Influence of the Static Contact Angle on the Liquid Film Coverage for Falling-Film Systems

The coverage of the liquid film over the horizontal tubes, particularly the wetting ratio, is important for gravity-driven evaporators and absorbers to achieve a better heat transfer mechanism. A 2D, two-phase CFD model was developed to examine the falling-film hydrodynamics and transient film coverage for Reynolds numbers ranging from 100 to 500, static contact angles spanning from 0°/30°/60°/90°, and tube-to-tube distance of 10 mm. The VOF method is used in this article to capture the liquid–gas interface. The findings showed that the complete spreading of the liquid film is difficult at low Reynolds numbers and high contact angles. The formation of dry regions on the tube wall as a result of insufficient liquid supply, liquid film breakage, and liquid film shrinkage. Furthermore, as the Reynolds number increases and contact angle decrease, the wetting ratio over the tube surface increases. It is worth noting that each contact angle must have a minimum Reynolds number in order to keep the surface completely wet. The research also revealed that for higher Reynolds numbers, the influence of contact angle on wettability of tube wall can be ignored. For the same Reynolds number, the liquid propagation time required to wet the tube surface increases as the contact angle value increases.


Introduction
Nowadays, most of industrial heat is wasted in to the environment without being properly utilized especially at the coastal areas (Zhang and Akiyama 2009).Economic and environmental factors constitute the primary reasons for a strong intention to improve thermal systems efficiency, which is primarily initiated by improving operating performance of these systems.Furthermore a higher heat transfer coefficient is expected in phase change heat transfer applications such as absorption, condensation and evaporation related areas.Horizontal falling-film systems have numerous applications in industry and scientific research.Horizontal tube array type falling-film heat exchangers have been actively utilized in process industries such as food industries (Cyklis 2017), desalination (Mabrouk and Fath 2015;Harandi et al. 2021), refrigeration (Narváez-Romo et al. 2017;Hong et al. 2019), chemical, and petroleum refineries (Sun et al. 2021) and natural cooling methods (Tenorio Ríos 2020).The functional advantage of the gravity driven falling-film heat exchangers over the flooded heat exchangers makes them a better choice (Karmakar and Acharya 2020;Fernández-Seara and Pardiñas 2014).Because of the improvement in the effective use of low-grade waste heat energy and the minimum operating temperature difference, these investigations have drawn the attention of many academics and researchers in this field.
In falling-film evaporators (FFE), the liquid exits the series of distributor holes in the form of a droplet, column, sheet, or a combination of droplet-column and column-sheet flow pattern and falls onto the tube surface, where it begins to spread across the tube wall.The liquid film flows downward from one tube to the subsequent tube under predefined operating conditions and the action of gravity.The liquid film hydrodynamics and transient film coverage are important considerations in gravity-driven falling-film systems for the development of a thin liquid profile.
The liquid film profile over the tube wall is the primary thermal energy carrier, and it is highly reliant on the wettability parameter to achieve the desired heat transfer and operating performance.Despite the numerous benefits listed above, gravity-powered FFEs have major downsides such as poor liquid film distribution, the formation of dry spots and flooding, mal-distribution and waste of fluid and energy (Jiang et al. 2018).All of the preceding factors are inextricably linked to the heat transfer mechanism (Yan et al. 2019).Dry spots on the tube walls, as well as uneven liquid film coverage, have all contributed to a significant deterioration in the evaporation process.According to the studies, the flow hydrodynamics, film coverage and flow characteristics are all directly connected to the wettability parameter.Thus, it is important to investigate microscopic flow mechanism, film coverage and flow hydrodynamics in order to improve flow characteristics and better design of FFEs.
The present study provides insight into the hydrodynamic behavior of the falling-film using an established 2D model, allowing us to extrapolate it to other application areas.

State of Art
The existing literature has revealed the following primary flow modes between horizontal tubes, including the droplet mode, column flow pattern, sheet flow mode and intermediate flow regimes are droplet-column mode, column-sheet mode (Hu and Jacobi 1996;Armbruster and Mitrovic 1994).Chen et al. (Chen et al. 2015a) investigated computationally and experimentally to examine the different flow regime transformations between the horizontal tubes.Kandukuri et al. (Kandukuri et al. 2022a) experimentally examined the various phases in the column flow regime as well as flow parameters such as axial film thickness and jet diameter using image analysis approach.
The distributor type, distributor height, and orifice spacing all had a strong influence on the inter tube flow regimes and their characteristics (Wang et al. 2013;Qu et al. 2019).Mohamed (Mohamed 2007) conducted experiments to investigate the effect of the test tube rotational speed on flow pattern transformations.The flow transformation begins at a lower Reynolds number (Re), when the falling liquid comes into contact with a horizontal spinning tube.
Nusselt first proposed falling film theory, assuming a sheet flow mode from one tube to the next.He proposed the undermentioned expression to find the liquid film thickness for different radial angles (Nusselt 1916).
The film Reynolds number can be expressed as follows, Hou et al. (Hou et al. 2012) used a displacement micrometer to measure the circumferential film thickness.All experiments were conducted with varying peripheral angles ranging from 15° to 165°, outside tube diameters ranging from 20 to 32 mm, and different tube spacing.The liquid film thickness was found to be the thinnest between 90° and 115°.They developed the following correlation to assess the thickness of the liquid profile for different peripheral angles based on Nusselt's Eq. ( 1).The values of coefficient (C) and exponent (n) are given in the circumferential angle ranges 0° < ≤ 90° and 90° < ≤ 180°.Wang et al. (2019) developed a numerical model to examine the insight details of the column flow regime.The study also explored the effect of Re varying from 221 to 295 and tube spacing ranging from 10 to 30 mm on falling-film flow mechanism.Based on the simulation results, the following correlation is established to quantify the peripheral distribution of film thickness.Maliackal et al. (2022a) devised a novel optical shadow method for measuring circumferential film thickness for fully wetted horizontal copper tube in the 5°-175° range.They also developed a Computational fluid dynamics (CFD) model to examine film coverage, and the results agreed well with the experimental results.Furthermore, different techniques have been employed in the existing literature to estimate the film thickness, such as the optical-electronic method (Zhang et al. 2000), double-fiber, optical probe (Zaitsev et al. 2003), laser-induced fluorescence technology (Chen et al. 2015b), laser confocal displacement meter (Han et al. 2015), and air-coupled ultrasonic transducer (Jayakumar et al. 2019).Qiu et al. (2015) performed numerical simulations to explore variation in radial film thickness on the surface of a fully wetted tube.It was revealed that a distinct visible liquid-free zone forms extremely close to the lower stagnation region.As the Re number increases, so does the size of the zone.Zhao et al. (2018) investigated numerically the propagation of liquid film over horizontal tube walls.It was revealed that the film thickness on the tube surface increased with Re while decreasing with increase in liquid temperature, tube diameter, and liquid sprayer height.Han et al. (2017) i model to study the falling-film flow mechanism over the tube surface for different tilting and sloshing conditions.Both the tilting angle of the tube and the sloshing conditions influenced the liquid film coverage.de Arroiabe et al. ( 2018) carried out simulations to analyze the flow mechanism and to estimate the wetted area in the LiBr-H 2 O absorber.It was observed that each Re has a maximal static contact angle to continuously wetting the entire tube surface.Ji et al. (2017) implemented a 2D numerical model to study the variation in film thickness with the Re and contact angle.Ding et al. (2018) investigated numerically to explore the spreading of droplets and liquid jet regimes over the tube surface.The contact angle influences the wettability factor and liquid profile coverage for the droplet mode and column flow regimes.Maliackal et al. (2022b) explored the liquid film hydrodynamics and film formation on copper tubes, thermal spray coated tubes, and metal foam wrapped tubes using experiments and a 2D CFD model.The research also revealed variations in heat transfer mechanisms among coated, metal foam wrapped tubes.Tahir et al. (2020) implemented a 2D numerical model to analyze the impact of test liquid viscosity and surface tension on flow hydrodynamics.An increase of 72% rise in film thickness is observed with the fluid properties.Kandukuri et al. (2022b) established a 2D CFD model to visualize transient film coverage for Re from 100 to 1000.For the same tube-to-tube distance, the inter tube flow structures are strongly influenced by Re, and at higher Re, air voids can be noticed beneath the tube wall.

Constraints in the Literature Survey
Based on the aforementioned literature review, some research has been done on horizontal tube falling-film.Numerous experimental and simulation studies have been published in the open literature that focused on various factors such as inter tube flow patterns, tube configuration, distributor type, flow rate of a liquid/gas phase, falling liquid type, and fluid properties.The majority of numerical studies were performed on single tube to comprehend the falling-film distribution over the tube wall.However, substantial numerical simulations on a two-tube design model are still needed for further exploration due to a paucity of study.Despite the fact that several studies in the open literature have been reported the flow distribution and transient flow characteristics for different operating parameters.However, some important issues, such as the effect of static contact angle on falling-film hydrodynamics and wettability characteristics are not adequately addressed.Furthermore, the numerous studies considered in the literature did not thoroughly quantify the finer details of flow parameters such as film coverage, relevant information on the development of peripheral dry spots, and data on inter tube flow structures to the best of authors understanding.The design and functioning of FFEs, involve careful consideration in order to prevent flooding or poor wetting phenomena over the tube surface and to improve flow characteristics.

Focus of the Present Study
Experiments make it difficult to quantify the finer details of flow parameters, such as transient flow mechanism, liquid profile coverage, and tube wettability for different peripheral angles.Because of the challenges in carrying out experimental work for various parametric conditions, numerical simulations play an important role in the study of falling-film flow behavior.It is easier to examine the falling-film flow mechanism with the help of CFD software for different operating conditions and distinct instances.A 2D numerical model was established to computationally investigate the flow mechanism over the tubes.The geometric VOF method (Hirt and Nichols 1981) is employed in this article along with the Continuum Surface Force (CSF) approach (Brackbill et al. 1992).The present work looks at the systematic research on the effect of static contact angle (0° ≤ Ψ ≤ 90°) and Re (100 ≤ Re ≤ 500) on flow characteristics.The flow characteristics such as liquid film hydrodynamics, circumferential film coverage, liquid film spreading at upper and lower stagnation zones, and including wettability of the falling liquid over the tube walls were meticulously elucidated.Furthermore, inter tube flow structures and peripheral dry regions also presented and discussed in detail.

Geometry Analysis and Boundary Conditions
The half domain symmetrical geometry is generated using the ANSYS Fluent software, as depicted in Fig. 1.Because the problem and geometry is symmetric, only half of the solution domain is designated for ANSYS CFD calculations to save computing time.The width of the solution domain and the diameter of the inline horizontal tubes are both 31 mm.The liquid feeder height (H) is 2 mm for all domains, tube spacing (S) is 10 mm and a 1 mm liquid phase inlet orifice is provided at the top.The first tube near the liquid distributor is referred to as the stabilizing tube, and the second tube is referred to as the test tube.The liquid feeder height is kept at 2 mm from the stabilizing tube  to ensure uniform distribution of droplets or liquid film over the tubes by minimizing impact force (Liu et al. 2019).
The inlet water flow boundary is shown on the top surface in Fig. 2. It was designed as a velocity inlet for the liquid phase, with the selected fluid flowing downward.The remaining section on the top side was labeled as inlet pressure, while the section on the bottom side was labeled as outlet pressure.The model front and right sides have symmetrical boundaries.Table 1 outlines the properties of the test liquid.

Fluid Properties and Assumptions
The following are the assumptions made in this study to ease of the numerical analysis a) The liquid is incompressible.b) The surface of the tube is completely wetted with a liquid.c) All the properties of the fluid are constant.d) The surface of the horizontal circular tube is smooth and the effect of surface roughness is remiss.

Governing Equations
Hirt and Nichols (Hirt and Nichols 1981) implemented and proposed the VOF technique for accurately capturing the interface between the gas-liquid phases.In this study, the Pressure Implicit with Splitting of Operator (PISO) algorithm and the Geo-Reconstruct scheme are used.The working fluid is incompressible, and the fluid flow is transient and laminar.The governing equations for the mass, momentum and volume fraction can be defined as follows, Mass equation: The CSF model (Brackbill et al. 1992) is utilized in this study to add surface tension force to the momentum equation.In the CSF model the surface tension force term is as follows, The VOF method employs a phase indicator function, also known as a color function, to track the interface between two or more phases.When a control volume is completely filled with one of the phases, the indicator function returns a value of 1 or 0; otherwise, it returns a value between 1 and 0. As a matter of fact, the phase indicator function has volume fraction properties.In the VOF approach, if the volume fraction of the liquid phase is 1, then the cell belongs to liquid phase, for water volume fraction contours.The CSF method is implemented in this analysis to effectively characterize the gas-liquid phase interface, where surface tension impacts cannot be ignored.To account for the influence of the tube wall property, a wall adhesion model (Brackbill et al. 1992) is adopted in conjunction with the surface tension method.The circular tubes and test liquid in this work are assumed to be copper and water, respectively.

Methodology
In this study, the commercially accessible CFD software "ANSYS Fluent v 21.0" is used.The solution domain is interconnected by quadrilateral dominant mesh elements.The boundary layers of the circular tube wall and inter tube distance are finely meshed to precisely obtain the gas-liquid interface.The PISO coupled algorithm was employed for full pressure-velocity coupling, and the Pressure Staggering Option (PRESTO) scheme was adopted for pressure interpolation.In this study, the geometric VOF model was used in conjunction with the CSF technique, along with the Geo-reconstruct scheme, spatial discretization-Least Squares Cell Based, and Second Order Upwind scheme.To compute the solution, the transient method is combined with an adaptive time stepping scheme and a multi-phase specific method.The model settings are VOF explicit method, volume fraction cutoff is 10 -6 , and maximum iterations for time step are 40.The convergence criteria chosen for the developed model were 10 -6 for continuity and 10 -3 momentum equation.The initial time step size used was 10 -6 s and a total of 1.5 s duration for all simulations.This time step was chosen to ensure that the Courant number is less than or equal to 0.25, which is a VOF method stability criterion as per the Courant-Friedrichs-Lewy condition.In the following sections, the volume fraction contours are used to visualize the manner in which the liquid film covers the tube wall.Furthermore, the VOF value of 0.5 for the liquid-gas interface in the respective radial direction are captured and explored in this study.

Verification of Mesh Independence
The grid independence study is important factor in CFD modeling.The number of elements in the computational zone 23,543, 32,162, and 40,379 were chosen for the optimal mesh selection to analyze the film thickness distribution under the same input conditions, i.e.Re = 500 and tube distance of 10 mm, in order to examine mesh independence.Figure 3 depicts the liquid film profile is almost same and super imposed for the 32,162, and 40,379 grid elements.Thus, the grid number 32162 was preferred in this article as to save computing time.2012), as well as numerical findings by Ji et al. (2017).Based on the experimental results (Hou et al. 2012), the uncertainties of measured flow parameters such as sprinkle density and film thickness are ± 1% and ± 7%, respectively.For the established 2D model, the spreading of the film thickness trend for the Re = 574 and S = 10 mm is consistent with the experimental and simulation results from the literature.It is evident that there is a decrease in film thickness until 90° and a minimum value in the range of 90°-120°, after which there is an increase in liquid film thickness.The mean absolute percentage error for the simulation results and values reported experimental work is less than 10%.However, the simulation outcomes are in good agreement with Nusselt correlation.Thus, the established 2D model is competent to examine and analyze the film hydrodynamics over the horizontal tube surface.

The Effect of Static Contact Angle on Liquid Film Coverage
The development of dry spots, mal-distribution, and uneven liquid film coverage for a given Re has a negative impact on the operating performance of FFEs (Maliackal et al. 2022b;Tahir et al. 2020).Wettability of a liquid film has noteworthy influence on the efficiency of falling-film heat exchangers, which includes condensers and evaporators.Therefore, it is important to analyze the impact of static contact angle (Ψ) on wettability properties and film hydrodynamics.The present study examines the impact of the Ψ value from 0°-90° on the flow mechanism and the film coverage on the tube surface for the Re ranging from 100 to 500. Figure 6 depicts a schematic view of droplet wetting over a solid surface for different Ψ values.Figures 7, 8, 9, 10, and 11 illustrate the influence of Re and Ψ on film hydrodynamics such as wetting ratio, liquid film coverage, and liquid film breakage.Water volume fraction contours for various time frames were used in the following segments to comprehensively visualize the flow mechanism.The red region represents water, while the blue region represents air.
Figure 7 depicts the liquid film coverage for Re = 100 and various Ψ values ranging from 0° to 30°, 60°, and 90°.Water volume fraction contours were used to visualize the Increase in the wetting ratio.transient film coverage for different time frames in each case.Water flows downward from the feeder hole at a chosen velocity and under the influence of gravity.After reaching the upper stagnation zone, it extends circumferentially.The droplet formation can be seen beneath the stabilizing tube, when Ψ = 0° and Re = 100 at 0.415 s.The developed droplet also known as the primary droplet, also mentioned this behavior (Killion andGarimella 2003, 2004).The developed primary droplet has a hemispherical shape at the liquid head, and due to the low Re, the droplet made contact with the test tube and began to spread as a thin film.At this moment, the temporary formation of a neck can be observed during the earlier wetting of the tube wall.Furthermore, the thin liquid film completely wets the stabilizing and test tube walls without any breakages at 0.56 s.At this stage, the tube has ideal wettability properties (Ji et al. 2017).Due to insufficient liquid sprinkle density, the developed neck becomes weak and unable to sustain over time.After 0.56 s, the liquid neck thins out, eventually breaking at the weakest point and forming tiny droplets (Kandukuri et al. 2022a).There are two primary reasons for a liquid neck shrinking between the tubes.One is that the developed inter tube droplet disengages faster than the gathering of new liquid.Second, the surface tension forces around the liquid neck begin to contract, and neck breaking occurs at the weakest point.
Another source of concern from Fig. 7 at 0.59 s is the neck breaking process occurred after one cycle of complete wetting of both the stabilizing and test tubes.There are no peripheral dry zones and film breakage on either tube wall due to ideal wetting conditions at Ψ = 0°, but the developed liquid profile is very thin due to low Re.
As can be illustrated from Fig. 7b, the liquid film first wets the stabilizing tube wall in the same way it happened in the previous case for the same Re and Ψ = 30°.The shape of the developed primary droplet for Ψ = 30° at 0.875 s is different, as clearly outlined in the following inter tube flow patterns segments.The developed neck lasted only a short time and began to break before complete wetting of a test tube at 0.95 s.It was interesting to note that the development of peripheral dry zones and film breakage close to the upper stagnation zone of the test tube can be seen in Fig. 7 at 1.065 s.Furthermore, the size of the dry zone for the test tube broadens with the time.At 1.205 s, the major portion of the test tube wall appears dry without a thin liquid film.However, the remaining liquid is retracted by the stabilizing tube because surface tension forces outweigh gravitational forces (Tahir et al. 2020;Killion and Garimella 2003), after the neck breaks in both stages described above.When the Ψ value is increased to 60° and 90°, the liquid film from the feeder hole reaches the upper stagnation zone.However, the film coverage over the tube perimeter is limited.It is noticed that the liquid film has limited coverage for Ψ = 60° and 90°.
Figure 8 portrays the coverage of the liquid profile for Re = 200 and Ψ values ranging from 0° to 90°.When Ψ = 0°, 30°, the liquid film wets both the stabilizing and test tubes without a breaking.Figures 8a and b show the development of a neck at 0.41 s and 0.425 s, respectively.The increased Re causes a continuous neck to establish between the stabilizing and test tubes, which remains intact.However, there is no neck breaking and no retraction process at this point due to sufficient liquid load.The Ψ value for the same Re influences the inter tube flow pattern in this case as well.
When the Ψ value is increased to 60°, the liquid film exiting the distributor broadens in the upper stagnation zone, as shown in Fig. 8c at 0.74 s.The shape of the inter tube flow pattern is similar to a droplet connected to a neck here, and it differs from the previous two cases.Following that, the liquid film wets the stabilizing tube and commences to break the film just after making contact with the next tube at 0.795 s.The primary reason for the initiation of liquid film breaking is shrinkage of the liquid at this Ψ = 60° value (Ji et al. 2017).
Furthermore, the droplet detaches beneath the stabilizing tube and begins to spread across the test tube at 0.835 s.It was interesting to find that the detached droplet began to spread as an uncoupled liquid film over test tube wall at 0.94 s.After a span, the liquid film from the upper portion of the stabilizing tube wall commences to break and commute as unattached liquid film.In this sequence, the liquid film uncovers the major portion of both tubes, but a tiny developed droplet is attached beneath the tubes at 1.325 s.This is also referred to as the periodic wetting process.Furthermore, as the Ψ value approaches 90°, the liquid film begins to shrink in the upper stagnation zone, limiting its wetting area.The liquid coverage area is minimal at this stage and does not wet the entire tube, as depicted in Fig. 8d.
Figure 9 clearly demonstrates the film coverage profiles for Re = 300 and Ψ values spanning from 0° to 90°.The falling liquid begins to spread across the tube walls when Ψ = 0°, 30°.The liquid film wets both the stabilizing and test tubes.There is no widening of the liquid film in the upper stagnation zone.Furthermore, the liquid film completely covers the tube walls without breaking.Because of a sufficient supply of liquid load, there is no neck breaking and retraction process.
When Ψ = 60°, the sequence in the wetting process as shown in Fig. 9c.During the early stages of the wetting process, the liquid film expanded at the upper stagnation zone and then spread.The liquid film first wets the entire perimeter of the stabilizing tube before moving on to the next tube at 0.325 s.However, the breaking in the film profile began at the lower portion of the stabilizing tube after the liquid film made contact with the next tube at 0.425 s.Following that, liquid film breaking, both individual liquid film profiles flow downward under the influence of gravity for a short period of time, as shown in Fig. 9c.The stabilizing tube appeared to be completely wet after a short span of time at 0.6 s, but not the test tube.The main reason for complete wetting of the stabilizing tube, which is placed close the liquid distributor.On the test tube wall, due to the shrinkage of the liquid film, it moved as uncoupled film and results in the formation of peripheral dry zones, as seen at 0.51 s and 0.6 s.The liquid coverage area for a given same Re, Ψ = 90° is limited and does not wet the entire tube.
Figure 10 demonstrates the transient film coverage for Re = 400 and different Ψ values.When Ψ = 0°, 30°, and 60°, the liquid film wets the entire tube.Because of the adequate liquid sprinkle density, there is no breaking and retraction process for the Ψ = 0°, 30°, 60°, as previously discussed.When Ψ = 30° at 0.22 s, 60° at 0.235 s, the development of a tiny droplet connected to the liquid film head is clearly visible at this point.Furthermore, as shown in Fig. 10b and c at 0.27 s, and at 0.295 s, the developed tiny droplet at the liquid head collapses beneath the test tube, resulting in the formation of air voids for a short time.
Another significant aspect for Ψ = 90° is the liquid film leaving position.As the Ψ value is increased to 90°, the liquid film first wets the stabilizing tube wall during the early stages of the wetting process.However, the liquid film leaves the stabilizing tube wall before reaching the lower stagnation zone at 0.215 s.The liquid film falls as droplets on the next tube after leaving the stabilizing tube wall, as shown in Fig. 10d at 0.225 s.The test tube, on the other hand, experienced the same thing during the first cycle of the wetting process at 0.265 s and 0.39 s.The liquid film eventually completely wets both tubes.
Figure 11 illustrates the different stages of liquid film coverage for Re = 500 and different Ψ values.In this case, the liquid film completely wets the entire tube for Ψ = 0°-90°.It was important to reveal that, when Ψ = 0°, the liquid film wets the both the tube walls completely without developing a tiny droplet, as depicted in Fig. 11a at 0.205 s.In the case of Ψ = 30°, 60°, the developed tiny droplet phenomenon can be seen clearly visualized, as illustrated in Fig. 11b and c at 0.18 s and 0.235 s.The developed tiny droplet triggers the formation of air voids beneath the test tube.
When Ψ = 90°, the liquid film leaves the tube surface as a tiny droplet without reaching the lower stagnation zone, as can be seen in Fig. 11.The liquid film leaves the tube perimeter without covering the entire tube in the initial stage of the wetting process and spreads progressively to wet the entire tube.Figures 12,13,14,15,and 16 illustrate the inter tube flow patterns for different Re and Ψ values.According to the existing literature, the transition of flow regimes from tube to tube is based on Re (Hu and Jacobi 1996;Armbruster and Mitrovic 1994;Chen et al. 2015a;Kandukuri et al. 2022a).It is noticed that the inter tube flow regimes are primarily the result of Re.Furthermore, the present study unearthed the effect of Re and Ψ on flow structures.The inter tube flow mode structures have a significant impact on the liquid film formation process on the subsequent bottom tube.The flow pattern is droplet when Re = 100.When Ψ = 0°, the droplet has a hemispherical cap at the liquid head due to droplet extension in the vertical direction, as depicted in Fig. 12.Furthermore, when Ψ = 30°, the droplet shape changed to a spherical shape and attempted to detach from the liquid head.This phenomenon was ignored for the Ψ = 60°, 90°, due to non-wetting of tube walls.

Inter Tube Flow Regime
Figure 13 depicts the flow structures beneath the stabilizing tube wall for Re = 200 and Ψ = 0°, 30°, and 60°.When Ψ values range from 0° to 30°, 60°, the shape of the developed droplet narrows for = 30° and becomes spherical front followed by a weak neck for Ψ = 60°.As the Ψ values increases for the same Re, the droplet development time increases due to shrinkage of the film over the tube wall (Ji et al. 2017).As a result, the shape of the droplet turns into narrow.This stage is ignored for the Ψ = 90°.
The inter tube flow regimes for Re = 300 and Ψ = 0°, 30°, 60° and 90° as depicted in Fig. 14.The droplet made contact with the next tube and began spreading on the tube wall when Ψ = 0°.The neck is formed between the tubes as a column flow pattern is established in this sequence.The flow structure is spherical front connected by a small column of liquid film when Ψ = 30°.Furthermore, Ψ values increased to 60°, it is noticed that the small volume of the liquid droplet connected to the liquid head.At this point, the flow pattern is attempting to transition from column flow pattern to droplet flow pattern.Due to poor wettability, this phase is ignored for the Ψ = 90°.
The continuous neck is developed for all values in the Re = 400, 500 cases, as seen in Figs. 15 and 16.When Ψ = 0°, 30°, the spherical front is followed by the fluid column.When the Ψ value is increased to 60°, 90°, a greater volume of fluid is collected beneath the tubes and causes the formation of tiny droplets.Furthermore, when Re = 500 and Ψ = 60°, 90°, the formed tiny droplets collide beneath the tube, resulting in the formation of air voids and a recirculation zone in the flow field.The formation of air voids beneath the tube can have a negative impact on the heat transfer mechanism.the propagation time is defined as the time required for the liquid film to completely wet the stabilizing and test tubes in a single cycle.The present study also revealed, that the propagation time required to wet the tube surface increases as the Ψ value for the same Re.Furthermore, as Re increases, the wetting time decreases.The propagation time of 0 indicates that the wettability is low because the tube wall is not covered by a liquid film.For Re = 100, the wetting time difference is large for Ψ = 0°, 30°, whereas for Re 200-500, the wetting time difference is small.All of the investigations revealed a significant difference in wetting time for the Ψ = 60°, 90°, as shown in Figs. 17 and 18.Furthermore, Re is the primary reason for changes in the wetting time for different operating conditions.As the liquid load increases, the inertial forces of the liquid film take precedence over adhesion and viscous forces (Tahir et al. 2020;Liu et al. 2019).As a result, the liquid film spreads quickly.In this process, a liquid film covers the tube wall in the shortest amount of time.The aforementioned water volume fraction contours revealed that the Re and Ψ have significant influence on the liquid film propagation.The development of dry spots over the tubes may have a negative impact on the operational performance of FFEs.For a given instance, the wetting ratio over the tubes can be defined as ratio of wetted perimeter to the total perimeter.

Liquid Film Wetting Time
The results indicated that the wetting ratio over the tubes increases with the increase of Re and decreases with Ψ.When the Re = 100, wetting ratio is greater for the Ψ = 0°, than for other contact angles of the same Re.As illustrated in Figs. 17 and 18, the liquid film starts to cover the tube perimeter for Ψ = 0°, and a continuous liquid film.Furthermore, when Re = 200, the liquid film wets the entire tube for both Ψ = 0°, 30°, and the wetting ratio is 1.
When the Re = 300, the continuous liquid film is developed for the Ψ = 0° and Ψ = 30°.It was important to find that the development of dry spots for the Ψ = 60°, despite of liquid film covering the entire tube after a span.The wettability is low for the other Ψ = 90°.When Re = 400, 500 the wetting ratio is 1 for the Ψ = 0°, Ψ = 30°, Ψ = 60°, and Ψ = 90° which represents the liquid film covers the whole tube surface.
Overall the study indicated that there is a minimal Re for each Ψ value in order to keep the surface completely wet.The research also revealed that at higher Re, the Ψ value influence on the wettability factor can be disregarded.Lower Re makes it difficult to spread the liquid film, and shrinkage of the liquid film at higher Ψ value causes film breakage and peripheral dry spots.

Conclusion
A 2D and two phase model was developed to examine the gravity driven falling-film flow distribution over the inline horizontal tubes for the Re ranging from 100 to 500, Ψ value varying from 0°/30°/60°/90° and inter tube spacing of 10 mm.Overall, this research revealed that the Ψ value and Re have a substantial impact on liquid film hydrodynamics and can be considered in FFEs for droplet mode and column flow regimes.The proposed approach is an important tool for designing efficient gravity driven FFEs by improving flow characteristics.The transient flow mechanism and liquid film spreading under various operating parameters were analyzed in detail with key conclusions as follows: i.It is difficult to meticulously distribute the liquid film for low Re and higher Ψ values.
The development of peripheral dry regions on the tube surface due to inadequate supply, liquid film shrinkage and breakage.The wetting ratio over the tube surface increases with the increase of Re and decrease of Ψ value.ii.It is worth noting that each contact angle has a minimum Re in order to keep the surface completely wet.The study also demonstrated that for higher Re, the contact angle influence on wettability over the tube surface can be ignored.iii.The Ψ values have had a significant impact on the inter tube flow structures for a given Re.The formation of air voids near the lower stagnation zone have adverse effect on the FFEs performance.iv.The liquid film spreading ability for a given Re increases with the increase of Ψ value as well as flow time increases with the Ψ value and decreases with Re.

Fig. 1 a
Fig. 1 a Schematic of the inline horizontal tube array b Sketch of computational solution domain

Fig. 2
Fig. 2 2D Half symmetrical model with boundary conditions

Figure 4
Figure4depicts the comparison of simulation results with the experimental values reported byHou et al. (2012), as well as numerical findings byJi et al. (2017).Based on the experimental results(Hou et al. 2012), the uncertainties of measured flow parameters such as sprinkle density and film thickness are ± 1% and ± 7%, respectively.For the established 2D model, the spreading of the film thickness trend for the Re = 574 and S = 10 mm is consistent with the experimental and simulation results from the literature.It is evident that there is a decrease in film thickness until 90° and a minimum value in the range of 90°-120°, after which there is an increase in liquid film thickness.The mean absolute percentage error for the simulation results and values reported experimental work is less than 10%.However, the simulation outcomes are in good agreement with Nusselt correlation.Thus, the established 2D model is competent to examine and analyze the film hydrodynamics over the horizontal tube surface.

Fig. 3 Figure 5
Fig. 3 Comparison of film thickness for different mesh numbers

Fig. 4 Fig. 5
Fig.4Comparison of numerical outcomes with the published literature(Nusselt 1916;Hou et al. 2012;Ji et al. 2017) value of the static contact angle.

Fig. 6 Fig. 7 a
Fig. 6 Schematic of droplet wetting over the solid surface

Figures
Figures 17 and 18 show the propagation time for the stabilizing tube and test tube, as well as the time taken into account for liquid film inter tube commuting.In this study,

Fig. 17
Fig. 17 Variation in propagation time for stabilizing tube

Table 1
Working fluid properties Test liquid Density (kg/m 3 ) Viscosity (Pa s) Surface tension (N/m)