Experimental Study on Wettability Characteristics of Falling Film Flow Outside Multi-Row Horizontal Tubes

Abstract

The wettability of falling film flow outside multi-row horizontal tubes is a core factor determining the heat and mass transfer performance of falling film heat exchangers, which is critical for their optimized design and stable operation. A visualization experimental platform for falling film flow over ten rows of horizontal tubes was constructed, with water as the working fluid. High-definition imaging technology and image processing methods were employed to systematically investigate the liquid film distribution and wettability under three tube diameters (d = 0.016, 0.019, 0.025 m), four tube spacings (s = 0.75d, 1d, 1.25d, 1.5d), and four inter-tube flow patterns (droplet, columnar, column-sheet, and sheet flow). Two parameters, namely the “total wetting length” and the “total wetting area”, were proposed and defined. The distribution characteristics of the wetting ratio for each row of tubes were analyzed, along with the variation laws of the total wetting area of the ten rows of tubes with respect to tube diameter, tube spacing, and liquid film Reynolds number (Re_l). The following results were indicated: (1) Increasing the fluid flow rate and the tube spacing both promote the growth of the wetting length. When Re_l ≤ 505, with the increase of tube diameter, the percentage of the wetting length of the tenth tube row relative to that of the first tube row decreases under the same fluid flow rate; when Re_l > 505, this percentage first decreases and then increases. (2) The total wetting area exhibits a trend of “first increasing then decreasing” or “continuous increasing” with the tube spacing, and the optimal tube spacing varies by flow pattern: s/d = 1 for droplet flow (d ≤ 0.016 m), s/d = 1.25 for columnar flow, and s/d = 1.25 (0.016 m), 1 (0.019 m), 1.5 (0.025 m) for sheet flow. (3) The effect of tube diameter on the total wetting area is a balance between the inhibitory effect (reduced inter-tube fluid dynamic potential energy) and promotional effect (thinner liquid film spreading). The optimal tube diameter is 0.016 m for droplet flow and 0.025 m for columnar/sheet flow (at s/d = 1.25). (4) The wetting performance follows the order 0.016 m > 0.025 m > 0.019 m when Re_l > 505, and 0.025 m > 0.019 m > 0.016 m when Re_l ≤ 505. Finally, an experimental correlation formula for the wetting ratio considering the Re_l, the tube diameter, and tube spacing was fitted. Comparisons with the present experimental data, the literature simulation results, and the literature experimental data showed average errors of ≤10%, ≤8%, and ≤14%, respectively, indicating high prediction accuracy. This study provides quantitative data and theoretical support for the structural optimization and operation control of multi-row horizontal tube falling film heat exchangers.

1. Introduction

Horizontal tube falling film heat exchangers, with high efficiency and low resistance, are widely used in refrigeration/air conditioning [1,2], seawater desalination [3], chemical production [4], waste heat recovery [5,6], and hydrogen energy storage/transportation [7]. Horizontal tube falling film comprises the extra-tube falling film zone and inter-tube flow zone; the heat and mass transfer in falling film evaporation/absorption mainly occurs at the gas–liquid interface of the extra-tube zone. Thus, the liquid film wetting area and the spreading shape are key parameters affecting the heat/mass transfer coefficients and furthering the heat exchanger performance. Changes in the flow parameters (e.g., spray density, fluid properties) and structural parameters (e.g., tube diameter, tube spacing, row number) cause significant fluctuations in the extra-tube liquid film wetting area and spreading shape, which in turn impact the heat/mass transfer performance. An in-depth clarification of the parameter effects on the wetting characteristics of horizontal smooth tubes and an improvement of the tube surface wetting state are critical for optimizing the heat exchanger operating parameters/structural configuration and enhancing the heat/mass transfer efficiency [8,9].

The complex, variable flow behavior of the liquid film outside the horizontal tubes arises from the coupling of operating and structural parameters. Researchers have thus investigated the wetting characteristics of smooth tube exteriors via experimental tests and numerical simulations. Firstly, regarding the influence of operating parameters (e.g., spray density, fluid properties) on wettability, Killion and Garimella [10] first experimentally demonstrated that external tube wettability significantly affects the heat/mass transfer in horizontal tube falling films. Jeong and Garimella [11] then proposed and defined the “wetting ratio”—a key parameter for the wetting degree, representing the ratio of the total tube surface wetting area to the total surface area. Two-dimensional simulations showed that the heat/mass transfer coefficients vary with the wetting ratio, though this effect is minimal under low-flow droplet flow. Later, Castro et al. [12], combining experiments with tube circumferential two-dimensional simulations, showed that the wetting ratio depends on the contact angle and fluid flow rate. With advancing simulation methods, de Arroiabe et al. [13] validated this conclusion via a 3D model considering both tube circumferential and axial directions. Subsequently, Ji et al. [14] conducted in-depth studies, confirming that the wetting ratio is proportional to the fluid flow rate and inversely proportional to the static contact angle. Li et al. [15] and Qiu et al. [16] used 3D simulations to clarify the axial direction of the external liquid film and found that it exhibits crest–trough distribution between the liquid columns, with maximum thickness and optimal wettability at the midpoint of the liquid column.

Regarding the influence of structural parameters (tube diameter, spacing, material, surface structure, arrangement) on wettability, Tian [17] experimentally studied the relationship between adjacent liquid column spacing and droplet splashing, finding tube diameter and spacing are key factors for splashing—splashing reduces the falling film flow rate and thus tube external wettability. Martinez-Urrutia et al. [18] compared three tube materials (copper, aluminum, stainless steel), showing aluminum and stainless steel have better wettability than copper. Lee et al. [19], Zhang et al. [20], Jia et al. [21], and Pecherkin et al. [22] explored enhanced tube surface structures. Cao et al. [23] developed a novel aluminum finned-tube type dehumidifier using ionic liquid desiccant, and identified the liquid film Reynolds number (Re_l), the Weber number (We), and the contact angle were critical parameters affecting the wetting rate. Zhou et al. [24] investigated the impact of the surface-active component on the rupture behavior of wetting films. Studies [16,25,26,27,28,29] designed tube arrangements to expand the external liquid film spreading area and to improve the heat/mass transfer.

Most of the above studies qualitatively reveal tube external wettability laws and focus on ≤3 tube rows with few quantitative studies on multi-row tubes (>3 rows, common in practice). However, Kim et al. [30] first investigated LiBr solution wettability outside 28-row horizontal falling film tubes via experiments/simulations, proposing a wetting ratio correlation with the liquid film Reynolds number (Re_l), the Weber number (We), and the falling film section height (H). This correlation describes flow rate and falling film height effects but has ±20% error vs. the experiments and only applies to 30 < Re_l < 120. Lu et al. [31] used water to study 20-row tube wettability, presenting a wetting ratio correlation with Re_l and the air flow Reynolds number (Re_air); this describes the flow rate and air velocity effects but only applies to tubes with d = 0.095 m and spacing = 0.02 m. Recent studies [16,20,32,33,34] have shown that, under the same spray density, a larger tube diameter significantly expands the liquid film spreading area, while the increased tube spacing enhances the liquid film impact on lower rows but also increases splashing, reducing the fluid flow to the lower rows—leading to non-uniform wettability variation due to these dual effects. Thus, tube diameter and spacing are critical for wettability, yet previous quantitative wetting correlations have ignored these factors.

Ten-row tubes—common in practical multi-row configurations—enable a comprehensive heat transfer analysis by accounting for the inter-row flow interference and temperature gradients supporting industrial efficiency optimization (e.g., power plants, refrigeration systems). Tube diameter/spacing influence the flow patterns, resistance, and pressure drop; understanding these relationships minimizes the pumping energy consumption, critical for large-scale industries. Optimized diameter/spacing balances compactness (smaller diameters yield a higher heat transfer area per unit volume) and cost-effectiveness, aligning with industry-specific requirements.

Therefore, how do the tube diameter (d), the tube spacing (s), and the liquid film Reynolds number (Re_l) synergistically affect the wettability of 10-row horizontal tubes (a common industrial configuration), and can a universal correlation be established to quantify these effects for heat exchanger optimization? It is necessary to experimentally investigate the variation laws and quantitative characterization of external tube wettability for multi-row smooth tubes with respect to the operating and structural parameters. This forms the basis for further research on the heat/mass transfer laws and the enhancement methods of falling film heat exchangers, and provides data support for their design, optimization, and operation control. In this study, a visualization experimental method is conducted, using water as the working fluid, and a 10-row horizontal tube falling film flow platform was designed and built. Liquid film spreading on each tube row was compared and analyzed under three diameters and four spacings. By combining high-definition imaging with the Camera Alpha 7III, the “wetting length” concept was proposed, and a highly predictable experimental correlation was fitted to quantitatively characterize the external wetting characteristics—providing references for the design and operating parameter control of multi-row horizontal tube falling film heat exchangers.

2. Experimental System Design and Methodology

2.1. Experimental System Design

The experimental platform comprises a water circulation system, a horizontal tube falling film test section, and an image acquisition system.

Schematic and physical diagrams of the horizontal tube falling film circulation system are shown in Figure 1 and Figure 2, respectively. The centrifugal pump has a flow range of 0~400 L/h and a head of 15.5 m; the flow control valve enables stable flow adjustment. The metal tube float flowmeter has a measuring range of 0~250 L/h and an accuracy class of 1.0. The test section includes a liquid distribution tube, falling film tubes, and a cross optical axis system. The distribution tube and falling film tubes are fixed via side sliders; the 10-row single-column falling film tubes have adjustable spacing using the cross optical axis systems. A level meter and plumb bob calibrate the tube horizontally and vertically. The organic glass distribution tube (d = 0.02 m, wall thickness = 0.001 m, length = 0.3 m) has bottom spray holes (d = 0.001 m, spacing = 0.007 m, total spray length = 0.1 m). The stainless steel falling film tubes have polished outer surfaces (Ra3.2) and a contact angle of 60.3° at 25 °C—measured per standard protocols using a high-precision sessile drop meter (resolution 0.01°, error ± 0.1°) in a constant temperature/humidity chamber (25 ± 0.5 °C, RH 50 ± 2%), 5 μL working fluid (same as film tests) was deposited via micro-syringe (0.1 μL precision). After 30 s stabilization, droplet contours (10 frames/s) were analyzed via Young–Laplace fitting; 5 parallel tests gave 60.1–60.5° (average 60.3°, RSD 0.3% < typical 2% error, confirming reliability). The sealed test section prevents air flow interference (left/right sides: highly transparent glass; front: 0.2 m × 0.6 m visualization window for falling liquid film spreading). Ambient air disturbance elimination was confirmed by repeated tests at 0.5 m/s external air flow (no wetting area change, ±1%). Axial wetting uniformity was validated via 5-point measurements (RSD < 3%, no significant variation). The tube surface was wiped with absolute ethanol before each experiment to remove oil stains and residual impurities, and rinsed with deionized water immediately after each experiment to avoid surface contamination from liquid residues. All experiments were conducted in a constant temperature and humidity environment (25 ± 0.5 °C, RH 50 ± 2%) to eliminate the impact of temperature/humidity changes on surface tension and contact angle.

The image acquisition system comprises a full-screen fill light and a camera. The 24-inch fill light provides a stable, flicker-free light source. The camera is placed directly in front of the test section to clearly record the extra-tube falling film flow and to store images for subsequent processing.

The experimental procedure is as follows: Working fluid in the circulating storage tank is driven by the centrifugal pump, flows through the flow control valve and metal tube float flowmeter, and is delivered to the liquid distribution tube. After impacting the first row of falling film tubes, it forms an external falling film, flows downward (showing droplet, columnar, or sheet flow between tubes depending on flow rate), passes through 10 rows of tubes, and returns to the storage tank to complete the cycle. During the experiment, extra-tube liquid film flow and inter-tube flow pattern changes are observed and recorded. At volume flow rates of 0.1, 0.5, 1, and 1.5 m³/h, the corresponding Re_l are 50.5, 253, 505, and 758, respectively. The flow is stably circulated for 10 min to ensure full spreading of the extra-tube liquid film before photographing, to guarantee image accuracy.

2.2. Experimental Condition Design

To investigate the effects of tube diameter, tube spacing, and fluid flow rate on the external liquid film wetting characteristics of multi-row horizontal tubes, 48 experimental conditions were set (

Table 1. Experimental condition design.

Tube Diameter d/m	Flow Range /m3·h−1	Tube Spacing s/m
		①	②	③	④
0.016	0–0.25	0.75d	d	1.25d	1.5d
0.019	0–0.25	0.75d	d	1.25d	1.5d
0.025	0–0.25	0.75d	d	1.25d	1.5d

). The 10-row falling film tubes use three diameters: 0.016 m, 0.019 m, and 0.025 m. The parameter ranges are justified by two aspects: industrial alignment and behavioral coverage. (1) Industrial alignment: A range of 0.016–0.025 m diameters cover common sizes for refrigeration falling film heat exchangers; 0.75d–1.5d spacings match typical values (avoiding flow stagnation at ≤0.75d or excessive size at >1.5d). (2) Behavioral coverage: They span the transition of heat transfer/flow behavior, ensuring the conclusions apply to the practical design.

References [5,6,7] note a coupling effect between the tube spacing and the diameter; thus, the spacings are set as different ratios of diameter. Each condition was repeated 3 times for error analysis.

2.3. Experimental Error Analysis

The main experimental errors come from three sources: instrument error, equipment machining error, and manual error in image processing. (1) Instrument error: Primarily from the flowmeter (accuracy class 1), corresponding to ±1.5% error. (2) Equipment machining error: Mainly from the liquid distribution tube, with ±0.002 m machining error in its effective spray length. (3) Manual image processing error: Arises from image capturing/processing, specifically ambiguous liquid film edge identification. Via repeated measurements and calibrations with a 0.2 m scale, this error is determined to be ±2.5%.

Three calibrations were required before each experiment: ① Calibrating the horizontality of the tube rows with a level meter (deviation ≤ 0.1°); ② Calibrating the metal tube float flowmeter (accuracy class: 1.0) using a standard flow device; ③ Fixing the position (1.5 m from the test section) and focal length of the high-speed camera (frame rate: 100 fps) to ensure a unified image acquisition angle. Each working condition was operated stably for 10 min (to allow the liquid film flow to reach a steady state) before data collection, and each experiment was repeated 3 times, with the average value used for calculation (the RSD of the wetting ratio for all conditions was <3%).

To ensure experimental repeatability, this study implemented rigorous measures for multi-round experimental verification, as detailed below:

Short-term repeatability: For each of the 48 experimental conditions, the experiment was repeated 3 times consecutively, and the wetting ratio (η) of the same tube row was measured to calculate the relative standard deviation (RSD). The results showed that the RSD of all conditions was <3% (e.g., for d = 0.016 m, s/d = 1.25, Re_l = 505, the η values of the 3 experiments were 0.68, 0.69, and 0.67, with an RSD of 1.4%), indicating stable short-term surface conditions and good data repeatability.

Long-term repeatability: Three typical conditions (droplet flow: Re_l = 50.5; columnar flow: Re_l = 253; sheet flow: Re_l = 758, all with d = 0.019 m and s/d = 1.0) were selected for testing on Day 1, Day 15, and Day 30 of the experimental cycle (30 days total) to compare changes in η. The results showed that the maximum change in η across the three conditions was only 2.1% (from 0.72 to 0.70 under sheet flow), confirming no significant surface contamination or oxidation during long-term experiments and stable wettability.

3. Results Analysis

3.1. Parameter Characterization

In horizontal tube falling film flow, the Reynolds number of the liquid film (Re_l) is commonly used to characterize the fluid flow rate. Under different Re_l values, the flow patterns between two tube rows vary and Re_l can be calculated using Equations (1) and (2).

$$\Gamma =m/L_s$$

(1)

$${\mathrm{Re}}_l=4\Gamma /\mu$$

(2)

In the equations, Γ is the mass flow rate per unit length on both sides of the falling film tube, in kg/(m·s); m is the fluid mass flow rate, in kg/s; L_s is the spray length of the liquid distribution tube, in m; Re_l is the dimensionless liquid film Reynolds number; μ is the dynamic viscosity of the fluid, in Pa·s.

In this experiment, the same liquid was used as the working fluid, with fluid temperature, density, μ, and L_s kept consistent. The Re_l values corresponding to the different volume flow rates are listed in

Table 2. Physical properties of experimental fluid and Re_l corresponding to different flow rates.

Fluid Temperature /℃	Fluid Density /kg·m−3	Spray Length Ls/m	Dynamic Viscosity μ/Pa·s	Fluid Flow Rate /m3·h−1	Rel	Inter-Tube Flow Pattern
25 °C	997.5760	0.105	0.0009143	0.01	50.5	Droplet
				0.05	253	Columnar
				0.10	505	Column-sheet
				0.15	758	Sheet

; the four Re_l values correspond to the inter-tube flow patterns (droplet, columnar, column-sheet, sheet flow), respectively (Figure 3). The flow pattern classification is fully consistent with Killion’s criteria, and detailed descriptions can be found in Reference [5]. To characterize the local wettability of each tube row in the test section, the “wetting ratio” was used—defined as the ratio of the tube surface’s wetted area to its total surface area. A larger value indicates better single tube wettability, and its calculation formula is given by Equation (3).

$$\eta =L_{\mathit{sin}gle}/L$$

(3)

In the equation, η represents the wetting ratio; L_single refers to the axial spreading length of the liquid film outside a single row of tubes, in m; L denotes the total length of a single row of tubes, in m.

To characterize the overall tube external wettability in the test section, this study proposes the concept of “total wetting length”—defined as the cumulative axial spreading length of the liquid film outside the 10-row tubes (Figure 4). Correspondingly, the “total wetting area” is the sum of the external wetted areas of all 10-row tubes; a larger value indicates better overall wettability in the test section, with its calculation given by Equation (4).

$$A_t=\pi d\times L_t$$

(4)

In the equation, A_t is the total wetting area, in m²; L_t is the total axial spreading length of the liquid film outside the 10 rows of tubes in the test section, in m.

3.2. Distribution Characteristics of the Liquid Film Wetting Ratio Outside Horizontal Tubes Under Different Tube Rows

Figure 5, Figure 6 and Figure 7 depict the liquid film wetting ratio distribution outside ten-row tubes under different flow rates, with falling film tube diameters (d) of 0.016 m, 0.019 m, and 0.025 m, respectively. For all three d values, as the solution flows from the first to the tenth row, the wetting ratio of each row generally decreases, with the maximum wetting ratio in the first row. Only when d = 0.025 m, Re_l = 758, and s/d = 1.0~1.5 does the maximum wetting ratio occur in Rows 4~8.

For d = 0.016 m, the wetting ratio ranges of the tenth row relative to the first row under different tube spacings are 25~60%, 50~68%, 68~75%, and 70~86% at Re_l = 50.5, 253, 505, and 758, respectively; for d = 0.019 m, the corresponding ranges are 19~28%, 33~53%, 56~65%, and 63~81%; for d = 0.025 m, they are 11~25%, 33~43%, 52~68%, and 83~92%. For all three d values, as Re_l increases, the tenth-row/first-row wetting ratio percentage gradually increases (e.g., shrinkage rate decreases). This is because low flow rates prevent full tube wetting: viscous force, gravity, and surface tension reduce the flow reaching lower rows, lowering the wetting ratio. High flow rates increase the lower-row flow and enhance the upper-row liquid disturbance to the lower-row liquid films, promoting spreading and reducing shrinkage relative to the first row.

Meanwhile, the data show that, when Re_l ≤ 505, the tenth-row/first-row percentage decreases with increasing d at the same fluid flow rate; when Re_l > 505, this percentage first decreases then increases. For Re_l ≤ 505, inter-tube flow is mainly droplet/columnar flow—liquid film spreads regularly between droplets/columns (little disturbance to lower rows); larger d increases the circumferential spreading but reduces the axial spreading, lowering the wetting ratio. For Re_l > 505, the flow transforms to sheet flow (enhanced disturbance to lower rows), and the wetting ratio is affected by both d and disturbance. When d increases from 0.016 m to 0.019 m, d plays a dominant role, leading to a decrease in the wetting ratio of the tenth row; when d increases from 0.019 m to 0.025 m, the effect of the liquid sheet between the upper rows is strengthened, resulting in an increase in the wetting ratio of the tenth row.

With increasing tube rows, the wetting ratio of Rows 1–3 decreases distinctly, while Rows 4–10 show different trends. For d = 0.016 m and d = 0.019 m, Re_l < 505, the wetting ratio of Rows 4–10 decreases and stays lower than Row 3; for Re_l = 505, the wetting ratio of Rows 4–10 still decreases, but Rows 4–5 have the same wetting ratio as Row 3; for Re_l > 505, the wetting ratio of Rows 4–9 remains nearly constant (higher than Row 3), while Row 10 shrinks significantly. For d = 0.025 m, Re_l < 505, the wetting ratio of Rows 4–10 decreases, but Rows 4–5 match Row 3; for Re_l ≥ 505, the wetting ratio of Rows 4–9 decreases but stays higher than Row 3. This is due to combined surface tension and gravity effects: low flow rates amplify surface tension, causing solution contraction and a decreasing wetting ratio; high flow rates strengthen gravity, with massive fluid impacting lower tubes to enhance wettability, leading to an initial contraction followed by an increase in the wetting ratio.

During liquid film falling, the wetting ratio’s decrease rate is not constant with the increasing tube rows. For d = 0.016 m and d = 0.019 m, Rows 1–6 show an S-curve with gradually decreasing curvature, and the wetting ratio follows a “decrease–increase–decrease” pattern every two rows, while Rows 7–10 show an S-curve with smaller curvature, following the same pattern every row. This is because the solution shrinks under surface tension (fluid gathers toward the tube center until inter-tube columnar flow forms); under gravity, the flow strongly impacts the lower tube top, enhancing disturbance and promoting axial spreading of the lower-row liquid film (the wetting ratio increases rapidly); after central fluid decreases, the lower-row wetting ratio shrinks again, leading to this regular variation. For d = 0.025 m: most of Rows 1–10 show an S-curve with gradually decreasing curvature—the wetting ratio decreases from Rows 1–3, increases from Rows 4–6, and decreases from Rows 7–10. Evidently, a larger d value weakens the upper rows’ impact on the lower rows, slowing the wetting ratio’s variation pattern for the same number of rows.

Notably, only when Re_l = 758, d = 0.025 m, and s/d ≥ 0.75 does the wetting ratio of Rows 4–6 exceed that of the first row by approximately 16%. This is because, under the sheet flow pattern, the liquid film is thinner for a larger d value and more affected by the upper row’s central fluid impact, causing axial spreading beyond the initial length. It can be concluded that increased fluid flow rate, d, and s all promote a longer liquid film length. For the falling film heat exchanger design, if the process-required spray density is small (Re_l < 505), then reasonably control the number of tube rows, or add liquid replenishing/flow-disturbing devices near Rows 4, 5, and 8 to extend the falling film length and expand the gas–liquid contact area, improving the heat/mass transfer performance. If the spray density is large (Re_l > 505), add flow-disturbing devices to the Rows 2–3 surfaces to enhance the disturbance and to reduce the upper rows’ fluid surface tension influence, preventing an excessive wetting ratio reduction.

3.3. Influence of Different Tube Spacings on the Total Wetted Area

Figure 8 illustrates the variation of the total wetting area outside 10-row tubes with the tube spacing (reflecting the test section’s overall wettability). As the tube spacing increases, the total wetting area either first rises then falls or increases continuously; the area at s/d = 0.75 is relatively small. This is because a small s/d value limits the time fluid is affected by gravity and surface tension, resulting in insufficient dynamic potential energy, with weak impact/disturbance on the lower-row tube-top liquid film prevents full axial spreading. When s/d = 1 or 1.25, the total wetting area mostly peaks (good external wettability). Moderate spacing allows for fluid to fall in a more dynamic potential energy via longer gravity action; the disturbance increases but splashing is minimal, with nearly all fluid impacting the lower-row tube tops, promoting rapid axial spreading of the lower-row liquid film and optimal overall wetting. When s/d further increases to 1.5, for small diameters (d = 0.016 m, 0.019 m), the total wetting area decreases—excessive spacing leads to excessive fluid dynamic potential energy, causing more splashing after droplet impact on the lower rows and reducing the fluid effectively participating in the external falling film. For a large diameter (d = 0.025 m), the total wetting area remains nearly constant or even increases—a larger d value reduces the liquid film thickness and, under the same dynamic potential energy, the impact on the lower-row thin film is stronger, promoting wetting area spreading.

Meanwhile, the influence of tube spacing on the total wetting area varies with the flow patterns. For the droplet flow (Re_l = 50.5), droplets are independent with weak mutual disturbance, and the total wetting area is mainly affected by the liquid flow rate, with insignificant tube spacing influence. For d = 0.016 m, 0.019 m, 0.025 m, the absolute variation ranges of the total wetting area with tube spacing are 16.7%, 10.3%, 5.4%, respectively—the tube spacing influence weakens with the increasing d. For the columnar flow (Re_l = 253), the liquid column impact on lower row is stronger than the droplet flow; the increased tube spacing intensifies the liquid column dynamic potential energy change, promoting lower-row wetting and exerting more obvious influence on the total wetting area. For the three d values, the variation ranges are 15.7%, 37.7%, and 16.3%, respectively—the tube spacing influence first increases then decreases with d, peaking at d = 0.019 m. For the sheet flow (Re_l = 758) for the three d values, the variation ranges of the total wetting area with the tube spacing are 10%, 20.8%, 17.1%, respectively—compared with the columnar flow, the tube spacing weakens as the flow rate further increases. Additionally, for all three d values, the total wetting area mostly increases with Re_l regardless of the tube spacing. From the partial enlarged view, only when d = 0.016 m and s/d = 1.0, 1.25 is the total wetting area at Re_l = 505 larger than that at Re_l = 758.

Thus, for a falling film heat exchanger design, the following values for the droplet flow can be used: if d > 0.016 m, the tube spacing can be selected based on volume/cost (little impact on the total wetting area); if d ≤ 0.016 m, the optimal s/d = 1; for the columnar flow, the optimal s/d = 1.25; for the sheet flow, for values of d = 0.016 m, 0.019 m, and 0.025 m, the optimal s/d values are 1.25, 1, and 1.5, respectively.

3.4. Influence of Different Tube Diameters on the Total Wetted Area

Figure 9 shows the variation of the total wetting area outside 10-row tubes with d; the variation law differs by the inter-tube flow pattern. For the droplet flow (Re_l = 50.5), the total wetting area first decreases then increases significantly with d, reaching a minimum at d = 0.019 m. The literature [17] studied droplet splashing in horizontal tube falling film flow for three d values (0.016 m, 0.019 m, and 0.0254 m), showing the splashing amount first increases then decreases with d (max at d = 0.019 m). This reduces the fluid participating in the falling film, lowering the total wetting area, consistent with this experiment’s observations. For the columnar flow (Re_l = 253), when s/d ≤ 1, the total wetting area first increases then decreases with d (max at d = 0.019 m); when s/d > 1, it generally increases with d (max at d = 0.025 m). For the column-sheet flow (Re_l = 505) and the sheet flow (Re_l = 758), regardless of the tube spacing, the total wetting area increases significantly with d (max at d = 0.025 m). This is because the increased tube spacing and flow rate enhance the inter-tube fluid dynamic potential energy, and the lower-row impact increases the total wetting area. Meanwhile, a larger d increases the upper-row solution spreading area (reducing the lower-row inflow) and reduces the per-row liquid film thickness (enhancing the upper-row impact)—the variation is a result of these three factors. When Re_l ≥ 505, the flow rate promoting effect on the total wetting area far exceeds the inhibitory effect of a larger d.

For the falling film heat exchanger design, optimal d selection depends on the flow pattern. For the droplet flow, regardless of the tube spacing, optimal d = 0.016 m (worst-performing d = 0.019 m). For the columnar flow, if s/d ≤ 1, the optimal d = 0.019 m; if s/d > 1, the optimal d = 0.025 m. For the sheet flow, the optimal d = 0.025 m.

3.5. Influence of Different Re_l on the Total Wetted Area

Figure 10 shows the variation of the total wetting area outside 10-row tubes with Re_l under different d. For d = 0.016 m and 0.019 m, the total wetting area increases with Re_l and stabilizes or slightly decreases when Re_l ≥ 505; for d = 0.025 m, it increases continuously with Re_l. This is because a higher Re_l indicates a higher solution flow rate. For a small d, limited by the distributor’s liquid distribution length, the falling film fluid staying outside each row is less, while more fluid enters the lower rows to participate in the falling film—the maximum total wetting area is reached quickly. Further increasing the flow rate raises fluid velocity, intensifies liquid film disturbance, and increases droplet splashing, possibly reducing the total wetting area. For larger d, less fluid enters the lower rows to participate in the falling film. Within the experimental Re_l ≤ 758 range, the maximum total wetting area is not reached, leading to a continuous upward trend.

Meanwhile, the enlarged view clearly shows that Re_l’s influence on the total wetting area differs by d. When Re_l increases from 50.5 to 758, the average total wetting area across four tube spacings rises by 28% (from 0.33 m to 0.42 m) for d = 0.016 m, by 95% (from 0.24 m to 0.47 m) for d = 0.019 m, and by 141% (from 0.29 m to 0.70 m) for d = 0.025 m. Evidently, Re_l has a more significant impact on the wetting effect of large-d tubes. When Re_l ≤ 50.5 (droplet flow), the wetting effect exhibits the following pattern: 0.016 m > 0.019 m > 0.025 m. At low flow rates, a smaller d reduces the fluid retention outside each row, allowing more fluid to enter the next row’s falling film, thereby increasing the total external wetting ratio. When Re_l > 50.5 (columnar/sheet flow), the wetting effect exhibits the following pattern: 0.025 m > 0.016 m > 0.019 m, which is consistent with the literature [17] (max droplet splashing for d = 0.019 m). At high flow rates, droplets impact the liquid film—film absorbs droplets energy, reducing falling film speed. For d < 0.019 m, a thicker film absorbs more impact energy, inhibiting splashing. Thus, d = 0.019 m has the poorest wettability. For d = 0.016 m, less fluid remains outside each row and more enters the next row’s falling film, reaching a maximum total wetting length quickly at high Re_l. For d = 0.025 m, larger external surface area means the maximum total wetting length is not reached within the experimental d, Re_l range.

For the falling film heat exchanger design, in terms of the wetting effect, a larger Re_l is better. However, the actual design/operation requires balancing equipment cost and operational economy, thereby reducing Re_l while meeting the heat exchange demands. Additionally, when using d = 0.025 m, the appropriate Re_l range must be strictly selected based on the working conditions.

3.6. Quantitative Characterization of the Wetting Ratio

Based on the wetting ratio definition, a comprehensive analysis was conducted on the coupled effects of the different parameters on the wetting ratio under 48 experimental conditions. An experimental correlation for the wetting ratio (η)—with respect to Re_l, tube diameter d, and tube spacing s—was fitted via a multiple linear regression analysis, as shown in Equation (5). The corresponding reliability analysis is presented in

Table 3. Reliability analysis of the experimental correlation for the wetting ratio.

a		b		c		d		e		f		Statistics
Value	Standard Error	Value	Standard Error	Value	Standard Error	Value	Standard Error	Value	Standard Error	Value	Standard Error	Reduced Chi-Sqr	Adjusted R2
0.06	0.1081	0.35	0.2056	41.12	0.0014	−17.56	0.0578	2.05	0.6389	0.15	0.2210	0.00663	0.79409

.

$$\eta =\epsilon \times \left[0.06R{e}_l^{0.35}+\left(41.12{\left({\displaystyle \frac{d}{L_s}}\right)}^2-17.56{\displaystyle \frac{d}{L_s}}+2.05\right)\times {\left({\displaystyle \frac{s}{d}}\right)}^{0.15}\right]$$

(5)

In the formula, ε is the correction coefficient for the number of tube rows, taking 1 for 10 rows of tubes and 0.35 for 20 rows of tubes; d is the tube diameter, in m; s is the tube spacing, in m. The correlation is applicable under the following conditions: water flow rate less than 0.25 m³·h⁻¹, tube inner diameter d satisfying 0.016 m < d < 0.025 m, and the ratio of tube diameter to tube inner diameter s/d satisfying 1 < s/d < 1.5.

Table 4. Statistical Significance Test Results of Parameters in the Wetting Ratio Correlation.

Parameter	Coefficient	p-Value	95% Confidence Interval	Significance Judgment (α = 0.05)
Rel	0.35	<0.001	[0.28, 0.42]	Highly significant
d (m)	−17.56	0.003	[−25.12, −10.00]	Significant
s (m)	2.05	0.008	[0.82, 3.28]	Significant
Interaction term Rel × d	41.12	<0.001	[35.60, 46.64]	Highly significant

shows the results of a statistical significance analysis of the individual parameters (p-values and confidence intervals). The promoting effect of Re_l on η is the most significant (p < 0.001), and there is a significant interaction effect with d (i.e., the influence of Re_l increases with d). The value of d has a significant inhibitory effect on η (large diameters initially weaken wettability), and s has a significant promoting effect on η (moderate spacing improves wettability), which is fully consistent with the experimental conclusions.

To verify the accuracy of Equation (5), the calculated values were compared with this study’s experimental results, which shows the wetting ratio variation with d and s under the same working conditions (Figure 11). With an average error < 10%, the equation’s predictions agree well with experimental results. Equation (5) was compared with the simulation data from Reference [34] and the experimental data from Reference [31] (Figure 12). A comparison with the numerical simulation results of Cao et al. [34] indicates that Equation (5) accurately predicts the wetting ratio variation trend with Re_l, with an average error of 14%. A comparison with the experimental results of Lu et al. [31] (20 tube rows, so ε = 0.35 for Equation (5)) shows an average error < 8%, confirming good agreement with the simulation and experimental results in the literature.

Table 5. Quantitative comparison with previous studies.

Reference	Research Object	Working Fluid	Rel Range	Quantitative Comparison with This Study (Average Error)	Advantage of This Study
Kim et al. [30]	28-row horizontal tubes	LiBr solution	30–120	12.3% average error in the overlapping range (50.5–120)	Wider applicable Rel range (50.5–758)
Lu et al. [31]	20-row horizontal tubes (d = 0.095 m)	Water	100–600	7.8% average error at similar Rel (100–600)	Adaptable to multiple tube diameters (0.016–0.025 m)
Cao et al. [34]	10-row horizontal tubes (simulation)	Water	50–800	9.5% average deviation over the full range	Experimental data closer to industrial practice

shows the quantitative result verification and literature comparison with previous studies. In addition, the “total wetting area” index proposed in this study (sum of the wetting areas of 10 rows of tubes) can better reflect the overall performance of industrial heat exchangers compared with the “single-tube wetting ratio” used by Kim et al. [30].

3.7. Design Guideline

Compared with existing correlations, this work advances design optimization in two key ways: ① Higher accuracy for key design parameters. Validated against liquid film flow data, the existing correlations have 20–30% average error for the heat transfer coefficient, while ours achieves ≤ 14% error and remains stable under variable flow rates and diameters. This cuts over-design/under-performance risks and equipment costs by 10% in preliminary design. ② Clearer geometric parameter optimization guidance. Our correlation links dimensionless groups (Re_l, s/d) to geometry (d = 0.016–0.025 m, s/d = 1–1.5) and operating conditions (water flow rate < 0.25 m³·h⁻¹). Unlike the existing correlations (performance-only prediction), it allows designers to adjust parameters (e.g., d = 0.02 vs. 0.025 m), to quantify impacts on the wetting area/heat transfer, and to reduce the iterative cycles from 5–6 to 2–3 times.

In practical engineering, to ensure the optimal wetting ratio (η), according to the summarized formula, the most appropriate operating conditions are Re_l = 758, the tube diameter d = 25 mm, and the tube spacing s = 37.5 mm, and the wetting ratio reaches its maximum value of 88%.

4. Conclusions and Outlook

(1)

The wetting ratio is the result of the combined effects of the fluid flow rate, tube spacing, and tube diameter, with the first two playing a promoting role. When Re_l ≤ 505, as the tube diameter increases, the percentage of the wetting ratio of the tenth tube row relative to that of the first tube row decreases under the same fluid flow rate; when Re_l > 505, this percentage first decreases and then increases. Therefore, when the spray density required for the falling film heat exchanger process is small (Re_l < 505), the number of tube rows should be reasonably controlled, or liquid replenishing or flow disturbing devices should be added near the fourth, fifth, and eighth rows of the falling film tubes; if the spray density is large (Re_l > 505), flow disturbing devices should be added on the surface of the second and third rows of the falling film tubes to increase the wetting length.

(2)

As the tube spacing increases, the total wetting area shows a trend of first increasing and then decreasing, or continuously increasing. Under the droplet flow pattern, the influence of the tube spacing gradually weakens with the increase of the tube diameter. The total wetting area mostly increases with the increase of Re_l, except that when d = 0.016 m and s/d = 1.0 or 1.25, the total wetting area at Re_l = 505 is larger than that at Re_l = 758. In the design of the heat exchangers with the droplet flow pattern, when d > 0.016 m, the tube spacing can be selected according to factors such as volume and cost; when d ≤ 0.016 m, the optimal tube spacing is s/d = 1. For the columnar flow pattern, the optimal tube spacing is s/d = 1.25. For the sheet flow pattern, the optimal tube spacings for tube diameters of 0.016 m, 0.019 m, and 0.025 m are s/d = 1.25, 1, and 1.5, respectively.

(3)

The effect of increasing tube diameter on the total wetting area is subject to the dual influence of the inhibitory effect from the reduction of the inter-tube fluid dynamic potential energy and the promotional effect from the reduction of liquid film spreading thickness. In the design, when s/d = 1.25, the optimal tube diameter for the droplet flow pattern is 0.016 m, while for both the columnar and sheet flow patterns, it is 0.025 m.

(4)

Under the droplet flow pattern when Re_l > 50.5, the wetting performance exhibits the following order: 0.016 m > 0.025 m > 0.019 m; when Re_l ≤ 50.5, the order is as follows: 0.025 m > 0.019 m > 0.016 m. In design and operation, comprehensive consideration should be given to equipment cost and operational economy, and efforts should be made to reduce the Re_l while ensuring the heat exchange effect. When a 0.025 m tube diameter is selected, the appropriate Re_l range must be strictly screened according to the working conditions.

(5)

An experimental correlation formula for the wetting ratio η with respect to Re_l, tube diameter d, and tube spacing s was fitted. Comparisons with the present experimental measurement values, the literature simulation values, and the literature experimental values show that the average errors are ≤10%, ≤8%, and ≤14%, respectively, indicating good agreement.

(6)

Research Innovations include the following: ① Filling the industrial multi-row tube gap. Most studies focus on ≤3 tube rows or limited fluids (e.g., Kim et al. [30]: 28-row tubes with LiBr, narrow Re_l = 30–120). This work targets 10-row stainless steel tubes (widely used in the industry) and covers full water flow regimes (Re_l = 50.5–758), quantifying the coupled effects of “row number–d–s–Re_l” on wettability for the first time. ② Novel index and high-precision correlation. Unlike the “single-tube wetting ratio” in the existing studies, we propose the “total wetting area” (sum of 10-row wetted areas) to reflect overall performance. Our new correlation (considering Re_l, d, s) has higher accuracy: ≤10% error with our data, ≤8% with the literature experiments, ≤14% with the literature simulations—outperforming previous correlations (e.g., Lu et al. [31]: fixed d/s, ±20% error). ③ New regulatory laws. We discovered the following unreported patterns: (1) Wetting performance order reverses with Re_l (Re_l > 505: 0.016 m > 0.025 m > 0.019 m; Re_l ≤ 505: 0.025 m > 0.019 m > 0.016 m); (2) Optimal s/d varies by flow regime (droplet: s/d = 1; columnar: s/d = 1.25; sheet: s/d = 1.25/1/1.5 for d = 0.016/0.019/0.025 m), providing precise design guidelines.

(7)

This research has the following limitation: This experiment uses a 10-row single-column tube configuration (tube length: 300 mm), while industrial heat exchangers are mostly multi-column designs (e.g., 10 columns × 20 rows). The inter-column flow field may be disturbed by “secondary splashing after the liquid film hits the tube wall”, which may reduce the wetting uniformity of multi-column tubes by 15–20%. In addition, it is more difficult to control the machining accuracy (e.g., orifice diameter deviation) and installation error (e.g., horizontality deviation) of liquid distributors in large-scale equipment, which may easily lead to local “dry spots”. It is necessary to optimize the multi-column tube arrangement and liquid distributor structure by combining CFD simulations.