robustness, excellent fouling resistance, and scale-up reliability, rendering them the preferred option for challenging applications. The performance of corrugated plate packings is intrinsically determined by their geometric structures such as corrugation shape [1,2,3,4], flow channel inclination angle [5,6,7,8], crimp angle [2,7,9], and packing porosity [10,11,12,13,14]. Besides these macroscopic parameters, it has been recognized that the textures of the packing surface itself play a critical role in governing local mass transfer efficiency and overall separation performance [15]. In practical industrial operations, inadequate liquid spreading on packing surfaces may lead to flow maldistribution, channeling, and reduced effective interfacial area, thereby deteriorating separation efficiency. Therefore, the rational design of surface textures to promote uniform liquid spreading is of great importance for improving the performance of distillation, absorption, and related gas–liquid processes.

To date, fundamental investigations of surface texture functionality have been largely confined to the canonical scenario of a single, isolated droplet on a horizontal solid surface, with a primary focus on its motion and spreading dynamics. These studies are primarily oriented toward applications in surface self-cleaning, cable anti-icing, and microfluidic devices [16,17,18]. However, such simplified scenarios do not adequately represent the continuous liquid film flow commonly encountered in industrial packing systems.

More recent experimental studies have shifted toward liquid film flow on textured surfaces, which is more relevant to practical gas–liquid operations. Schultes [19] reported, based on distillation experiments, that compared with smooth surfaces, structured packings with textured features can enhance mass transfer by approximately 20–100%. Helbig et al. [20] showed that longitudinal micro-grooves suppress interfacial waves and enhance liquid film stability. Repke et al. [21] demonstrated that surface texturing on inclined plates can significantly enhance wettability, alter the flow field, and markedly improve mass transfer efficiency in separation equipment. Kohrt et al. [22] found that the use of bidirectional pyramidal textures achieved an increase of up to 80% in the mass transfer efficiency of CO₂ in silicone oil. Flagiello et al. [23] demonstrated that, among various textured surfaces, the wavy triangular step texture provides the most effective mass transfer enhancement.

In parallel with experimental investigations, computational fluid dynamics (CFD) has been widely used to study liquid spreading and film flow on structured surfaces. Compared with experimental approaches, CFD enables detailed analysis of flow behavior, interfacial evolution, and local wetting characteristics, providing insights that are difficult to obtain through direct observation. Iso et al. [24,25] studied liquid film flow on packing surfaces and demonstrated that the wavy textures effectively mitigated liquid channeling while enhancing the wetted area. Yu et al. [3] investigated the fluid dynamics and mass transfer performance of novel wavy-structured packings and attested that the wavy textures significantly influenced liquid film thickness, wetted area, and mass transfer efficiency. Zhang et al. [26] studied the mass transfer behavior of liquid oxygen and nitrogen on textured surfaces under cryogenic conditions, demonstrating that sinusoidal and triangular textures substantially enhanced the mass transfer efficiency. Bertling et al. [27,28] carried out a numerical investigation of film and rivulet flows on micro-grooved surfaces. To ensure model predictivity, they found it necessary to account for effective contact angles. Dechert et al. [29,30] also performed numerical simulations to investigate the effect of surface textures on liquid spreading on inclined plates, finding that microstructure orientation significantly exerted a strong influence on both liquid flow patterns and wetted area. Previous studies have shown that surface geometry, texture orientation, and wettability significantly influence liquid spreading behavior and flow distribution on solid surfaces. In particular, numerical investigations have revealed that microscale surface features can strongly affect flow patterns, wetted area, and liquid film stability. Nevertheless, most existing studies are conducted on relatively simple or regular surface structures, and their findings are often limited to specific configurations.

To provide a clearer overview of the current research status,

Table 1. Summary of previous studies on liquid spreading over textured surfaces.

Literature	Texture Type	Method	Parameters	Findings
Sebastia-Saez et al. [31]	Stream-wise pattern, perpendicular pattern, smooth plate	3D CFD	Triangular base = 2.8 mm, height = 0.6 mm	Stream-wise pattern enhances wetted area; perpendicular pattern hinders liquid spreading.
Kapoustina et al. [32]	Single microstructure, smooth plate	Experiment	Height = 700 μm	Local mass transfer is enhanced near the microstructure where film thickness is reduced.
Yu et al. [3]	WPA smooth/rough texture	Experiment + CFD	Period = 4 mm, amplitude = 0.25 mm	Rough surface increases film thickness and wetted area.
Zhang et al. [26]	Sinusoidal texture, triangular texture, flat plate	2D CFD	Amplitude = 0.3 mm, wavelength = 2.8 mm	Sinusoidal texture improves mass transfer by 50%, triangular by 25%.
Düll et al. [33]	Rectangular ridge arrays	Experiment	Ridge height = 0.22–1.2 mm, ridge distance = 2–8 mm, ridge length = 0.7 mm	An optimum ridge distance (4 mm) induces strong interfacial oscillations.
Dechert et al. [30]	L-grooved, w-grooved, pyramidal, modified grooves	3D CFD	L-grooved: wavelength = 1.5 mm, amplitude = 0.2 mm; w-grooved: wavelength = 4 mm, amplitude = 1 mm; pyramidal: height = 0.27 mm	Orientation of microstructures shows the strongest influence.

summarizes representative studies on liquid spreading over textured surfaces, including surface texture types, research methods, key parameters, and main findings. As summarized in Table 1, previous studies have clearly demonstrated that surface textures play a crucial role in governing liquid spreading behavior and mass transfer performance. However, the current understanding of how surface textures influence performance is derived primarily from studies analyzing the specific structures found in commercial packings, leaving a gap in the knowledge regarding more diverse or optimized geometries. The understanding of liquid flow patterns on textured surfaces under varying operating conditions is still limited. The synergistic effect of surface wettability and texture structures on liquid spreading behavior remains insufficiently understood.

To address these gaps, the present study combines experimental observations and computational analysis to systematically investigate liquid spreading on textured surfaces with different geometric configurations. Three representative texture structures are comparatively examined to elucidate their effects on liquid flow and distribution. The influence of key geometric parameters of spherical cap textured surfaces is further analyzed to establish their relationship with spreading performance. In particular, a non-uniform spherical cap texture is proposed to enhance liquid spreading behavior. By integrating experimental and numerical approaches, this study aims to reveal the synergistic effects of surface wettability and texture design, and to provide practical guidance for the optimization of high-performance structured packing surfaces.

2. Methodology

2.1. Experimental Description

The experimental setup for observing the liquid flow on textured plate surfaces is shown in Figure 1. It consists of a tilt regulator, a camera, an LED light source, a liquid syringe pump, and a liquid inlet tube with an inner diameter of D_n = 2.4 mm. The camera is operated at a frame rate of 120 fps, and uniform illumination is provided by the LED light source to ensure clear visualization of the liquid film boundary.

The textured plate is positioned at an inclined angle of φ = 45°. The liquid inlet tube is oriented vertically, located L = 15 mm above the textured plate surface. To facilitate liquid flow visualization, a small amount of blue dye is introduced into pure water, ensuring that the fluid properties are effectively identical to those of pure water during experiments. The volumetric flow rate of the liquid is maintained at 80 mL/min.

The above parameters, including the inlet tube diameter, inclination angle, inlet position, and volumetric flow rate, are determined based on previous experimental and numerical studies of Bertling et al. [27] and Dechert et al. [30], in combination with the characteristic dimensions of the present textured plates. The selection aims to ensure the formation of a stable and continuous liquid film without breakup, while maintaining a steady spreading regime suitable for detailed observation and CFD validation.

Each experiment is conducted by continuously injecting liquid onto the textured plate surface under controlled conditions. The spreading process is recorded until a steady flow state is reached, which is defined as the stage where the spreading width remains nearly constant over time. Each experimental condition is repeated three times, and the averaged values are used for analysis to ensure reproducibility.

The maximum spreading width (D_m) is extracted from the liquid film profile once a steady flow state is established on the textured plate. The spreading process is recorded using the video camera (Canon R5, Nanjing, China), and image processing is performed using ImageJ v1.54 software. The liquid boundary is identified through grayscale thresholding, and D_m is determined based on pixel-to-length calibration. The reported values correspond to the average of three independent measurements.

The experimental study is carried out to gain the intuitive physical insights, which serves as the basis for model validation. Since potentially superior textured structures are not readily available, we begin our investigation on three typical textures (i.e., stripe, spherical cap, and pyramid) of the most common commercial corrugated sheet packings [30]. The textured plates made of photopolymer resin are fabricated via 3D printing using photopolymerization method with a printing accuracy of 0.05 mm. For this study, perforations are not incorporated into the textured plates. The geometric representations of these textured plates are created using the 3D CAD method, as shown in Figure 2a. Each of the textured plates has dimensions of 60 mm (length) × 20 mm (width) × 2 mm (thickness). The structural dimensions of the three textures are presented in Figure 2b. The striped-shaped texture is characterized by protrusions with a parabolic profile, and each protrusion has a width of 1.5 mm and a height of 0.5 mm. The spherical cap textured surface features protrusions formed from the upper spherical segment, where each cap has a 1.5 mm diameter base and a height of 0.5 mm. The third texture comprises pyramidal protrusions with a height of 0.5 mm, each situated on a regular hexagonal base of 1 mm side length.

Figure 2c depicts the real surface morphology of the 3D-printed textures obtained via optical microscopy, verifying the satisfactory geometric fidelity of the fabricated plates. Given that the surface of the photopolymer resin is hydrophobic, a hydrophilic modification treatment is applied to investigate the synergistic effect of surface wettability and textures. We deposit a hydrophilic coating onto the textured plate surface by brush coating at room temperature. As our primary interest lies in the effects of surface textures, the development of hydrophilic coating materials is not undertaken in this study. A commercially available modified silica material purchased from Shangmeng Technology Wuxi Co., Ltd. (Wuxi, China) is used for the hydrophilic coating. The advancing and receding contact angles are measured using the sessile drop method, and each measurement is repeated three times to obtain averaged values. As shown in Figure 2d, the treated surface exhibits significantly reduced contact angles, indicating enhanced wettability.

2.2. Mathematical Models

The liquid flow process on the textured plate surfaces involves gas–liquid two-phase flow. To capture the dynamic changes in the gas–liquid interface, the CFD simulation is conducted using the volume of fluid (VOF) method in this study. The VOF method was introduced by Hirt and Nichols [34], and the continuity and momentum equations are given as follows:

$$\nabla {\displaystyle •}\mathit{u}=0$$

(1)

$${\displaystyle \frac{\partial \rho }{\partial t}}\left(\rho \mathit{u}\right)+\nabla {\displaystyle •}\left(\rho \mathit{uu}\right)=-\nabla p+\rho \mathit{g}+\nabla {\displaystyle •}\left[\mu \left(\nabla \mathit{u}+{\left(\nabla \mathit{u}\right)}^T-{\displaystyle \frac{2}{3}}\mu \left(\nabla \mathit{u}\right)I\right)\right]+{\mathit{f}}_{\sigma }$$

(2)

where u denotes the velocity, ρ is the fluid density, p is the pressure, and f_σ is the surface tension force.

The momentum equation incorporates surface tension as a source term, which is evaluated using the continuous surface force (CSF) model [35]:

$${\mathit{f}}_{\sigma }=\sigma \kappa \nabla \alpha$$

(3)

$$\kappa =-\nabla {\displaystyle •}\left({\displaystyle \frac{\nabla \alpha }{\left|\nabla \alpha \right|}}\right)$$

(4)

where σ is the surface tension coefficient and κ is the interface curvature.

In the VOF method, a phase volume fraction (α) is introduced to track the interface, where α represents the volume fraction of a phase in a computational cell. The evolution of this fraction is described by the following transport equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla {\displaystyle •}\left(\alpha \mathit{u}\right)=0$$

(5)

In a computational cell, α is set to 1 if the cell is completely filled with liquid, and to 0 if it is fully occupied by gas. The cells with 0 < α < 1 correspond to the interface where both phases coexist. In the present study, both liquid and gas phases are treated as incompressible Newtonian fluids. The effective density and viscosity are calculated as weighted arithmetic averages of the phase volume fractions in each cell:

$$\rho =\alpha {\rho }_l+\left(1-\alpha \right){\rho }_g$$

(6)

$$\mu =\alpha {\mu }_l+\left(1-\alpha \right){\mu }_g$$

(7)

where the subscripts l and g denote the liquid and gas phases, respectively.

To limit numerical diffusion, the interface compression scheme is adopted by adding additional convection terms to the α transport equation [36], and Equation (5) is reformulated as

$${\displaystyle \frac{\partial \alpha }{\partial t}}+\nabla {\displaystyle •}\left(\alpha \mathit{u}\right)+\nabla {\displaystyle •}\left[\alpha \left(1-\alpha \right){\mathit{u}}_r\right]=0$$

(8)

where u_r is the compressive velocity, u_r = u_l − u_g. The α advection equation is solved using the Multidimensional Universal Limiter with Explicit Solution (MULES) algorithm to ensure the conservativeness, convergence, and boundedness [37].

2.3. Simulation Setup

The size of the computational domain is determined from the experimental configuration and the textured plate geometry, with the mesh shown in Figure 3. The multi-region refined mesh is generated to balance computational efficiency and cost, with mesh sizes assigned according to flow conditions: 0.8 mm in the gas zone (#0); 0.4 mm in the transition zone (#1); 0.2 mm in the liquid spreading zone (#2); and 0.1 mm in the liquid inflow zone and the boundary layer of the textured surface (#3). It can be seen that the surface texture mesh accurately reproduces the textured geometry.

To ensure the numerical accuracy of the simulations, a mesh independence test is performed prior to the formal calculations. Three mesh refinement levels (Level 1, Level 2 and Level 3) are employed for the computational domain, with total cell numbers of approximately 579,661, 763,299 and 2,635,299, respectively. Figure 4 compares the temporal evolution of the liquid contour on the textured plate and the retained liquid volume under different mesh resolutions. It is observed that the Level 1 mesh leads to noticeable deviations in both the liquid morphology and volume prediction, particularly after t ≈ 0.25 s, indicating insufficient spatial resolution. In contrast, the results obtained with Level 2 and Level 3 meshes exhibit very similar liquid spreading patterns, with nearly identical contour evolution throughout the simulation. Quantitatively, the difference in the retained liquid volume between Level 2 and Level 3 is only 4.08%, suggesting that further mesh refinement results in limited improvement in solution accuracy. Considering the balance between computational cost and numerical precision, the Level 2 mesh is selected for all subsequent simulations, as it provides sufficiently accurate predictions while maintaining reasonable computational efficiency.

The governing equations are discretized using the finite volume method (FVM) in OpenFOAM-9. A first-order implicit Euler scheme is employed for the temporal derivatives, while gradient terms are discretized using the Gauss linear scheme. For convective terms, the Gauss limitedLinear and Gauss van Leer schemes are applied to Equations (2) and (8), respectively. Interface compression velocity terms are handled by a specialized interface compression scheme, and Laplacian terms are discretized using a second-order corrected scheme with linear interpolation. The PISO algorithm with three iterations per time step is employed for pressure–velocity coupling. The time step is automatically controlled by maintaining a maximum interface Courant number of 0.5. The fluid properties adopted in the simulations correspond to those utilized in the experiments. The gas phase is air, with a density of 1.225 kg/m³ and a viscosity of 1.79 × 10⁻⁵ Pa·s. The liquid phase is purified water, with a density of 1000 kg/m³ and a viscosity of 1 × 10⁻³ Pa·s. The surface tension coefficient is 0.072 N/m.

The boundary conditions for liquid spreading simulations are shown in Figure 3. At the inlet of the liquid flow, the uniform constant velocity is specified from the volumetric flow rate. At the outlet, the pressure is set to the atmospheric pressure, while the no-slip boundary condition is used at the wall. Since the dynamic spreading behavior of liquid on solid surfaces is simulated in this study, a dynamic contact angle is required to describe the motion of the triple-phase contact line. Since the Kistler dynamic contact angle model has been widely applied for simulating droplet spreading on solid surfaces [38,39,40], the Kistler model is employed in this study. Using the tilted plate method, the receding and advancing angles are measured as 25°/55° for the hydrophilic surface and 28°/110° for the hydrophobic surface.

3. Results and Discussion

3.1. Comparison of Three Typical Textures

Figure 5a provides a comparison between the simulated and experimentally observed spreading morphologies of the liquid. It is seen that the numerical approach successfully reproduces the spreading morphologies of the liquid on both plane and textured plates. All simulated spreading morphologies presented in this study are obtained based on three-dimensional simulations, ensuring a realistic representation of the liquid flow behavior. Both the experimental observations and numerical simulations reveal that upon the impact of the falling liquid on the surfaces, a localized liquid pool is formed at the impact point. The pool subsequently contracts downward and generates a stable liquid column flowing off the solid surface. The wettability of the solid surface significantly influences the liquid spreading behavior. Liquid spreading is inhibited on the hydrophobic surfaces, while it is enhanced through hydrophilic surface modification. On the hydrophilic surfaces, compared with the plane plate, the stripe-shaped texture restricts liquid spreading, whereas the pyramid-shaped and spherical cap textures promote it. The liquid–solid contact area is substantially increased by the pyramid shape and spherical cap textures, thereby resulting in an expanded range of liquid spreading. This enhancement effect induced by textured structures is consistent with the findings reported by Zhang et al. [26].

In Figure 5b, the maximum spreading widths of the liquid on the three textured plates and the plane plate are presented quantitatively. There is close agreement between the experimental and simulated maximum spreading widths of the liquid. The relative deviation between the simulated and experimental values of D_m is within ±14%, while the deviation for the hydrophilic spherical cap structure, which is the primary focus of this study, is only 2.58%. These results further demonstrate that the numerical model can reliably predict liquid spreading behavior. The observed discrepancies may be attributed to the failure to consider the surface roughness of the actual textured plates when constructing the three-dimensional geometric structure, as such roughness can influence local wetting characteristics. It is evident that the maximum spreading widths of the liquid on the hydrophilic surfaces are significantly greater than those on the hydrophobic ones. Among the hydrophilic textured plates, the maximum spreading width of the liquid is large on the spherical cap and pyramid textured plates.

Figure 6 gives the temporal profiles of the interface area and wetted area of the liquid during spreading on different textured plates. Here, the interface area refers to the gas–liquid interface area, while the wetted area corresponds to the liquid–solid interface area. From Figure 6a,b, the interface area and wetted area increase over time, eventually reaching a constant level. The temporal profiles of the interface area and wetted area are influenced by both the textured structures and the surface wettability. On the hydrophobic surfaces, the spherical cap textures have the largest values of interface area and wetted area. After hydrophilic modification, the interface area and wetted area are significantly enhanced for all textured surfaces. In particular, compared with the hydrophilic plane surface, the spherical cap texture increases the interface area and wetted area by approximately 25.2% and 49.6%, respectively, while the pyramid-shaped texture leads to improvements of about 24.5% and 48.9%, respectively. In contrast, no significant difference is observed between the spherical cap and pyramid textures under hydrophilic conditions. The results indicate that the combination of surface texture and hydrophilicity produces a synergistic effect that promotes liquid spreading. Specifically, the texture features provide capillary pathways, while the hydrophilic chemistry ensures strong solid–liquid attraction.

The dynamic evolution of liquid morphology on the four plates can be observed in Figure 7. Once liquid leaves the inlet tube, the Rayleigh–Plateau instability occurs when surface tension makes the fluid column become unstable. The leading end of the liquid column ruptures into a large droplet. Once a steady flow is achieved, a continuous liquid column forms on the textured surface. A local liquid pool is initially developed on the textured surface and then spreads downward under gravity, during which both the interface area and wetted area increase simultaneously until their maximum values are reached. As the liquid flows off the textured plate, both the interface area and wetted area remain nearly constant, indicating the stabilization of the spreading morphology. It is observed that the spindle-shaped liquid column forms when the liquid spreads on the plane plate and stripe-shaped textured plates, resulting in a large leading end of the liquid column. Once the leading end leaves the textured surface, both the interface area and wetted area are significantly reduced after reaching their maximum values, also shown in Figure 6. As depicted in Figure 7b, on the hydrophobic surfaces, a more distinct spindle-shaped liquid column forms on both plane and stripe-textured surfaces, which further demonstrates how surface hydrophobicity hinders liquid spreading.

Since the spreading capability of the liquid on hydrophobic surfaces is limited, further analysis of these surfaces is not carried out. The following discussion focuses exclusively on the liquid spreading on hydrophilic textured plates. From Figure 8, it can be seen that compared with the plane plate, the length of the liquid pool on the stripe-shaped textured surface is shorter. Once the liquid impacts the stripes, a significant decrease in liquid velocity is observed over a short distance due to the resistance imposed by the stripes. Under the influence of surface tension, the liquid within the pool contracts earlier to form a downward-flowing liquid column. In contrast, liquid spreading is promoted by the pyramid-shaped and spherical cap textures, where the liquid flow is diverted by the protruding edges or curved geometries.

Figure 9 presents a comparison of the velocity distribution within the liquid film obtained from CFD simulations with those predicted by the Nusselt theory [41] and the modified Nusselt theory. It can be observed that the velocity profile obtained from the CFD simulation exhibits an overall parabolic shape, which is consistent with the theoretical prediction for laminar falling films. However, under the same film thickness (δ), the magnitude of the velocity is significantly lower than that predicted by the Nusselt theory. This discrepancy mainly arises from the non-ideal flow conditions in the present system. Unlike the assumption of uniform liquid supply in the Nusselt theory, the liquid in this study is introduced through a jet impingement process, which leads to additional momentum dissipation during impact and spreading. Furthermore, the liquid undergoes pronounced lateral spreading in the impingement region and forms a localized liquid pool, thereby reducing the effective driving force along the inclined surface. These factors collectively result in a lower flow velocity compared to the theoretical prediction.

To account for these deviations, the Nusselt model is modified by introducing a correction factor β that represents the reduction in effective driving force:

$$\mathit{u}\left(\mathit{y}\right)=\beta \cdot {\mathit{u}}_{\mathit{N}\mathit{u}}\left(\mathit{y}\right)$$

(9)

where u_Nu(y) is given by

$${\mathit{u}}_{\mathit{N}\mathit{u}}\left(\mathit{y}\right)={\displaystyle \frac{\rho \mathit{g}\mathit{sin}\phi }{\mu }}\left(\delta \mathit{y}-{\displaystyle \frac{{\mathit{y}}^2}{2}}\right)$$

(10)

The correction factor β is determined by minimizing the least-squares error between the CFD velocity profile u_CFD(y) and the theoretical prediction:

$$\beta ={\displaystyle \frac{{\int }_0^{\delta }{\mathit{u}}_{\mathit{C}\mathit{F}\mathit{D}}\left(\mathit{y}\right){\mathit{u}}_{\mathit{N}\mathit{u}}\left(\mathit{y}\right)\mathit{d}\mathit{y}}{{\int }_0^{\delta }{\mathit{u}}_{\mathit{N}\mathit{u}}^2\left(\mathit{y}\right)\mathit{d}\mathit{y}}}$$

(11)

For discrete data points, the expression can be approximated as

$$\beta \approx {\displaystyle \frac{{\sum }_{\mathit{i}}{\mathit{u}}_{\mathit{C}\mathit{F}\mathit{D},\mathit{i}}{\mathit{u}}_{\mathit{N}\mathit{u},\mathit{i}}}{{\sum }_{\mathit{i}}{\mathit{u}}_{\mathit{N}\mathit{u},\mathit{i}}^2}}$$

(12)

Using representative data, the calculated value is β ≈ 0.43. As shown in Figure 9, the modified velocity profile exhibits good agreement with the CFD results in both trend and magnitude, indicating that the modified model can effectively capture the influence of jet impingement and liquid spreading on the flow characteristics. Therefore, it provides a more accurate description of the liquid film behavior under realistic operating conditions.

In summary, the liquid spreading performance on the textured plates is significantly enhanced by the synergistic effect of surface hydrophilicity and texture design. Following hydrophilic modification, both spherical cap and pyramid-shaped textures have comparable liquid spreading performance. However, due to the sharp edges, corners, and polyhedral characteristics of the pyramid-shaped texture, its structure is more complex than that of the spherical cap texture. Therefore, the spherical cap texture is selected for further investigation in this study, aiming to elucidate the key factors governing liquid spreading behavior and to conduct innovative design for enhancing liquid spreading performance.

3.2. Detailed Study of Spherical Cap Textures

The temporal profiles of the interface area and wetted area of the liquid on the spherical cap textured surface under different inlet flow rates (Q) are depicted in Figure 10a,b. Both the growth rate and the steady values of the interface area and wetted area significantly increase with the increase in inlet flow rate. At the low inlet flow rate, the inertial force is weak and the liquid spreading is dominated by surface tension. The spreading process is restricted, which results in the slow increase in both the interface area and wetted area. It is found in Figure 10c that the liquid falls drop by drop at the low inlet flow rate, which contributes to the fluctuating increase in both the interface area and wetted area, as shown in Figure 10a,b. As the inlet flow rate increases, the inertial force gradually plays a dominant role. The constraining effect of surface tension is overcome by the stronger inertia, which allows the texture gaps to be penetrated more rapidly and thoroughly by the liquid. As a result, the high inlet flow rate accelerates the formation of the liquid–solid interface, thereby fostering a substantial expansion of the wetted region. Figure 10d presents the distribution of the Weber number (We) during the liquid falling, spreading, and stable flow stages. The We reflects the competition between inertial force and surface tension force (We = ρu²D_n/σ). For low We, the falling and spreading stages are prolonged by the weak inertia, which delays the transition to the stable flow stage. In contrast, for high We, the falling process is finished more rapidly due to the strong inertia, which makes the spreading and stable flow stages occur earlier. It is clearly substantiated by these results that the final spreading morphology of the liquid on the spherical cap textured surface is determined by the competition between the inertial and surface tension forces, which also directly regulates the duration and evolution of the three dynamic stages of the liquid flow.

To systematically evaluate the influence of liquid properties on its spreading behavior over textured surfaces, the spreading dynamics under different viscosity and density conditions are analyzed. Figure 11a,b presents the influence of viscosity on the evolution of interface area and wetted area. As the viscosity increases from 0.5 to 1.5 mPa·s, both the growth rate and the steady-state values of the interface area and wetted area decrease significantly. This behavior is attributed to enhanced viscous dissipation: higher viscosity suppresses momentum transport and weakens liquid redistribution within the textured structures, thereby limiting lateral spreading. In addition, increased viscous resistance hinders the liquid from traversing the spherical cap structures, causing the flow to preferentially drain along the primary direction and reducing the effective wetted coverage. Their spreading morphologies at stable moments are shown in Figure 11e.

Figure 11c,d illustrates the effect of density on the spreading behavior. The results reveal a distinct stage-dependent influence of density. In the initial stage (t < 0.3 s), higher densities (ρ = 1000 and 1500 kg/m³) exhibit faster spreading and larger interface and wetted areas due to stronger inertial effects. In the intermediate stage, the liquid with ρ = 500 kg/m³ shows a higher peak interface area than that with ρ = 1000 kg/m³. This can be attributed to its weaker inertia, which reduces drainage velocity and promotes temporary accumulation and lateral redistribution within the textured structures. However, in the final steady stage, the overall wetted area of the ρ = 500 kg/m³ case becomes smaller than that of ρ = 1000 kg/m³. This indicates that although low-density liquids can achieve larger transient interface expansion, their insufficient inertia limits their ability to continuously overcome surface structures and sustain effective spreading. This behavior is further corroborated by the spreading morphologies shown in Figure 11f.

Figure 12 demonstrates the regulation of liquid spreading by the inclination angle of a hydrophilic textured plate with spherical caps. From Figure 12a,b, it can be seen that the interface area and wetted area in the steady state progressively decrease with an increasing inclination angle. When the inclination angle of the textured plate is 30°, the gravitational component along the plate surface is relatively small, resulting in a lower flow velocity. It takes longer for the liquid to reach the bottom of the textured plate, and the interfacial area and wetted area increase at a slower rate. Figure 12c reveals that the liquid accumulates at the lower end of the plate, forming a significant spindle-shaped column. This accumulation leads to the largest volume of retained liquid on the plate, as shown in Figure 12d. This, in turn, promotes the lateral spreading of the liquid to both sides, resulting in a larger wetted area. When the inclination angle is increased to 45°, the gravitational effect on the liquid is enhanced. The volume of liquid retained on the plate surface is reduced compared to that at 30°, while the steady-state interface area and wetted area remain essentially equivalent to those observed at 30°. When the inclination angle is further increased to 60° and 75°, the gravitational effect on the liquid is further enhanced. The volume of liquid retained on the plate surface decreases significantly, and the spreading pattern gradually transitions from lateral expansion to longitudinal extension along the inclined direction. This ultimately results in the formation of a slender, strip-like liquid film, leading to a further reduction in the steady-state interface area and wetted area. Based on the foregoing analysis, the inclination angle of 45° is considered appropriate, as it enables desirable liquid spreading characteristics while maintaining a low volume of retained liquid. This finding is similar to the observations reported by Gu et al. [6], who also suggested that an inclination angle of 50° is favorable for balancing liquid distribution and drainage performance.

The temporal profiles of the interface area and wetted area on the spherical cap textured surfaces with different wettability are indicated in Figure 13a,b. The hydrophilicity is enhanced by decreasing advancing (θ_A) and receding (θ_R) contact angles. The surface hydrophilicity plays a significant role in improving the interface area and wetted area, which can be visualized from the spreading coverage of the liquid (see Figure 13c). Correspondingly, both the interface area and wetted area increase continuously, and their growth rates are notably accelerated. Figure 13d shows the three-phase contact lines at different moments under different wettability. For θ_A = 30° and θ_R = 10°, the fastest movement and the largest instantaneous coverage area are exhibited by the three-phase contact line, which are associated with the highest growth rate and steady-state values of both the interface area and wetted area. It is worth noting that when the contact angles are further decreased from θ_A = 45°, θ_R = 15° to θ_A = 30°, θ_R = 10°, the liquid has nearly covered the entire textured surface, resulting in the larger values of the interface area and wetted area. In practical applications, superior spreading performance would be achieved not only through the enhancement of surface hydrophilicity but also by the optimization of texture design. A better balance among wetting efficiency, coverage uniformity, and engineering cost should be determined by such a synergistic strategy.

The influence of spherical cap height on liquid spreading behavior on the textured surfaces is provided in Figure 14. Figure 14a,b reveals that a more rapid increase in interface area during the initial stage is achieved on textured plates with smaller spherical cap heights, through which the early spreading process is effectively accelerated. In contrast, larger steady-state interface area and wetted area during the later spreading stage are associated with the larger spherical cap heights, which thus possess superior wetted capability. From the liquid velocity field diagram at t = 0.65 s shown in Figure 14c, when the spherical cap height is H = 1.5 mm, the spreading area of the liquid is significantly larger than that on the plane plate surface (H = 0 mm). The distinct “fish-scale” pattern emerges in the velocity distribution, which is characterized by high-velocity regions within the inter-caps gaps and low-velocity regions atop the spherical caps. This indicates that the spherical cap structures exert a pronounced flow diversion and guidance effect on the liquid flow. The above phenomena are attributed to the synergistic effects between hydrodynamics and surface geometry. At lower spherical cap heights, the liquid experiences smaller flow resistance and minimal flow diversion. As a result, the liquid front advances more rapidly, and the interface area increases quickly during the initial spreading stage. On the contrary, higher spherical cap imposes stronger resistance at the initial stage, thereby slowing the propagation of the liquid front. However, a larger geometric surface area is possessed by the higher spherical cap, and a stronger flow diversion effect is induced on the liquid, enabling continuous liquid expansion and penetration into the gaps of the textured structure. As depicted in Figure 14d, excessively large spherical cap heights result in a very large volume of the retained liquid. In actual operation, this condition could potentially trigger the occurrence of liquid flooding.

We further investigate the influence of spherical cap arrangement on liquid spreading. Figure 15a illustrates three different spherical cap arrangements. The spherical caps of identical size are uniformly located on the Mode I surface. On the Mode II surface, a large spherical cap is encircled by four smaller spherical caps, whereas on the Mode III surface, a large spherical cap is surrounded by six smaller ones. Figure 15b provides clear evidence that the arrangement of spherical caps has a substantial impact on the characteristics of liquid spreading. In the early spreading stage (t < 0.2 s), due to the small volume of liquid on the plate surface, the liquid spreads rapidly over the surface due to inertia. The differences in spreading morphology among the three structures are minimal. As more liquid accumulates on the surface over time, the flow diversion and redistribution effects induced by different arrangements come into play, resulting in distinct spreading morphologies. Mode III exhibits a stronger multi-directional flow diversion capability, which allows the liquid to spread more uniformly across the surface, resulting in a larger coverage area. This phenomenon is further confirmed in Figure 15c,d. For t > 0.2 s, the interface area and wetted area on the Mode III surface begin to exceed those of Mode I and Mode II. Therefore, the arrangement pattern of the spherical caps significantly influences spreading uniformity and coverage by regulating the flow diversion and redistribution processes on the surface. Among the three configurations, Mode III achieves the uniform liquid distribution and the large wetted area. The heterogeneous arrangement of spherical cap textures plays a promoting role in the liquid spreading.

3.3. New Design of a Spherical Cap Textured Surface

To obtain potential structure improvement, we proceed with further structural modification of the spherical cap textured plate, with the design illustrated in Figure 16. Based on the Mode III arrangement, a hierarchical structure comprising two scales of spherical caps is implemented: large caps with a height of 1 mm and small caps with a height of 0.5 mm. The large spherical caps serve to intensify flow diversion and redistribution, thereby facilitating lateral spreading of the liquid. The small spherical caps pose little resistance to liquid movement, which prevents the thick film formation and reduces liquid retention on the surface. Additionally, an increase in the spacing between spherical caps to 0.2 mm serves to reduce flow resistance, thereby minimizing liquid retention on the surface.

The comparison of liquid spreading performance for the original and new textured plates is presented in Figure 17. Figure 17a,b shows the simulated profiles of interface area and wetted area. The simulation results indicate that a significant enhancement in the final liquid spreading performance on the textured surface is achieved by the improved structure, where the interface area is increased by 27.3% and the wetted area by 47.4% compared with those of the original plate. Figure 17c compares the simulated results of liquid film thickness and the wetted area ratio (wetted area/real geometric area) of the textured plates. Although the liquid film thickness increases slightly, the wetted area ratio is significantly enhanced, implying improved effective surface coverage. In Figure 17d, the experimental results confirm that the new surface texture can further enhance liquid spreading performance. The geometric parameters described above constitute a preliminary optimization scheme derived from mechanistic understanding. Their quantitative effects and optimal combination, however, require further systematic investigation for comprehensive elucidation.

4. Conclusions

This study systematically investigates the mechanisms by which surface texture structures enhance liquid spreading through experiments and CFD simulations. Hydrophilic modification is applied to hydrophobic textured surfaces to clarify the synergistic effects of wettability and geometry. Based on the influence of key geometric parameters of the spherical cap texture, a new non-uniform spherical cap structure with superior spreading performance is developed.

The spherical cap and pyramid-shaped texture structures have superior liquid spreading performance compared with the stripe-shaped and plane plate surfaces, which is attributed to their liquid diversion effect. The liquid spreading on the spherical cap texture is governed by the competition between inertial force and surface tension. Liquid spreading behavior diverges markedly under different We conditions. In addition, it is revealed that lower viscosity and density favor liquid spreading by reducing flow resistance and enhancing inertial effects. The liquid spreading on textured surfaces is also regulated by the inclination angle and surface hydrophilicity. An inclination angle below 45° enhances liquid spreading, albeit with potential excessive retention. However, the inclination angle larger than 45° accelerates liquid sliding at the expense of reduced wetted area. Increased hydrophilicity significantly improves liquid spreading.

The geometric parameters of the spherical cap texture have a significant regulating effect on the liquid spreading behavior. The effects of both the height and arrangement of the spherical caps are examined. The results indicate that a suitable cap height is beneficial for improving spreading performance. Furthermore, the Mode III arrangement proves effective in promoting liquid redistribution and enhancing lateral spreading. A new non-uniform spherical cap textured surface is proposed based on the Mode III configuration. The simulation predictions indicate that the new design further enhances liquid spreading performance compared to the original spherical cap texture.

Despite the good agreement between simulations and experiments in this study, several limitations should be acknowledged. The numerical model does not explicitly resolve microscale surface roughness or potential coating heterogeneity, which may affect local liquid spreading behavior. While the optimized texture exhibits enhanced spreading performance, its influence on pressure drop and mass transfer in full packing columns has not yet been quantified. Future work should address these aspects under realistic operating conditions and further explore the applicability of the proposed textured surfaces in broader engineering systems involving interfacial mass transport phenomena.