Breakup of a liquid rivulet falling over an inclined plate: identification of a critical weber number

Transcription

1 0 0 Breakup of a liquid rivulet falling over an inclined plate: identification of a critical weber number Rajesh K. Singh,*, Janine E. Galvin, Greg A. Whyatt and Xin Sun National Energy Technology Laboratory, Albany, OR, USA Pacific Northwest National Laboratory, Richland, WA, USA *Corresponding Author: rajesh-kumar.singh@netl.doe.gov; rajeshsingh.@gmail.com ABSTRACT We have numerically investigated the breakup of a rivulet falling over a smooth inclined plate using the volume of fluid method. Rivulet breakup is a complex phenomenon dictated by many factors, such as physical properties (viscosity and surface tension), contact angle, inertia, plate inclination, etc. An extensive simulation was conducted wherein these factors were systematically investigated. Regimes for a stable rivulet and an unstable rivulet that leads to breakup are examined in terms of a critical value of the Weber number We that delineates these regimes. A higher We cr implies a higher flow rate is required to maintain a stable rivulet. The impact of liquid properties is characterized by the Kapitza number (Ka). Variation of We with Ka shows two trends depending on the Ka value of the liquid. Liquids with lower Ka values, corresponding to high viscosities and/or low surface tensions, show linear variation and smaller value of critical Weber number. In other words, the lower the liquid Ka value the more stable the rivulet will tend to be with changes in liquid inertia. While liquids with higher Ka values exhibit higher values of We and quadratic variation of We with Ka. This behavior is more pronounced with increasing contact angle ( ). Higher contact angles promote rivulet breakup so that inertia must be higher to suppress breakup, consequently We increases with increasing. The effect of plate inclination on breakup shows that We decreases with increased inclination angle ( ) owing to higher effective liquid inertia. However, the effect is negligible beyond 0. The effect of the inlet size reveals that We decreases with inlet cross-sectional area, but the corresponding solvent flow rate for rivulet breakup remains unchanged. A phenomenological scaling for critical Weber number with Kapitza number and contact angle is presented, which may offer insight into rivulet breakup.

2 Introduction Fossil-fueled based power plants account for 0 percent of all CO emissions, [] and a need exists to mitigate these emissions to control global warming. Post-combustion carbon capture by chemical absorption using countercurrent gas liquid flow in structured packing is a popular and efficient technology for reduction of CO emission from power plants []. The structured packed column provides a large surface area for mass transfer between the phases with minimum pressure drop across column []. These structured packed columns are several meters in length (diameter: 0 m and heights h: 0 0m) [, ]. In contrast, the size of the structured packing unit is much smaller, typically around 0 cm, and the liquidfilm on the packing is typically less than a millimeter in thickness. Thus, a wide disparity in length scales exists in the structured packed column. It is not computationally feasible to perform simulations at the length scale of the column size while accounting for the actual geometry of the internal packing and the local gas-liquid interactions at a film level. Accordingly, multiscale modelling must be used to resolve these differences in scale []. Gravity-driven film flow down an inclined plate provides a simple configuration for understanding the hydrodynamics at the scale of the liquid film. As shown previously [, ], film falling down an inclined plate exhibits different flow patterns such as full film, rivulet, and droplet. These flow features are dependent upon various parameters, such as, solvent physical properties, solvent flow rates, plate surface texture, plate inclination, etc. In this paper, the volume of fluid method (VOF) is used to examine the transition from rivulet to droplet flow by breakup of the rivulet. A better understanding of this flow regime transition and how the various parameters play a role may be useful in the operation and design of a structured packing column where rivulets may occur. It is also relevant to other areas where breakup of a rivulet into droplets plays a key role in design and performance, such as in microfluidics, inkjet printings, etc. Rivulets flowing down an inclined plate exhibit a wide variety of instabilities, such as, interfacial waves, braiding, meandering, breakup of a rivulet into droplets and transition of rivulet flow from laminar to turbulent []. These instabilities are complex phenomenon dictated by many factors such as, contact line motion (the point where the interface meets the solid surface), contact angle, surface tension, gravity, viscosity and inertia [, ]. Surface tension generally acts to destabilize the rivulet whereas gravity and viscosity tend to stabilize the rivulet []. The classical Rayleigh Plateau instability observed in freestanding liquid jets, wherein a falling stream breaks up into smaller drops, is an example of a capillary instability which is driven by surface tension. In the case of a rivulet the analysis becomes more complex due to the additional solid-liquid interaction [0]. Over the past several decades rivulet stability been studied using a variety of techniques: analytically using linearized stability analysis [,,, ], the

3 0 0 0 energy minimization method [-], or the perturbation method []; experimentally [-0], and numerically [, ]. The stability of a static rivulet on a horizontal surface in the absence of gravity at different contact line conditions (e.g., fixed or moving contact lines with either fixed contact angles or smoothly varying contact angles) was investigated in the pioneering study of Davis []. A condition for stability in terms of a critical wavenumber was presented as a function of the equilibrium contact angle. This study was further extended by Young and Davis [] to explore the effects of contact line motion and slip at the surface on the stability of rivulet. The rivulet size and contact-line motion strongly influence the stability of rivulet. A wider rivulet or a rivulet having immobile contact lines is prone to kinematic wave instabilities, whereas a narrower rivulet with moving contact lines is disposed to capillary breakup. Both of these studies [, ] were conducted in the absence of gravity. The stability of a transverse rivulet (rivulet width is greater than its length in the flow direction) has been studied in the presence of gravity [, ]. In this case gravity tends to stabilize the rivulet by suppressing the instability. Inertia is another important factor affecting rivulet behavior and the manner of instability. Inertia may give rise to kinematic instabilities (i.e., formation of surface waves at the interface) [], braiding [] and meandering instabilities in the rivulet [, ]. Braiding (a stationary flow pattern that varies in width and height) has been attributed to competition surface tension which tends to narrow the rivulet and fluid inertia which acts to widen the rivulet []. Under this condition, contact lines are less mobile and capillary instability is suppressed. The rivulet is only subjected to kinematic wave instability where capillary effect acts against instability whereas inertia promotes it []. A meandering rivulet, wherein a rivulet destabilizes or deviates from a straight line, is generally observed upon increasing flow rate. The rivulet shape is irregular in general and given sufficient time may become stationary or destabilize and break down depending on flow conditions []. The mechanism by which the straight rivulet destabilizes into a meandering rivulet is not fully understood. Kim et al [] report that the meandering instability is a competition between inertia and surface tension but is also related to the tangential velocity difference across the interface and effects of the dynamic wetting. While Birner et al [] has reported that the meandering instability arises from disturbances in the flow rate and that a rivulet will usually tend to meander unless the flow rate is maintained at a highly constant value. More recently Couvreur and Daerr [0] emphasize the role played by geometry of the contact line (including roughness of the contact line and wetting hysteresis) on the critical meandering flow rate. While interesting, such instabilities are not the primary focus of the current effort which is to study breakup of the rivulet. Rivulet breakup is dictated by a balance of various forces, such as viscous, capillary, gravity, inertia and the force due to solid-liquid interaction [0]. Two mechanisms for rivulet breakup are

4 0 0 0 recognized: capillary wave growth, a counterpart to the Rayleigh-Plateau instability found in free surface liquid jets, and end pinching []. In capillary wave breakup, surface tension gives rise to a capillary pressure gradient that leads to thinner and thicker regions similar to the Rayleigh-Plateau instability. This perturbation grows and eventually the rivulet may break up into small droplets. The size and spacing between the droplets depends on the contact angle; higher contact angles (corresponding to hydrophobicity) lead to more closely spaced larger droplets [, ]. As noted earlier, such breakup phenomenon tends to occur in narrow rivulets having mobile contact lines []. It is worth noting that increasing the fluid inertia tends to mitigate this type of instability [, ]. End-pinching, the second mechanism, has been observed in breakup of a rivulet of finite length along a horizontal surface in the presence of gravity [, ]. The tip/end of the rivulet retracts because of surface tension and the subsequent gathering of fluid into the tip results in rapid bulb formation at the end. The bulbous end then pinches off from rivulet and a droplet appears. This instability differs from the aforementioned capillary breakup instability in that it is not a simultaneous occurrence along different regions of the rivulet. Instead, perturbations at the ends of the domain are observed propagate toward the uniform bulk resulting in a smaller daughter droplet that appears as a consequence of the pinch-off process. Recently, Wilson et al [] investigated the breakup of a thin rivulet into daughter-rivulets (multiple small rivulets) using the lubrication approximation and a prescribed a minimum value of rivulet semi-width above which breakup occurs. The capillary wave breakup mechanism for a rivulet on a surface has been studied numerically using the finite element method [] and the phase-field method [], analytically [0, ], and experimentally []. Good agreement with classical Rayleigh prediction was found in the experimental study for rupture of a polymer rivulet. Despite the knowledge gained by these studies, many have either simplifying assumptions or underlying limitations in their analysis. For example, the numerical studies based on an Arbitrary Lagrangian-Eulerian (ALE) technique by Ubal et al [] were limited to investigating onset of breakup phenomenon. Since this method cannot follow changes in domain topology, their simulations could not continue beyond breakup. Thus study of post break phenomena was not possible. The phase field simulation for rivulet breakup neglected inertia [], while the analytic stability analyses that rely on a lubrication approximation fail to correctly predict the rivulet behavior at large contact angles []. Furthermore, stability analysis becomes even more complex when trying to accommodate the effects of inertia as the simplifications made possible by lubrication theory are no longer appropriate []. Multiphase flow simulations using the volume of fluid method (VOF) method provide a tool to fully explore these effects and are therefore useful for investigating rivulet breakup dynamics.

5 0 0 In this paper, we consider breakup of a rivulet on an inclined plate over a wide range of parameters including physical properties (viscosity and surface tension), contact angle, and plate inclination angle. We employ VOF formulation as an interface capturing technique for multiphase flow study. In section, a mathematical model and numerical formulation are presented. Section discusses the problem setup including discretization and the different solvent properties examined. Simulation results on stability of the rivulet are presented in section. A new correlation is developed for a critical Weber number demarcating the transition from a stable to an unstable rivulet that leads to breakup and subsequent droplet formation. The results of the simulations are then summarized.. Mathematical formulation and Numerical method The volume of fluid (VOF) multiphase method [] is used to study the breakup of a rivulet flowing down a smooth inclined plate Recall that unlike an interface tacking method, such as the Arbitrary Lagrangian-Eulerian technique (ALE), the VOF method can overcome changes in the interface topology which will occur during breakup of a rivulet []. In the VOF method the entire flow field is treated as a single phase. Therefore, the conservation equations are solved for a single shared field. While a brief sketch of the VOF is provided here a more detailed explanation can be found in the Fluent theory guide []. The governing equations involved in the simulation are shown: u 0, () ( u) T ( uu) p u u g F. () t Here, is the fluid density, p is the pressure, g is gravitational acceleration, and is the fluid viscosity. The surface tension force, F, produces a jump in the normal traction across the interface. The conservation equations () and () are solved by a finite volume method. The surface tension force (F) is expressed as a volume force through the continuous surface force model (CSF) []. This force is distributed across the thin interfacial layer: F f g l Here is the interfacial tension (which is constant in the CSF model), is the local curvature of the interface and is the gradient in the liquid phase volume fraction (f). The interface between the phases is captured by solving an additional transport equation () for the liquid phase volume fraction (scalar f). Accordingly, the value of f varies from 0 to (0 corresponding to cells with all gas and corresponding to all liquid). ()

6 0 0 f t u. f 0 () In the case of a two-phase simulation, equation () is only solved for the secondary phase, and volume fraction of the primary phase is computed by satisfying constraint f. The behavior of the interface between two fluids in contact with the wall is influenced by wall adhesion []. In the VOF model used here this behavior is, by default, dictated through the specification of a static contact angle ( ). A dynamic contact angle model, discussed next, was also invoked for comparative purposes in select cases. In either case (static or dynamic), the boundary condition is not imposed at the wall itself, but instead the contact angle that the fluid is assumed to make with the wall is used to adjust the surface normal in cells near the wall. Specifically, the normal vector at the interface is adjusted near the wall by equation(): nˆ nˆ cos nˆ sin () wall t where and are unit vectors normal and tangential to the wall, respectively. The vector lies in the wall and normal to the contact line (where the interface meets the wall) and is computed using following expression nˆt f f () In practice, the contact angle may vary as the rivulet evolves (e.g., as it spreads or wets the surface). So that a dynamic contact angle, measured at advancing or receding contact line velocities, may differ significantly from the static value []. Still the dynamic contact angle is not well understood and a complete theory on contact angle hysteresis is still under development [, 0]. Nevertheless, a dynamic contact angle model would provide a more accurate description of the wetting and spreading process. In this effort, the correlation by Yokoi et al. [0] for dynamic contact angle was selected and implemented using a User Defined Function (UDF). In this case, the dynamic contact angle becomes a function of the velocity of the contact line ( ). Their work is based on Tanner s law [], and it is reproduced here: n i i U CL / Ca min e, mda if UCL 0, k A / Ca max e, mdr if UCL 0, kr () Here, Ca is capillary number Ca / ; and are empirical constants for advancing and receding cases, respectively, that depend on the surface material; and is the equilibrium contact angle.

7 0 0 0 The quantity refers to the maximum dynamic advancing angle, while refers to the minimum receding angle. In measurements, these correspond to limits in the contact angle as increases and decreases, respectively. Thus, the value of the dynamic contact angle lies within the interval. As evident this model requires several parameters for characterization of the dynamic contact angle:,,,, and. The selection of these constants are ( 0, 0 ) from literature [0], and contact angles from experiments (.,., and 0. The flow simulations were conducted in Fluent.0 [] based on the finite volume method using an implicit transient formulation. The conservation equations (EQs & ) were solved using a pressure-based solver. Specifically, the PISO algorithm [] was employed for coupling between velocity and pressure. The second-order upwind scheme was used in the spatial discretization of all equations. For the volume fraction transport equation an explicit formulation was employed. An algebraic multigrid (AMG) method was used to accelerate convergence of the solver. Interface capturing was achieved with the geometric reconstruction method using the piecewise linear interface calculation (PLIC) []. Convergence of the solution was assumed when the sum of the normalized residual for each conservation equation was less than or equal to 0 -. The stability of the transient simulation was controlled by the CFL condition with a value of Courant number of 0.0. Therefore, a very small time step ( t), varying from 0 0 sec, was required to satisfy this condition. Consequently, the simulations were computationally expensive.. Problem setup and solvent properties We investigate rivulet breakup and identification of flow regimes for rivulet flow down a smooth inclined plate. A schematic of the simulation setup is presented in Figure (a) showing the inlet and outlet boundaries, the side walls, and the top and bottom (plate) boundaries. The flow domain consists of a smooth plate (0 0 mm ) inclined 0 to horizontal (i.e., inclination angle, =0 ) with a height of mm (see Figure (b)). Air (density ( ) =. kg/m and viscosity (µ) =. 0 - Pa.s) and industrially used solvents for carbon capture were used as working fluids (see Table ). Air was considered as a stagnant phase. A number of alkanolamine-based solutions are being used as solvents for carbon capture processes [, ]. Therefore, various aqueous alkanolamines at different concentrations were used to represent the liquid phase in order to cover a wide range of physical properties. Consistent with previous work [], the fluid properties are studied in terms of the Kapitza number (Ka), which has a fixed value for each solvent and is independent of the flow rate.

8 Ka g / () The Kapitza number has been extensively used in the film flow community to investigate interfacial wave phenomena [, ]. As evident a wide range of Ka numbers are investigated. However, it should be noted that surface tension plays a relatively minor factor in changing the Ka number compared to viscosity as the values of surface tension vary within a relatively narrow range. Table : Physical properties of the solvents at C and atm Solvent (mpa s) (Kg/m ) (mn/m) Ka Water 0.. 0% MEA. []. [] 0% MEA. [] [] 0 0% MEA. []. [] 0 0.0xMPZ [0] % MDEA x MPZ[0] x MPZ [0] x MPZ [0] x MPZ [0] MEA: monoethanolamine MPZ: -methylepiperazine MDEA: N-methyldiethanolamine %: Percentage by weight x: mole fraction The liquid was introduced through a small inlet of dimension mm located in the middle of the inlet face (see Figure (a)). The inlet was defined by a uniform and constant velocity perpendicular to the boundary. The remaining part of the inlet cross section was set as a pressure outlet boundary with zero gauge pressure. The influence of inlet size was examined for a single case, which is discussed in more detail below. In brief, however, the solvent flow rate corresponding to rivulet breakup was found to be insensitive to inlet size. The solvent then exits through the outlet face in the presence of gravity. The outlet and top boundaries were set as pressure outlets with zero gauge pressure. Unless otherwise stated, the plate and side walls were set as no-slip walls with a static contact angle ( ). The contact angle is a varying and complex quantity [] and the value corresponding to each solvent was not readily available in the literature. Note that a solid surface may exhibit different values of

9 0 0 0 with change in (e.g., varying solvent) []. Similarly, for a given liquid, the contact angle may also vary with a change in solid surface. As a result, a static contact angle value of 0, corresponding to water on steel, was used in the preliminary simulations. However, the effect of varying contact angle ( =0 0 ) on the rivulet breakup and subsequent droplet formation was also investigated. As already noted most cases were conducted using a static contact angle (SCA). A dynamic contact angle model (DCA; see Equation () ) was implemented but the simulation campaign could not be readily rerun due to the computational expense involved. As noted earlier however, a dynamic contact angle provides a more accurate description of spreading and wetting processes. In view of this, the impact of a DCA was explored in select cases for comparative purposes. It is worth noting that the DCA model involves several parameters that are not readily available and therefore need to be tuned. Ultimately, the SCA and DCA simulations yielded similar results regarding identification of a critical Weber number that represents the transition from rivulet to droplet flow. The critical Weber number based on the DCA was slightly lower. This aspect is discussed in more detail in the Results section. The Weber number is a representation of the fluid inertia and it has been extensively used in the analyses of this type of flow []. Here it is computed according to the inlet conditions as: We U D () H where, U is the inlet velocity and is the inlet hydraulic diameter. Recall the hydraulic diameter is defined as / where is the cross-section area and is the wetted perimeter of the cross section. For a rectangular inlet this becomes /, where W and h are width and height of the inlet, respectively. Meshing is a critical step in constructing a numerical simulation that will impact the convergence and accuracy of the simulation. The flow domain was discretized with a non-uniform structured grid for efficient computations. The mesh density inside the rivulet and near the interface is finer than the region adjacent to the surrounding gas (see Figure (a)). Accordingly, a very fine grid was used near the plate and along the center of the flow domain (see exploded view). A grid independence study was also conducted to determine a reasonable mesh while maintaining grid independent predictions. Only the grid across the width of the inlet was varied, as opposed to the whole domain, as breakup of the rivulet occurs proximate to the longitudinal centerline of the flow domain where the inlet was located. The shape of the rivulet at the onset of breakup was evaluated for different grid resolutions using water with 0 and 0 and a Weber number of 0.. At this Weber number a water rivulet undergoes the breakup process. The grid resolution is varied from 0 grid points (GP) across the width of the inlet region to 0

10 0 0 GP. The shape of the rivulet does not substantially change during breakup with resolutions greater than GP (see Figure (b)). The value of the critical Weber number that corresponds to the transition from rivulet to droplet flow was also examined for this setup. The predicted value of remains approximately the same following grid resolutions of GP or higher (see Table ). Hence, a resolution consisting of GP across the inlet region) was chosen as the optimum mesh. This number of GP corresponds to.m cells over the entire simulation domain. It is worth noting that in the investigation of varying inlet size the number of grid points across the inlet is also varied to maintain a consistent grid size according to these grid independence results. Thus, the total number of cells in those cases will increase as the inlet size is increased. Overall, mesh sizes ranging from.. M cells were used, which falls within the range reported by similar numerical studies [0, ]. Table : Variation of the critical Weber number ( with grid resolutions for water rivulet at Grid points at Number of Cells inlet width 0. M 0.. M 0.0. M M 0.0. Experimental setup In this work, experiments were carried out for rivulet flow down a smooth thick polycarbonate plate inclined at a 0 angle to the horizontal (see Figure (a)). The results from the experiments were used for validation purposes. Three liquids were used in the experiments including water and two standard silicon oils with ~0cP and 00cP viscosity. The liquid was introduced to the top surface of the plate by flowing over a 0 mm wide weir formed by cutting a square pocket into the top of the polycarbonate plate (see Figure (b)). When testing the water flow, the fluid entered at the back of the pocket via a ¼ metal tube and then passed through two 0 mesh screens to eliminate turbulence and provide a smooth fluid surface at the weir. A single 0 mesh screen was found sufficient for testing of the ~0 cp standard silicon oil, while no screen was used when testing the ~00cP fluid as the fluid appeared smooth and calm at the top of the weir without the screen. The liquid flow rate was set by setting the speed on a gear pump. The fluid was diverted to a cup for a set period and the weight gain determined to determine the flow. The test was then run without adjusting the pump speed. The temperature of the 0

11 0 0 0 circulating fluid was measured using a type K thermocouple and used to estimate the actual viscosity at test temperature. To aid visibility of the viscosity standards while flowing down the plate, a dye (Sudan III) was added to the standard silicon oil only that provides a red coloration to the flow. The camera was also angled to catch the reflection of an overhead fluorescent light to better determine the position of the flow boundaries. Snapshots of the flow and a corresponding short video were recorded for these liquids at different flow rates. A -cm square grid placed on the back side of the plate for dimensional reference was used to extract the rivulet width.the properties of the fluids were also measured. Surface tension of the silicon oils were measured using a Kruss K- Tensiometer. The instrument performance was checked before measurements using pure water and pure ethanol and then checked after measurements using pure water. The high viscosity of the standards interfered with the ability of the instrument to make measurements using its automated program requiring the measurements to be made manually. The densities of the silicon oils were determined by measuring the weight gain in filling a ml volumetric flask. Note, the density and surface tension was measured after the addition of the Sudan III dye. The measured surface tension values for the viscosity standards was conducted after adding the dye. Data without the presence of the dye was not taken to check the influence of the dye on surface tension. However, the measured values re close to what is expected based on literature (i.e., 0.0 vs N/m literature for 0 cp and 0.0 vs 0.00 N/m literature for 00cP). So, if there is an effect of the dye on the surface tension it is probably not significant in this case. Results and discussions: In the following section, simulation results on the breakup of a rivulet on a smooth inclined plate are presented. A critical Weber number (We cr ) is identified which effectively represents the flow rate for which rivulet breakup occurs and coincides with a transition in the flow regime (from continuous rivulet to droplet). As already noted, rivulet dynamics over an inclined plate are complex and dictated by many factors, such as, solvent properties ( and ), contact angle ( ), plate inclination angle ( ), etc []. Accordingly, the simulation results are presented in terms of the following representative dimensionless parameters: Weber number (We), Kapitza number (Ka) and dimensionless time (t * ). The Weber number and Kapitza number were defined in equations () and (), respectively. To nondimensionalize time, a time scale was defined as / by using the inlet hydraulic diameter ( ) as the length scale and the capillary velocity ( / ) as the velocity scale [].. Comparison with experiments In our previous study of a fully wetted film over a smooth inclined plate [], VOF- predictions for wetted area and film thickness were found to compare well with the experimental results of Hoffmann et

12 0 0 0 al [] and Nusselt theory [], respectively. To further ensure reliability of the current simulations, the predicted results are qualitatively and quantitatively compared with those obtained from the experiments conducted as part of this effort. CFD simulations were conducted for water as well as for two highly viscous general purpose silicon oils: 0cS and 00cS. In this setup, the simulations were conducted for a sufficiently long time as to achieve pseudo-steady state for wetted area of the plate and solvent mass flow at the exit. Once the rivulets achieved pseudo-steady state, the rivulet width was measured and compared with that from the experiment. As shown in Figure, the predicted shape of the interface (defined using an iso-surface at volume fraction of 0.) rivulet matches well with the experiment for water (Ka=) at two flow rates. 0 and. 0 / (corresponding to Weber numbers of 0.0 and 0., respectively). In this case, predictions from both static (SCA) and dynamic contact angle (DCA) simulations are shown. Both experiment and simulations exhibit a similar stationary braiding pattern in the flow [] where the rivulet varies in width and height. The predicted edge of the rivulet (rim) and the maximum height (hump) are approximately the same position as observed in the experiment Comparing the rivulet shape obtained from the dynamic and static contact angle simulations shows they are not significantly different. Once the rivulets achieved pseudo-steady state, the contact line in these straight rivulet cases is relatively immobile. So, it is unsurprising that the dynamic and static contact angle simulations provide similar predictions. A static contact angle of 0 was specified in these cases. The dynamic contact angle model requires a number of empirical parameters wherein assumptions are made the evolution and hysteresis of the contact angle []. Here values were selected as to achieve good agreement with experimental results resulting in 0, 0,.,., and 0. These values were used in all cases employing a DCA model. In cases of increasing flow rate the rivulet can become significantly diverted and a meandering rivulet may emerge (ref -see introduction for citations). This was observed experimentally for the case of water and increasing flow rate. In such cases of a moving contact line the contact angle may be expected to vary. Specification of a static contact angle would provide an inaccurate description of the process and a dynamic contact model becomes necessary. Therefore, simulations with the DCA model were employed for comparative purposes. The results are shown in Figure for two flow rates and (b) m /sec (corresponding to Weber numbers of 0.0 and 0.00, respectively). Given the dynamic nature of this phenomenon, the results presented correspond to a single snapshot in time wherein the experimental results and simulation predictions show similar behavior. As evident the flow rates reported in the meandering rivulet cases (Figure ) are lower than those of the braiding rivulet examples

13 0 0 0 (Figure ). This may be somewhat expected as inertia is reported to reduce contact line motion (thereby reducing the dynamic contact angle) []. The simulation predictions for the two general purpose silicon oils, 0cS (Ka=, = kg/m, µ=. mpas and =. mn/m) and 00cS (Ka=.0, = kg/m, µ=. mpas and =. mn/m), at varying flow rates were also examined. For these viscous liquids a stable rivulet (without any undulation) was found at all flow rates investigated. Variation in rivulet width with the Weber number (We) is presented in Figure. As in the experiment, the width of the rivulet is measured at a location 0mm down from the inlet as depicted by the red line in the inset of Figure (a & b). As shown, the predicted rivulet width matches well with the experimental measurement at different Weber numbers for both solvents. For 0cS silicon oil, Figure (a), a small discrepancy in the results is seen at low flow rates; however, the difference is 0%. Agreement between the CFD predicted rivulet shape and that obtained from experiments is also confirmed by the insets. Based on these and previous results, VOF is considered an appropriate method for the present multiphase flow study. Indeed, VOF has been widely used by others for studying the different flow regimes exhibited by film flowing down an inclined plate: droplet, rivulet, and film (fully wetted plate) [,,, ].. Rivulet breakup An earlier study on interfacial and wetted areas of a stable rivulet found that the interfacial area was greater than the wetted area and that this was due to the rivulet morphology (curvature at the rim along the contact line and to slight rippling at the surface) []. In the case of a structured packing column for solvent absorption, uniform films are thought to maximize mass transfer area, and therefore, breakup into droplets would be considered unwanted phenomenon. Understanding the stability properties of the flow in such a system and the mechanisms involved in breakup and dewetting is of industrial interest. In this view, rigorous simulations are conducted to identify the rivulet and droplet flow regimes. As reported in the literature, transition of flow regimes from a stable film to a rivulet and to a droplet is observed with decreasing liquid flow rate []. At low-flow rates, surface tension forces dominates over inertia, which acts to destabilize the film resulting in rupture/breakup []. Accordingly, simulations were initially conducted at a higher flow rate wherein a stable rivulet is observed. The solvent flow rate was then gradually reduced to identify the transition from rivulet to droplet. As earlier, the effects of the flow rate are presented in terms of the Weber number. Figure shows the interface shape for water (Ka=) at different flow rates. Recall the interface is defined by an iso-surface at 0.0. As the flow rate is decreased the width of the rivulet becomes narrower (We=0. & 0.). A further decrease in flow rate reveals smaller rivulets and droplets (We=0. & 0.). Therefore, breakup occurs between We=0. and 0.. The transition between flow

14 0 0 0 regimes is shown to depend on a critical Weber number (We cr ). The value of the critical Weber number (We cr ) was determined by observing animations of the predicted interface and monitoring the exit mass flow rate for either intermittent or steady rivulet flow (indicating droplet or rivulet flow, respectively) with iterative refinement in the value of Weber number. To show the breakup phenomenon more clearly, the temporal evolution of the interface for water (Ka=) at We=0. is depicted in Figure (a) where time is non-dimensionalized as. Here end-pinching can be observed to lead to breakup of the rivulet. As evident a bulbous end has developed at t * = along with the onset of wrist formation. The tip of the rivulet begins to retract because of surface tension and liquid gathers at the tip. Eventually, a bulbous shape forms at the tip of rivulet that further grows (see Figure (b)). The wrist formation proximate to the bulbous end, observed clearly at t * =, continuously retracts to become thinner with time. A neck appears at t * =, which is a precursor to end pinching. To better explain this breakup mechanism, capillary pressure at the interface was computed and is shown in the Figure (b). A local maximum negative capillary pressure exists at the neck owing to its larger curvature. Due to the pressure gradient, fluid is accelerated at the neck and driven toward the end of the rivulet. The accelerated fluid at the neck leads to further reduction in the neck radius. Eventually breakup occurs (t * =) as the bulbous end pinches off and a small cornered droplet (a drop that deviates from the classical spherical shape) forms. Specifically, the advancing contact line at the front of droplet remains rounded whereas the rear contact line at the trailing edge transforms into corner. Thus the corner at the trailing edge of the droplet develops due to contact line motion []. The breakup process repeats when the bulbous end reappears at the tip of the rivulet (see the figure (a) at t * =0). The effect of inlet size on the observed behavior of rivulet breakup was also examined. Specifically, the height of the inlet ( ) was kept constant as mm while the width ( ) was varied. Figure shows that the value of the critical Weber number decreases with increasing inlet size. In contrast, the corresponding volumetric flow rate was found to remain nearly constant for all cases (see the inset of the Figure ). This behavior is consistent with our previous findings concerning the dynamics of a rivulet falling over an inclined plate. In that case, inlet size had no effect on the developed width of the rivulet for a given flow rate and contact angle []. The decrease in We cr with increase in inlet width can be explained by considering the definition of the We number (equation()) and the change in inlet velocity with inlet size. In particular, as the inlet size is increased the hydraulic diameter also increases, however, the magnitude of the square in inlet velocity decreases more significantly so that decreases.. Effect of the solvent properties:

15 0 The breakup of a rivulet into droplets is a complex process dictated by many factors including the physical properties of the fluid. As mentioned earlier the fluid properties are studied in terms of the Kapitza number. The breakup of a rivulet and subsequent regime transition is first explored in We Ka space at a fixed value of the contact angle and inclination angle: γ 0 and θ 0. The impact of these latter two parameters is studied subsequently. Figure 0(a) indicates the occurrence of two flow regimes and their transition as it depends on liquid inertia. Specifically, the curve represents a critical Weber number (We ) with values of Weber number above the curve reflecting the presence of a stable rivulet and below the curve a droplet or breakup of larger rivulet into smaller daughter rivulets. The region for droplet formation increases with increasing Kapitza number as shown in Figure 0(a). In other words, higher flow is required to maintain a stable rivulet with increasing Ka value as explained further below. A vertical line is drawn in Figure 0(a) that divides the plot into two regions of variation in the critical Weber number (We cr ) with Kapitza number. Below Ka ( 00), Figure 0(b), We cr linearly varies with Kapitza number. This is referred to as region I. Above Ka ( 00), Figure 0(c), We cr shows a quadratic variation with Ka, We ~Ka. This is referred to as region II. The trend in variation of We cr is expressed here: We AKa Ka 00 cr = AKa BKa >00 (0) 0 0 where, A and B are empirical constants. Recall high values of Ka imply high interfacial surface tension and/or low viscosity. Higher values of the surface tension give rise to enhanced capillary forces and capillary forces act as a destabilizing effect that then requires higher inertia forces to stabilize the rivulet. In short, surface tension tends to have a destabilizing effect, which inertia counteracts []. On the other hand, low value of Ka corresponds to high viscosity and/or low surface tension. Such solvents show enhanced wetting characteristics. At these Ka values, viscous forces tend to dominate acting as a dissipative factor that partially suppresses the breakup instability thereby enhancing the stability of the rivulet []. Therefore, a stable rivulet can be achieved at lower inertia. The simulations also exhibit slightly different morphologies during the breakup process depending on the Ka value. The end-pinching of highly viscous solvents (Region ) show a small satellite drop along with a bigger daughter droplet. This behavior is shown in the inset of the Figure 0(b) for Ka=0. The smaller satellite drop remains pinned to the plate, it is unable to move, because of its light weight []. For higher Ka values (Region II) a satellite drop is not predicted in the breakup of the rivulet (see the inset of Figure 0(c) for Ka=).

16 Effect of inclination angle Besides solvent physical properties, the effect of plate inclination angle on the breakup of a rivulet is also examined. Inclination angle has been shown to impact various flow features such as film thickness, wetted area and interfacial area []. As the flow distribution in a structured packing is a key factor in the absorption efficiency of the column it is important to understand how the inclination angle affects the breakup of rivulet and subsequent flow regime transition. Accordingly, the inclination angle was varied from 0 to ( =0 ) at a fixed contact angle ( =0 ) for all solvents studied. By changing the inclination angle the gravitational effect on the rivulet also alters. In particular, an increased value of results in higher film velocity even at same inlet flow rate, and in turn, higher inertia. Recall, inertia acts to stabilize the rivulet against opposing capillary forces that tend to destabilize the rivulet [0]. As expected, We cr is reduced with increasing as shown in Figure for all Ka values studied. For example, We cr decreases by ~ percent when is increased from 0 to 0. This result is also consistent with previous observations based on linear stability analysis based on the lubrication approximation that reveals the stabilizing effect of gravity [, 0]. Similar to the variation of interfacial area with inclination angle [], the critical Weber number also becomes insensitive to changes in beyond 0. For example, the value of We cr for 0 and are almost identical in Figure. This is not surprising considering that does not significantly vary in those limits.. Effect of contact angle: So far the flow simulations were generally restricted to a fixed contact angle ( ) of 0. The contact angle between the fluid and the surface is another important quantity that reflects the wetting properties of the surface. Smaller values of the contact angle were found to lead to increased interfacial area owing to the enhanced surface wettability []. Note that the contact angle is a characteristic of a given solid-liquid system in specific environment [0]. That is, a liquid can exhibit a different contact angle with a change of solid substrate. Moreover, for a given solid surface the value of vs were observed to follow a linear trend for a homologous series of liquids. A lower values of correspond to smaller contact angles []. Contact angle will also play a role on the stability of the rivulet. The curvature of the interface increases with increasing contact angle, which leads to enhanced capillary pressure and converging of the flow that leads to breakup of a rivulet []. This tendency toward breakup may be exacerbated for solvents with large Ka (i.e., characterized in general by high surface tension). To better understand the effects of contact angle on rivulet breakup and identification of the critical Weber number (We cr ), extensive simulations are conducted for a wide range of static values (0 0 ).

17 Variation of the critical Weber number (We cr ) with contact angle ( ) for different solvents at fixed inclination angle ( = 0 ) is plotted in Figure (a). As expected, We cr increases with increased value of for all solvents. Furthermore, variation of We cr with is steeper for higher values of Ka number. Recall, high Ka and values promote breakup requiring enhanced inertia to then moderate the instability. To examine this behavior more closely the critical Weber number is renormalized with the critical Weber number at 0 (We ): We We /We. Two regions in variation of We cr with are revealed (figure not shown). In both regions, the critical Weber number shows a scaling relation as We ~. Furthermore, the value of the exponent m is higher at lower values of Ka corresponding to lower values of. 0 We ~ ( cos ) cr m m ; Ka 00 () = ; 00 0 The scaling relation shown equation () is presented in Figures (b) & (c). It is worth noting some similarities between variation in the critical Weber number with contact angle observed here and variation in interfacial area with contact angle observed in some earlier studies. In particular, variation in interfacial area with contact angle also exhibited two distinct regions for a given set of parameters influencing the rivulet flow [, ]. In the previous effort, a scaling relation between interfacial area with contact angle as ~ was shown where A In is the normalized interfacial area and n is an exponent that depends on Ka []. Solvents having low Ka (<00) value showed steeper variation of with than those having high Ka ( 00. In the present case, however, the trends in variation of We cr with contact angle were found to be opposite: solvents having high Ka value showed stiffer variation of We cr with value contact angle than those having low Ka value. It is worth noting that the variation in the We cr with contact angle, and variation in with contact angle are physically consistent. A high Ka and values, the solvent will demonstrate less wettability and an enhanced tendency for instability. Therefore, higher inertia is required to stabilize the rivulet, so eventually We cr increases. Earlier the critical Weber number representing rivulet breakup was shown to vary as We ~Ka, where p= for Ka 00 and p= for Ka 00 for a fixed contact angle 0 (see Figures 0(b&c) and equation (0)). This scaling relation was found to hold for all contact angles, as shown in the inset of Figures. All curves collapse when the expressions for the critical Weber number as a function of the Kapitza number (equation (0)) and the contact angle (equation ()) are combined. The following phenomenological correlation is obtained as shown in Figure :

18 We cr ~ Ka( cos ) Ka 00 / ~ Ka ( cos ) > 00 () In this relationship, solvents with low values of surface tension (low values of Ka) show a linear variation in We cr with Ka and reduced sensitivity to the value of the contact angle as the term decays more quickly than the term /. In comparison, solvents with high values of surface tension (high values of Ka) show a quadratic variation in We cr with Ka and greater sensitivity to the value of the contact angle. So solvents with low values of surface tension (low values of Ka) correspond to lower values of We Cr. A lower value of We cr implies a more stable rivulet even at low flow rate. It is worth noting that low values of Ka generally correspond to highly viscous solvents, and that solvents with high viscosity are characterized by lower values of surface tension [-], and in turn, smaller contact angles. While this investigation employed a constant and static value for contact angle it is recognized that the contact angle is a varying and complex quantity []. Still the dynamic contact angle is not well understood and a complete theory on contact angle hysteresis is still in development [, 0]. Therefore, a comparison between the static contact angle (SCA) and a dynamic contact angle (DCA) model on rivulet breakup was conducted for a single case involving water (Ka=). The shape of the rivulet at the onset of breakup is presented in Figure corresponding to a SCA and DCA simulation at We = 0. (recall that rivulet break up is expected at this We number). A comparison reveals slight differences in the shape of the bulbous end with the DCA case predicting a nearly spherical bulbous end and the SCA case resulting in a more cornered shape. In addition, the length of the wrist adjacent to the bulbous end is longer for the DCA setup than that of the SCA. Besides comparing rivulet shapes at the onset of breakup, the critical Weber number was also compared. The value of corresponding to the DCA model was found to be 0% lower than the corresponding one for the SCA condition. Finally, it is worth noting that highly viscous solvents may exhibit small contact angles so that variation in the advancing and receding contact angle will be not significant.. Conclusion We conducted three dimensional multiphase flow simulations with VOF interface capturing method for studying the breakup of a rivulet flowing down a smooth inclined plate. Understanding the factors that lead to rivulet breakup is relevant to industries where the microscopic flow pattern plays an important role, such as in packings that are employed in separation and distillation processes. The CFD predictions compared well with experimental results of a rivulet flowing down an inclined plate using water as well as highly viscous liquids. Specifically, the predicted morphology (shape) and width of the rivulet matched well with those observed in the experiments. Rivulet breakup is a complex phenomenon.

19 0 0 0 This process is influenced by many factors including solvent properties, flow rate (inertia) and inclination angle. Contact angle is another important parameter representing the wetting characteristics of the system (interaction of the solvent and plate). In this effort rivulet breakup through end-pinching was examined wherein a rivulet tip retracts due to surface tension causing the liquid to collect into a bulbous end that is eventually pinched off []. This long term instability leads to breakup and formation of a droplet. An extensive CFD simulation campaign was conducted to explore the role of the aforementioned factors on the breakup phenomenon. A critical value of the Weber number (We cr ) that separates droplet and rivulet regimes was computed as a function of the Kapitza number. For We values lower than the We cr a droplet is observed, and conversely a rivulet is observed for higher values of We. The results show that the critical Weber number increases with increased Kapitza number. This trend can be understood because high viscosity dampens growth in the instability, subsequently hindering breakup. Recall, high values of solvent viscosity and/or low values of surface tension correspond to low Kapitza numbers. Therefore, low Ka solvents will tend to exhibit rivulet flow even at low flow rates. The exact nature of the trend in the variation of We cr with Ka depends on the Ka value. Solvents with lower Ka value show We cr varying linearly with Ka. While solvents with higher Ka value show We cr having a quadratic variation with Ka. The effect of plate inclination was also explored. The critical Weber number decreases with increased inclination angle ( ) owing to accelerated liquid velocity. However, the effect is negligible beyond 0. The role of contact angle ( on rivulet breakup was also systematically examined. As the contact angle is increased so too does We cr. (i.e., greater inertia is required to achieve stable rivulet flow). As with We cr versus Ka, We cr versus shows two trends for variation. Solvents with high Ka values show steeper variation of We cr with contact angle than those with low values of Ka (corresponding to low values of surface tension). By combining these analyses a phenomenological scaling for critical Weber number with Kapitza number and contact angle is presented. Acknowledgement This research was supported in part by an appointment to the National Energy Technology Laboratory Research Participation Program, sponsored by the Office of Fossil Energy, U.S. Department of Energy, through Carbon Capture Simulation Initiative (CCSI) and administered by the Oak Ridge Institute for Science and Education. The CFD simulations were performed on NETL supercomputer Joule.

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24 (a) (b) z g x Figure : (a) Prospective view of the computational flow domain showing details of domain. The solvent is flowing down from top of the inclined smooth plate. (b) Front view of the flow domain showing inclination angle and direction of the gravity.

25 (a) (b) 0GP in inlet GP in inlet GP in inlet 0GP in inlet Figure : (a) Discretization of the computational flow domain showing the mesh. A very fine mesh was used both near the plate to capture the film flow field, and in the center of the domain to capture the rivulet flow dynamics. (b) Shape of the rivulet at the onset of breakup for varying grid resolution using water (Ka=) at We= 0., 0, 0. After GP, the shape of the rivulet only marginally changes with further resolution.

26 (a) 0 (b) Mesh Screen Weir Square grid Rivulet Polycarbonate plate Metal tube Figure : (a) Schematic diagram of the test rig of an inclined polycarbonate plate inclined 0 to the horizontal, (b) Experimental arrangement showing the rivulet flow over the inclined plate. A -cm square grid was placed on the back side of the plate for dimensional reference.

27 (a) (b) Experiment CFD (DCA) CFD (SCA) Experiment CFD (DCA) CFD (SCA) Figure : Comparison of the CFD predicted rivulet shape using a dynamic contact angle (DCA) and static contact angle (SCA) with that captured from experiment for water (Ka=) with θ = 0 at two Weber numbers: (a) We=0. and (b) We=0.0 (corresponding to flow rates of (a). 0 - and (b). 0 - m /sec). The hump and rim shows the braiding rivulet due to instability. The dynamic contact angle is defined by γ A =, γ R =. and γ e = 0 while the and static contact angle (γ) is 0.

28 (a) (b) Experiment CFD Experiment CFD Figure : Comparison of the CFD predicted rivulet shape using dynamic contact angle simulation (DCA) with that captured from experiment for water (Ka=) with γ A =, γ R =. & γ e = 0 and θ = 0 at two Weber numbers (a) We=0.0 and (b) We=0.00 (corresponding to low rates of (a) and (b) m /sec). In both cases, a dynamic meandering instability is observed.

29 0 0 (a) Experiment CFD 0 B (mm) (b) We 0 Experiment CFD 0 B (mm) Figure : Comparison between the predicted and measured rivulet width, (a) 0cS (Ka= and γ = ) and (b) 00cS (Ka= and γ = ) general purpose silicon oils with θ = 0. Width of the rivulet was measured at 0 mm from inlet. Insets show the predicted and observed shape of the interface from the simulation and experiment, respectively, at a fixed We number with (a) We=.0 and (b) We=0.. We

30 We=. We=0. We=0. We=0. We=0. Figure : Shape of the water rivulet (Ka=) at decreasing inlet Weber numbers (We) showing the rivulet breakup process and formation of a droplet with decreasing flow rate. θ = 0 and γ = 0.

31 (a) t = 0 t = t = t = t = t = 0 (b) Neck Bulbous Tip Capillary Pressure Figure : (a) Temporal evolution rivulet shape (f=0.0) for water at a Weber number (We) of 0. wherein end pinching leads to formation of the droplet with θ = 0 and γ = 0. Time (t) is non-dimensionalized as t = tt/d H μ. (b) A color map of capillary pressure at the onset of breaking (t * =) shows the greatest pressure appears at the neck.