Modeling Drop-Drop Collision Regimes for Variable Pressures and Viscosities

Transcription

1 5th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 9-2 January 22, Nashville, Tennessee AIAA 22-7 Modeling Drop-Drop Collision Regimes for Variable Pressures and Viscosities E. Loth and K. Krishnan 2 University of Virginia, Charlottesville, Virginia 2294 The collision of two spherical droplets in a gas is considered in terms of the five primary phenomenological outcomes: slow coalescence (SC), bounce (B), fast coalescence (FC), reflexive separation (RS), and stretching separation (SS). The boundaries that separate these outcomes were investigated empirically in terms of gas pressure, gas viscosity and droplet viscosity effects. Such effects are not accounted for robustly in the previous models, but can be important for hydrocarbon drops in pressurized sprays associated with many fuel systems. For the slow coalescence/bouncing (SC/B), a boundary prediction was developed based on a correlation of experimental data with viscous effects and gas pressure. For the B/FC boundary, the boundary model incorporated effects of drop and gas viscosity, which can affect the stability of the interface in different ways. Additionally, drop viscosity modifications were incorporated into the Brazier-Smith model for the FC/SS boundary and into a simplified Ashgriz-Poo model for the FC/RS boundary. Based on a review of available drop-drop collision data, the present empirical models were found to be reasonable, though approximate, for a wide variety of test conditions, including variation in drop diameter ratio, gas pressure, as well as density and viscosity of both the gas and liquid. However, additional analysis and data are needed to generalize these predictions for other effects such as pre-collision droplet shape, gas or liquid temperature, and droplet charge. Nomenclature d = droplet diameter v = velocity = normalized collision angle Δ = droplet size ratio δ = deformed droplet width normal to the collision velocity θ in = impact angle µ = viscosity ρ = density σ = surface tension Ca = Capillary number Kn = Knudsen number Oh = Ohnesorge number St = Stokes number We = Weber number ( ) p = Particle/droplet properties ( ) f = Fluid/gas properties Professor, Mechanical and Aerospace Engineering, 22 Engineer s Way, PO Box 4746, AIAA Associate Fellow Member 2 Graduate Student, Mechanical and Aerospace Engineering, 22 Engineer s Way, PO Box 4746, AIAA Student Member Copyright 22 by Eric Loth. Published by the, Inc., with permission.

2 I. Introduction Droplet-droplet collisions in a gas occur when two droplets move towards each other in flight and come into direct contact. The impact of two droplets can be important for many energy systems, particularly fuel sprays where the initial break-up occurs in dense regions with many drop-drop interactions. The collision outcomes can yield agglomeration, separation or even further break-up and such outcomes are critical to determine the overall size distributions, which in turn can affect overall combustion performance. Models for predicting this outcome have typically focused on water droplets in atmospheric pressure conditions. However, combustion sprays tend to occur at high pressures with a range of gas densities and viscosities, owing to a range of gas temperatures and species composition. Furthermore, the droplet viscosity and surface tensions of fuels often differ significantly from that of water droplets. As such, predicting the collision result under conditions of variable pressure, species compositions and viscosity of the gas as well as variable viscosity and surface tension of the liquid is important to understanding the dynamics of such systems. The most common interaction is a binary collision whereby two particles collide (trinary and more complex collisions are generally rare). A binary collision typically results in significant fluid particle deformation, particularly if the particles are of similar size. The dynamics can be classified by five types of interactions, based on Qian and Law 2, (hereafter referred to as Q&L) for which examples are shown in Fig.. Roughly ranging from low to high collision speeds, these five types can be termed and described as.: SC: slow coalescence where the droplets move slows that the deformation is limited and the interaction times is long enough time for the interfaces to merge by diffusion; B: bounce where the impact speeds yield significant deformation but the interaction is too fast such that the drops never merge; FC: fast coalescence whereby there are significant deformation dynamics after the impact (typically with mode 2 shape oscillations) causing the interfaces to break and merge, after which the dynamics subside and a single combined drop is formed; RS: reflexive separation where the droplets impact at roughly head-on angles and temporary coalesce due to the high impact speeds, but the high inertias and dynamics also cause a shape reflection (similar to a mode 2 oscillation), which leads to separation with two primary drops and often a smaller drop from the filament; SS: stretching separation or off-center separation where the droplets impact at roughly grazing conditions and temporary coalescence but then separate due to shape reflections (often more complex than a simple mode 2 oscillation) that leads to two primary droplets and often many satellite drops from the filament. These five regimes are most strongly controlled by the incoming droplet diameters, droplet density (ρ p ), impact angle, the liquid-gas surface tension (σ), and the relative velocity defined in Fig. 2 as v v in v in p p large small () To characterize the effects of these parameters, the inertia associated with the collision can be normalized by the surface tension effects by the collision Weber number defined by the small drop diameter as: 2 ρpvp pdsmall We (2) p p σ This parameter is the strongest determinant of the impact outcomes. The second most influential parameter is the angle of impact, which is finite if the collisions are not head-on. This collision skewness can be characterized by the incoming impact angle (θ in ) between the particle-particle collision plane and the relative incoming velocity vector as defined in Fig. 2. This angle can be used to define an impact parameter: in cos θ (3) ( ) Thus, a head-on collision corresponds to (θ in 9 o ) while a nearly tangential grazing condition corresponds to (θ in o ). The third most influential non-dimensional parameter is the drop size ratio (Δ), which simply normalizes the small diameter (d small ) by the large diameter (d large ) as: 2

3 d small Δ (4) dlarge As the drop size ratio becomes much less than unity, the collision outcome will depend primarily on the properties of the small droplet size (since the large droplet will approach that of an infinite liquid bath). If one ignores viscous effects (which are often weak), the droplet collisions are thus primarily characterized by three non-dimensional parameters: We p-p, and Δ. For a fixed size ratio, the dependence of the Weber number and impact parameter on the collision outcome (SC, B, FC, SS, and RS) can be expressed in terms of a nomogram, e.g. Fig. 3 for a water droplet at high pressure. In this figure, lines are drawn to indicate the qualitative boundaries between the regimes. One may note that there is significant experimental uncertainty regarding the boundaries between these regimes, which is typical of such measurements and is attributed to the difficulty in prescribing and measuring the impact conditions as well as in controlling any pre-collision instabilities. The boundary between slow coalescence and bouncing (SC/B) is observed at very low Weber numbers for head-on collisions. The slow coalescence regime occurs when the liquid interfaces can be connected through a drainage process over a sufficiently long time. The drainage process is due to molecular diffusion of the two interfaces in close proximity, ca. angstroms (Q&L). Note that the droplets remain approximately spherical (Fig. ) throughout the slow coalescence collision process up until the interfaces merge. At near-grazing conditions, there is little experimental evidence of this process occurring and this may be attributed to reduced contact time for the collision process in such conditions compared to head-on collisions. In cases where the diffusion process does not occur fast enough for the interfaces to significantly merge (We p-p >We SC ), the droplets will tend to bounce apart. At higher impact velocities (ca. We B ), the droplet deformation becomes much larger. When their minimum centroid distance is reached, this leads to a condition where the droplet interfaces are nearly flat and a thin sheet of gas occurs between the two drops. If the gas interface integrity can be maintained, then the droplets will repel as two separate entities, i.e. they will bounce. One may expect that increased gas pressure and increased gas viscosity will serve to improve the probability of a bounce outcome in such conditions. However, instabilities along this thin gap will tend to provoke rupture of the interfaces leading to a merger of the two droplets, and this is identified as fast coalescence. As such, off-axis collisions with smaller normal velocity components are more likely to bounce so that the Weber number for the B/FC boundary will increase as the increases. When the particle Weber number becomes very large, the internal dynamics of the droplet after merger become more exaggerated. If they are not sufficient to cause re-separation, then fast coalescence (FC) is still observed. However, if the temporarily coalesced entity has very high internal kinetic energy, this leads to strong interface dynamics which can cause the droplets to re-separate (Fig. ). This especially occurs if the process is a head-on collision (leading to RS) or a near-grazing condition (leading to SS). Brazier-Smith et al. 3 showed that increasing the collisional Weber number for the stretching separation regime also increases the number of small satellite drops which are formed, due to increased instability of the temporary filament. At very high impact speeds (ca. We p-p >), a shattering separation occurs whereby the combined mass rapidly breaks up into several drops 4. However, this sixth regime and other rapid break-up regimes are beyond the scope of this study. In all of the five regimes studies herein, viscous effects can generally stabilize a drop collision and allow increased critical Weber numbers. In some cases, the gas viscosity may also influence the collision outcome, if a cushion of gas is formed. To characterize their influence, it is convenient to define the particle and gas Ohnesorge numbers based on their respective viscosities (μ f and μ p ) and densities (ρ f and ρ p ) and the interfacial surface tension (σ): μp Ohp ρσ d (5a) Oh p μ f f = ρσ f d Oh p ρ / ρ p p f f μ / μ These two dimensionless parameters relate internal and external viscous stresses to surface tension stresses. For drops in gasses, Oh p >Oh f so that one may expect Oh p to generally dominate the viscous influence for deformation. (5b) 3

4 II. Previous Experimental Studies and Boundary Predictions There have been several studies which have examined the physics and proposed quantitative models for some of the outcome boundaries. In order to quantify the boundaries for head-on collision (=), three critical Weber numbers can be defined: We SC as the SC/B intercept, We B as the B/FC intercept, and We FC as the FC/RS intercept. Since the FC/SS boundary does not generally occur for head-on collisions and has no intercept, it can be quantified by defining a fourth critical Weber number, We SS, as the FC/SS boundary at =.5. These boundaries have been primarily investigated for water droplets in air at standard pressure and temperature (p o =,32 N/m 2 and T o =293 K). The most common of these predictive models are discussed below and compared to experimental data at varying gas pressures and drop viscosities, as seen in Fig. 4. A. The SC/B Boundary: There are few, if any, studies which model the boundary between slow coalescence and bouncing for the conditions of the present study: deforming drops in a continuum gas. However, there are some related studies at different conditions. Bach et al. 5 considered very small drops impacting on a nearly planar interface. For these small sizes and very low impact velocities, the Knudsen number (Kn, which is a measure of molecular mean free path to the droplet diameter) was no longer small indicating the presence of non-continuum effects, while the Weber number was very small (<<) indicating negligible deformation. In contrast, the focus of the present study is on drops of hundreds of microns (Kn< -3 ) with significant Weber numbers (>) so deformation will be important but non-continuum effects are expected to be negligible. Another study considered the rebound of solid particles in a liquid bath, and showed that the rebound could be damped and even eliminated by the surrounding fluid viscosity. 6 This effect was found to be controlled by the impact Stokes number, which can be expressed as St =We /2 /Oh, whereby no bouncing occurred for St < 5 and this damping effect was weak or negligible for St >25. 6 Since drop-drop experiments in gas have small Oh f values, they also tend to yield large impact Stokes numbers (in the range of 25-5 for the experiments modeled herein), for which this effect is not expected to play a large role. B. The B/FC Boundary: Estrade et al. 7 proposed a prediction for the boundary between bouncing and fast coalescence, by assuming that bouncing will occur if the initial kinetic energy does not exceed that needed to create a minimum deformation limit. This prediction takes the following form: 2 Δ( +Δ )( 4φ -2) WeB/FC = 2 χ ( - ) (6a) ( )( ) χ= - 2-ψ +ψ /4 if ψ>. 2 ψ ( 3-ψ ) /4 if ψ. (6b) ψ ( -)( +Δ ) (6c) 2/3 /3 3 3 φ (6d) 4φ 4φ In this expression, φ is defined as the ratio of δ large /d large or the ratio of δ small /d small beyond which coalescence or separation occurs, with δ being the deformed droplet width normal to the collision velocity. The predictions based on the above model correlated reasonably well with the data of Estrade et al. 7 for methyl alcohol drops in air at one atmosphere. However, this model is only qualitatively correct when evaluated with water droplets. In particular, the We B/FC values tend to be substantially over-predicted for one atmosphere pressure (Fig. 4a) and under-predicted at eight atmospheres pressure (Fig. 4b). This indicates a significant sensitivity to gas pressure, which is absent from their model, but consistently observed by Qian & Law. 2 This effect is attributed to a gas cushion which can develop between the two droplet interfaces at the minimum gap distance. This cushion can form when the gas is not able to squeeze out fast enough (and thus may be a function of the gas viscosity) and tends to prevent interface coalescence which thus tends to promote bouncing. Furthermore, droplet viscosity can influence the B/FC boundary since it can dampen the dynamics of the liquid motion, which determine whether the interface will break. This influence is demonstrated by Fig. 4c whereby fast coalescence is less likely to occur for tetradecane droplets than water droplets at the same Weber number. However, these gas and drop viscosity effects are neglected 4

5 in the model given by Eq. (6), which may explain why the model is not robust to materiel changes in the gas or the liquid. C. The FC/RS Boundary: Ashgriz and Poo 8, hereafter referred to as A&P, developed a model for the appearance of reflexive, or head-on separation. This was based on the total surface energy of the drops compared to the internal reflexive kinetic energy of the fluid within the droplets. Their criterion states that reflexive separation will occur if the reflexive kinetic energy of the two combined droplets is greater than 75% of the surface energy. This postulate leads to the following form: ( 3 2/3 ) ( 2 ) We =3 7 +Δ -4 +Δ 2 2 FC/RS /2 ( ) ( ) ( ) ( ) η =2 -ξ -ξ - ( ) /2 3 η =2 Δ-ξ Δ -ξ -Δ ξ= +Δ 3 ( ) 2 Δ +Δ Δη+η 2 (7d) This model correctly predicts the qualitative trends observed in experiments, but is not always quantitatively consistent with the results for liquids with higher droplet viscosity, whose We FC values are under-predicted as shown in Fig. 4c. This can be attributed to the internal drop viscous losses which dampen the kinetic energy required for separation, an effect that was recognized and studied by Jiang et al. 9 Qian & Law 2 showed that this phenomenon can be reasonably expressed in terms of the droplet Ohnesorge number defined in Eq. (5a). In particular, they suggested that the intercept value can be modeled as We 68Oh + 5 (8) FC p It should be noted that these values were obtained by assuming a nearly vertical (i.e. blunt) intercept for the FC/RS boundary, but this is different than the shallow intercept behavior observed in experiments and predicted by the model of A&P (Fig. 4). D. The FC/SS Boundary: Several different relations have been proposed for the FC/SS boundary and of the most common models are discussed below. Brazier-Smith et al. 3 proposed a model by considering the necessary energy required to form two separate droplets. Their criterion involved the amount of rotational energy required to overcome the surface energy of the two particles in order to collide and re-separate. This led to the following boundary description: We -2-3 { + Δ -( + Δ ) -3 }( + Δ ) -6 Δ ( - Δ ) 4.8 = + FC/SS 2 2 2/3 /3 This model was an improvement over a similar model developed by Park, which utilized a relationship between the surface tension and the angular momentum of the particles at their initial point of contact. Arkhipov later developed a formulation based on the minimum-potential-energy variational principle. Assuming a system with constant angular velocity, the boundary suggested by Arkhipov can be described as: Δ ( ) We FC/SS = Δ 3 2 () A&P expressed some reservations about the models of Park and Arkhipov, and pointed out that the criteria proposed by Brazier were not always observed. Therefore, A&P suggested that separation was actually caused by the particle s linear momentum and proposed an alternative prediction to describe the boundary. In particular, they assumed that separation occurs if the total effective stretching kinetic energy is larger than the surface energy of the region of interaction. Their model can be expressed as: (7a) (7b) (7c) (9) 5

6 ( 3 2 ) ( )( )( 3 small arge ) Δ ( +Δ )-- ( )( φ small +Δ φarge ) 4+Δ 3+Δ - Δ φ +φ We FC/SS = - 2Δ-τ Δ+τ for h> d 2 ( ) ( ) 3 2 small 4Δ φ small = 2 τ 3 ( 3Δ-τ ) for h< d 2 small 4Δ 2 ( ) ( ) 4 2 arge φ arge = τ 2 ( 3-τ ) for h< d 2 arge ( )( ) ( small large )( ) 2-2-τ +τ for h> d 4 τ +Δ - h d +d - This model is more complex than those of Eqs. (9) or (), but is still only a function of and Δ. The predictions from Eqs. (9-) are quite similar and reasonable for water droplets at one atmosphere (Fig. 4a). Since this is the most commonly studied condition, their prediction models are perhaps the most commonly cited. Their predictions are also reasonable for water at eight atmospheres pressure (Fig. 4b) indicating that gas pressure effects are not substantial for this boundary. However, for hydrocarbons with increased droplet viscosity, the above inviscid predictions tend to under-predict the We FC/SS number for a given. This is attributed to increased dissipation of energy which can prevent the droplets from re-separating once temporarily coalesced. This tendency is consistent with experiments of Jiang et al. 9 for a wide variety of liquid/droplet properties. Brenn and Kolobaric 2 proposed a model to include the effect of particle viscosity on the FC/SS boundary which gave good results for high viscosity liquids (Oh p on the order of.5 and greater). However, their predictions yielded poor results for low viscosity liquids like water and alcohol. For example, the predictions for 3 μm water drops with an Oh p of.68 based on Brazier-Smith et al. 6 yield We FC values in excess of, while experimental We FC values are only on the order of Thus, a robust model for this boundary is not currently available. III. Modeled Boundary Relationships As noted above, predictions from previous models were found to be only qualitatively consistent for boundary predictions. This lack of fidelity motivated the present work, whose objective was to develop quantitative boundary models which take into account effects of droplet and gas viscosity, as well as the pressure and density of the surrounding gas. A second objective was to keep the models relatively simple so that they may be computed rapidly for a large number of droplet interactions. The experimental data used for comparison herein is first discussed, followed by proposed models for each of the four aforementioned boundaries. The formulations for these boundaries are developed in Figs. 5-7, and then evaluated with all available experimental data in Figs A. Experimental Data Used for Comparison To investigate the various collision regimes, a comprehensive review was undertaken to obtain quantitative nomograms from all relevant experimental studies. On the basis on the data quality and quantity of data in terms of identifying individual droplet collision outcomes (e.g. a bounce), the following data bases were selected: Q&L 2, Estrade et al. 7, A&P 8 and Jiang et al. 9. Table lists the droplet and gas properties employed for Figs. 8-6.However, the data has some limitations. In addition to the outcome uncertainty previously mentioned, some of the test conditions only reported a range of droplet diameters (as opposed to a mean droplet diameter for each point on the nomogram). To model these cases herein, the average droplet diameter was employed to compute the Oh p, Oh f, etc. The largest expected deviation from this average from was the case of Q&L who reported diameters ranging from 2 μm to 4 μm. The use herein of an average diameter of 3 μm resulted in an uncertainty of about ±4% in the We and ±2% in the Oh p. /2 (a) (b) (c) (d) (e) 6

7 B. The SC/B Boundary: The mechanism of slow coalescence requires that there be sufficient time for a liquid bridge to form and drainage of the liquid through that bridge before the drop collision time is exceeded. The boundary between slow coalescence and bounce has not been previously modeled, but it is known that droplet and gas properties are important mechanisms which control the slow coalescence phenomenon. Since the influence of slow coalescence is highest at the head-on collision condition, we first seek prediction of We SC, the intercept value of this boundary. It is helpful to consider the experimental trends in developing such a model. Several physical parameters were studied and the strongest functional correlation was found to be: We SC = f (μ p,σ,ρ p,p) (2) Foremost, the data of Jiang et al. 9 indicate that We SC is strongly influenced by droplet viscosity and surface tension. In particular, comparing the water droplets and tetradecane droplets from Figs. -4, the We SC increases (i.e. slow coalescence is more likely for a given Weber number) as the droplet viscosity decreases and surface tension increases. This trend is consistent with the mechanism of drainage, whereby decreased viscosity allows the interfaces to coalesce before the reflection period is exceeded. By examining additional data it was found that both trends were independently observed, which is consistent with decreasing the Ohnesorge number of Eq. 5a. A more surprising trend is the substantial influence of gas pressure as shown in Qian & Law 2. As the pressure is increased, We SC decreases indicating that bouncing is more likely. This suggests that a gas cushion can form between the drops which can make it more difficult for a liquid bridge to form across the interfaces. By examining all the available experimental data which provided quantitative values of We SC, the following non-dimensional grouping (based on the parameters of Eq. 2) was found to the give the best correlation of the trends as shown in Fig. 5a: σ ρ p.5 σ (3) WeSC =.5 = μ p p Oh p pd The correlation of Fig. 5a appears to be quite strong indicating that this effect is significant for the current conditions. In contrast, the capillary, Mach and Knudsen numbers (as suggested by Bach et al. 5 ) did not robustly correlate the observed experimental trends studied herein. This lack of correlation may result from the conditions investigated which focused on drops of several hundred microns with a Kn< -3. To approximate the relationship of the SC/B boundary for a general collision angle, the experimental variations were considered. These indicated that slow coalescence did not occur for grazing conditions (=) and that the boundary tended to decrease monotonically with. Since the data was relatively scarce and limited, a linear variation was employed: We SC/B =WeSC ( - ) (4) This simple variation is empirical, and further investigation is warranted to determine a theoretical basis for this variation. The SC/B boundaries from Eqs. (3) and (4) were predicted for all the data of Table and are shown in Figs The SC/B boundary performs well for atmospheric condition cases such as water droplets in air (Fig. 8), water drops in nitrogen (Fig. a) and tetradecane drops in nitrogen (Fig. 3b). In addition to that, the model accurately predicts the SC/B boundary for high pressure environments such as the water drops in helium at 4.4 atm and 7.5 atm (Fig. 2a and b) and tetradecane drops in 2.4 atm helium environment. However, the model could not be verified for some high pressure cases as slow coalescence was not observed in such experiments, e.g. Figs. c, 2c, 3c, 4c as well as 5b and 5c. There was the case of tetradecane drops in helium at.7 atm shown in Fig. 4a, suggests the boundary of SC/B is described by We SC as high as 2, but a lower We SC could also be argued due to the void of data between of.2-.8 at Weber number of 3-7. The only distinct mis-prediction for the present model is the case of the tetradecane droplets in ethylene at 8 am (Fig. 5a) and also shown in Fig. 5a. Here it can be seen that Eq. (3) dramatically under-predicts the intercept Weber number by an order of magnitude. This anomaly was suggested to be a result of vapor concentration by Estrade et al. 7 who expected that the same large effect would occur in terms of humidity for water drops. However, Ochs III et al. 3 investigated water droplets at different relative humidities and saw only a minor effect in terms of collision outcome (see Fig. 5a and Fig. 8). This suggests another mechanism is the primary cause for the increase in We SC. One possibility is that the high pressure (8 atm) in Fig. 5a coupled with slight temperature differences may have caused the ethylene to condense a thin liquid film around the tetradecane droplets. In this case, the slow coalescence process would be controlled by ethylene s surface tension (as opposed to that of tetradecane). If one 7

8 were to assume such a liquid film, the predicted We SC would be much higher and indeed it agrees well with the experimental data as shown in Figs. 5a and 5a (denoted by the blue line). Note that any condensation due to humidity on a water droplet would not change the surface tension, which is consistent with the lack of a change shown in Fig. 8. However, it is noteworthy that all the experimental data considered herein included liquids and gases at room temperature. If a diffusion principle is dominant, then one may expect temperature effects could be significant indicating that the empirical factor of Eq. (3) is not necessarily a constant. C. The B/FC Boundary: To model the boundary between bouncing and fast coalescence, an attempt was made to first identify the liquid and gas properties that primarily influence the intercept value (We B ) obtained from the experimental data. The primary physics of the interaction are inviscid, and thus controlled by a critical Weber number. As the droplets approach each other but have not merged, a flattening effect occurs on the drop surfaces whereby nearly disk-like surfaces are separated by a thin gas film. If the interfaces have small instabilities, they can come into direct contact and cause rupture. If fast coalescence occurs, these interfaces fully merge before the drops rebound away from each other. This merging phenomenon converts the kinetic energy of the droplets into surface energy, and this criterion can be used to determine the critical Weber number (We B ) needed if viscous effects are ignored. 2 However, fluid properties may also affect the B/FC transition. The strongest influence is that of droplet viscosity. Comparing Figs. a and 3b, an increase in the We B correlates with an increase in the droplet viscosity (μ p ), at the same pressure. This trend is further substantiated by comparing Figs. 2a and 4c and Figs. b and 3c between water and tetradecane droplets at similar pressures. This trend is also corroborated by the results of Jiang 9 through experiments conducted with hydrocarbon droplets with increasing droplet viscosity. The increase in We B is consistent with an increased resistance to interface merging. Since high droplet viscosity can dampen the dynamics of the interaction, it may prevent instability of the interface. As a result one may expect increases in Oh f to decrease the We B, consistent with much of the observed experimental trends. In terms of gas effects, it was also observed (through comparisons of Figs., 2, 3 and 4a-c) that the We B generally increases with decreasing gas viscosity. The We B observed for helium environment is lower than those observed for nitrogen environment which suggests an influence of the Oh f. The potential mechanism for the influence of gas viscosity may be related to the gas cushion which can stand between the flattened drop surfaces as they interact. If the gas viscosity is small, this cushion may not be maintained because the gas would quickly escape. In contrast, high gas viscosity can support this cushion which prevents merging of the interfaces, and thus makes bouncing more likely. As a result one may expect increases in Oh f to decrease We B, When the data of We B was examined and correlated with various forms of Oh p and Oh f, a surprising nonmonotonic relationship was found as shown in Fig. 5. In particular, it was found that the trends correlated well with the ratio of Oh p /Oh f, and that there appeared to be a maximum We B at an intermediate ration of Oh p /Oh f. This suggests that the drop interaction is most stable with a particular ratio of the fluid viscosities. In particular, very low drop viscosities allow higher drop dynamics which promote rebound velocities (decreasing We B ) while very low gas viscosities remove cushioning effects and allow interface instability which promotes merging (also decreasing We B ). Note that even when the gas viscosity is constant, very high droplet viscosity (Oh p >.) can reduce We B (as observed by Gotaas et al. 4 ) indicating that it the instabilities are related to the viscosity ratio, consistent with studies by Payr et al. 5 The equations describing a fit to the correlations of Fig. 5b are listed below: (6a) We B =2.25(Oh p /Oh f ) for Oh p /Oh f 7.5 We B =4(Oh f /Oh p ) 2 for Oh p /Oh f > 7.5 (6b) The influence on collision angle was obtained based on the We dependence on, since the form of Eq. (6) was found to be consistent with experimental trends. 6 This dependence was simplified to a single-closed form expression and combined with the intercept definition yielding: We B( + ) WeB/FC = (7) ( ) In general, the proposed model is reasonable in predicting the boundary separating the bounce and head-on separation (B/FC) boundary for atmospheric conditions such as Figs. 9, and 3b. Although it may have overpredicted the case of water drops in nitrogen, it is hard to segregate the effects slow coalescence and bounce in because of uncertainty (outcome overlap) of the experimental data. The empirical model performs extremely well for high pressure cases notably Figs. 2 a-b and 5 but shows only qualitative agreement in Fig. 3a and Figs. 4b-c. 8

9 It is also important to note that the above model performs well for highly viscous droplets reproduced in Figs. 6a-c. However, the non-monotonic behavior of Fig. 5b suggests that the viscous effects are complicated and may not be described simply by the ratio Oh p /Oh f. Theoretical analysis and further experimentation for a range of drop and gas viscosities is needed to understand the mechanisms responsible for this complex trend. D. The FC/RS Boundary: As discussed earlier, the boundary between fast coalescence and reflexive separation was treated theoretically by A&P for inviscid conditions and this gave reasonable agreement at low droplet viscosities. However, Q&L noted the influence of Oh p at high liquid viscosity was significant. This effect can be traced to the reduced rebound dynamics associated with increasing the droplet viscosity, as discussed by Bayer & Megaridis 6. This can reduce the velocity associated with stretching the merged droplets apart, and thus can stabilize the coalescence and increase We FC. Q&L proposed a linear relationship to take droplet viscosity into account for the intercept value (We FC ) assuming a blunt, nearly vertical, intercept curve. Their relationship is given by Eq. (8) and is shown as a dashed line in Fig. 6. Also included in this figure are the We FC values obtained by assuming a blunt intercept based on Q&L. It can be seen that increasing Oh p generally leads to increased We FC, i.e. increasing viscous effects tends to prevent the temporarily coalesced droplets from re-separating due to dissipation of the kinetic energy. While the blunt intercept values (open symbols of Fig. 6) obtained by Q&L an expression for the intercept by Q&L as given by Eq. (8). Since the model predicted by A&P (Eq. (7)) and demonstrated in most of the experimental data (e.g. Figs. 9-6) indicated an angled-intercept, the use of Eq. (8) generally led to an overprediction of the boundary Weber numbers once considered for a variety of collision angles. Therefore, the measurements from the nomograms were analyzed assuming an angled intercept and this is shown by the solid symbols of Fig. 6. As before, a linear relationship describes the trend but with somewhat different constants: We FC =582Oh p +2 (8) In general the agreement to this trend line on Fig. 6 is quite good. However, there is one notable exception. The We FC values obtained by Adams et al. are much higher than predicted, but this is attributed to the charge they imposed on the particles. Such a result would be consistent with findings of Czys and Ochs 7 who noted that charged droplets are more likely to coalesce even when the charges are of the same sign (since the difference in magnitude between charges can cause attraction). To obtain the collision angle dependence, the four-equation expression given by A&P was simplified to a single equation for computational convenience as: We We FC FC/RS = (9) -3 Δ ( ) Once again, this expression assumes that the larger droplet determines the outcome of the collision. The combination of Eqs. (8) and (9) were then evaluated with the measurements of Figs. 9-6 (which did not include any intentionally charged droplets). The effect of diameter ratio (Δ) is shown in Figs. 9 and, where the changes predicted by A&P s model are consistent with the experimental data. Based on Figs. and 2, one can note that We FC is relatively independent of the gas properties (density, pressure, etc.), but comparison with Figs. 2 and 3 indicates the significant influence of droplet viscosity for both predictions and measurements. E. The FC/SS Boundary: Similar to the phenomenon associated with the FC/RS, the primary mechanism for the boundary between fast coleasence and stretching separation is the inviscid dynamics, but droplet viscosity can also play a significant role. The inviscid Brazier-Smith et al. 3 and A&P models for the FC/SS boundary behaved reasonably well for water droplets, the experiments of Jiang et al. 9 and others suggest that the FC/SS boundary tends to larger Weber numbers as the particle viscosity (and thus Oh p ) increases. The recent experiments conducted by Gotaas et al. 4 with high viscosity particles suggested a similar trend with increasing viscosity although with a varying degree of dependence. To determine this dependence, measured values of We SS were evaluated as a function of Oh p for the available experimental data of Table. As with the We FC dependence, the We SS showed a consistent increase for higher particle Ohnesorge numbers. This trend is shown by Fig. 7 and can be approximated as We =63 Oh +2 for Oh.3 (2a) ( ) ( ) SS p p We =35 Oh +36 for Oh >.3 SS p p This empirical dependence was based on a wide variation of surrounding gas properties (pressure, density and chemical composition) and particle properties (diameter, viscosity, and surface tension). This result is also 9 (2b)

10 qualitatively consistent with data obtained by Brenn et al. 8 which focused on satellite formation. It should be noted that Gotaas et al. 4 suggested a power law model of the form below for highly viscous particles. We = 939(Oh ) for Oh >.4 (2).756 SS p p However, since one of the goals of this study is to provide simple models and since there is a lack of experimental data at very high Oh p values, the simpler linear model of Eq. (2) was deemed acceptable for the high viscosity region as shown in Fig. 7. To incorporate a collision angle dependence, the form of the Brazier-Smith boundary equation is employed for the present model as it is quite reasonable in the limit of very low Oh p values. The combination of Eqs. 9 and 2 to take into account increasing particle viscosity and diameter ratio becomes: We -2-3 { + Δ -( + Δ ) -3 }( + Δ ) -6 Δ ( - Δ ) We = SS FC/SS 2 2 2/3 /3 As expected, this model predicts the SS region boundary quite well for water droplets (Figs. -2) whilst maintaining an accurate prediction of the viscosity dependence for higher viscosity droplets as shown in Figs. 6 a- c. However, the FC/SS boundary is difficult to clearly quantify in many cases since some of the nomograms contain overlapping coalescence and separation outcomes, which is attributed to uncertainty in the individual drop test conditions. F. Limitations of the Proposed Models Besides the uncertainty of the experimental data (which translated into uncertainties in the coefficient and functional relationships as discussed above), some of the above cited limitations of the proposed models include lack of influence effects of non-continuum effects, electric charge, temperature of the gas and droplet, and heat transfer. In addition, Willis & Orme 9 also investigated the collision of ellipsoidal drops, which can be quite complex as compared to that of their spherical counterparts. They noted that collision process is sensitive to the particle shapes just before the collision. This can be an issue at significant aerodynamic Weber numbers (We p ) where droplets become non-spherical prior to collision. The aerodynamic Weber number can be related to the impact Weber number in terms of the density ratio:we p ρ f v p 2 d/σ=we p-p (ρ f /ρ p ). However, an initially spherical droplet which has recently undergone coalescence or separation will be significantly non-spherical after the interaction until the shape oscillations damp out. Therefore, an ensuing secondary collision could involve non-spherical collisions even when the aerodynamic Weber number is small. Other issues not modeled include effects of temperature gradient. In particular, Neitzel & Dell Aversana 2 noted that the critical pressure for coalescence (related to the drop vapor emission) decreases with increasing temperature gradient. The presence of surfactants, which can inhibit coalescence, is another issue. Models for coalescence rates especially in contaminated systems is still quite empirical due to the complexity of the drainage nano-physics. 2 Finally, the non-monotonic behavior noted in Fig. 5 for the boundary between bouncing and fast coalesce suggest competing effects me be present indicating a single non-dimensional parameter to describe both the pressure and viscous effects may not be sufficient. Thus significant further research studies (experiments, theoretical analysis, and detailed interface-resolving numerical simulations) for a variety of test conditions are warranted to more fully understand and quantify the collision mechanism details and to better model the outcome boundaries. IV. Conclusions The prediction boundaries developed herein takes into account the viscous effects of the droplet and surrounding gas through the use of the Ohnesorge numbers. This appears to be one of the first attempts to model the SC/B, B/FC, FC/SS regimes with such effects. Different influences of parameters are seen for different boundaries. The SC/B boundary has an influence of pressure whilst suggesting a minor influence, if any, of relative humidity. The B/FC boundary is controlled by the gas properties especially viscosity. The FC/RS and FC/SS boundaries are sensitive to the viscosity of the droplet and the drop size ratio. The models are applicable for various droplets and gas environments, particularly of the interest of combustion sprays, and variable conditions of pressure. It is also possible to extrapolate the models presented to conditions outside the range described in the literature. The versatility and robustness of the models also makes incorporation of these models into computational schemes a feasible option without a significant increase in computational time and resources. However, one major shortcoming of these results is that the test conditions are limited, e.g. they do not include temperatures away from standard conditions, effects of pre-collision droplet deformation, nor the influence of size for changes in pressure, gas (22)

11 composition and drop compositions. Additional experimental and numerical data are needed to integrate these effects over a wide range of droplets, density ratios, viscosity ratios, Weber and Reynolds numbers, etc. In this regard, recent numerical advances in treating deformable particle may be of substantial benefit.

12 Table. Gas and liquid properties where TetDec refers to tetradecane. Data source d small (µm) Δ Liquid ρ p (kg/m 3 ) µ p x -6 (kg/m-s) σ (N/m) Gas μ f x -6 (kg/m-s) Q&L 3 Water 2.73 Nitrogen 7.4 Q&L 3 Water 2.73 Helium 9.4 Q&L 3 TetDec Nitrogen 7.4 Q&L 3 TetDec Helium 9.4 Q&L 3 TetDec Ethylene 9.6 Estrade 2.5, Ethanol Air 8.2 A&P 3.5- Water 2.73 Air Heptane Air Decane Air 8.2 Jiang 3 Dodecane Air TetDec Air Hexadecane Air 8.2 Ochs Water 2.73 Air, Vapor 8.2 Law 24 TetDec Air 8.2 2

13 Slow coalescence Bounce Fast coalescence Reflexive Separation Stretchingseparation Figure. Different types of droplet-droplet interactions for a hydrocarbon liquid with increasing relative velocities. (Qian & Law, 997) 3

14 large particle (d large ) collision plane v small v large small particle (d small ) ϑ in v p-p =v large -v small Figure 2. Collisional geometry between a large particle of diameter d large and a small particle with diameter d small. The impact angle is 9 o for head-on collisions and approaches o for grazing collisions. 4

15 Bouncing Coalescing Separation.8.6 We SS SS.4 B FC.2 SC RS 2 3 We SC We B We FC We p-p Figure 3. Stability nomogram showing approximate droplet-interaction boundaries of water droplet interaction in a 7.5 atm helium environment. (Qian and Law, 997) 5

16 Bouncing Coalescing Separation Brazier-Smith (FC/GS) Ashgriz-Poo (FC/GS) Arkhipov (FC/GS) Estrade (B/FC) Ashgriz-Poo (FC/HS) We p-p Figure 4. Previous prediction models in nitrogen at: a) water drops in atm and b) water drops in 8 atm and c) tetradecane drops in atm. (Qian and Law, 997) 6

17 a) 25 2 Assume ethylene film b) We SC We B Q & L Water Q & L Tetradecane Jiang et al Hydrocarbons Ochs et al Water (Low humidity) Ochs et al Water (High humidity) Law et al Tetradecane Model 5 σ ρp μ p p Oh p /Oh f Q & L Water Q & L Tetradecane Jiang Hydrocarbons Estrade Ethanol A&P Water Ochs Gotaas et al. Low Oh Fit High Oh Fit Figure 5. Measurements and empirical correlations of boundary intercept between: a) slow coalescence and bouncing and b) bouncing and fast coalescence. 7

18 7 6 5 We FC Oh p Jiang et al. (blunt intercept) Adams et al (charged particle) Jiang et al. (angled intercept) Q&L Nitrogen (angled intercept) Q&L Helium (angled intercept) Estrade (angled intercept) Q&L (blunt intercept) fit Low Oh fit Figure 6. Influence of Ohnesorge number on intercept between fast coalescence and reflexive separation. 8

19 a) 5 4 We SS 3 2 Q&L Nitrogen Q&L Helium Q&L Ethylene Estrade A&P Low Oh fit Oh p b) Low Oh Data Gotaas et al. Low Oh fit High Oh Fit We SS Figure 7. Influence of Ohnesorge number on intercept between: a) fast coalescence and reflexive separation and b) fast coalescence and stretching separation for particles. Oh p 9

20 We p-p Figure 8. Water droplets in air at atm atrelative humidity of: ~34% (red) and >95% (green). The difference in Weber number is amplified so that both sets of data could be shown side by side for comparison. (Ochs et al., 989) 2

21 Bouncing Coalescing Separation FC/GS FC/HS B/FC SC/B.8 Δ= Δ= We p-p Figure 9. Droplet-droplet regimes for ethanol in air at one atmosphere. (Estrade et al. 999) 2

22 Δ= Δ= Δ= We p-p Figure. Droplet-droplet regimes for water drops in air at one atmosphere. (Ashgriz-Poo, 99) 22

23 atm N atm N atm N We p-p Figure. Droplet-droplet regimes for water drops in nitrogen. (Qian & Law, 997) 23

24 atm He atm He atm He We p-p Figure 2. Droplet-droplet regimes for water drops in helium. (Qian & Law, 997) 24

25 atm N atm N atm N We p-p Figure 3. Droplet-droplet regimes for tetradecane drops in nitrogen. (Qian & Law, 997) 25

26 atm He atm He atm He Figure 4. Droplet-droplet regimes for tetradecane drops in helium. (Qian & Law, 997) 26

27 a) b) c) Figure 5. Droplet-droplet regimes for tetradecane drops in a) % ethylene b) 5% ethylene-5 % nitrogen and c) % nitrogen in 8 atm. (Qian & Law, 997) 27

28 a) b) c) We p-p Figure 6. Droplet-droplet regimes for a) monoethylglycol, b)diethylglycol and c) triethylglycol in air at atm. (Gotaas et al., 27) 28

29 References Adams, J.R., Lindblad, N.R. & Hendricks, C.D. (968), The Collision, Coalescence, and Disruption of Water Droplets Journal of the Applied Physics Vol. 39, pp Qian, J. and Law, C.K., (997) "Regimes of Coalescence and Separation in Droplet Collision," Journal of Fluid Mechanics, Vol. 33, pp Brazier-Smith, P. R., Jennings, S. G. and Latham, J., (972) The Interaction of Falling Water Drops: Coalescence, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 326, No. 566, pp Qian, J., Tryggvason, G.and Law, C. K. (997) An experimental and computational study of bouncing and deformation in droplet collision AIAA , Aerospace Sciences Meeting and Exhibit, 35th, Reno, NV, Jan Bach, G. A., Koch D. L. and Gopinath, A. (24) Coalescence and bouncing of small aerosol droplets, Journal of Fluid Mechanics, 58, pp Loth, E. (2) Point-force Collision Models for Solid and Fluid Spherical Particles, 7 th International Conference on Multiphase Flow, Tampa, FL. 7 Estrade, J.-P., Carentz, Hervé, Lavergne, G., and Biscos, Y., (999) Experimental investigation of dynamic binary collision of ethanol droplets a model for droplet coalescence and bouncing, International Journal of Heat and Fluid Flow, Vol. 2, pp Ashgriz, N. and Poo, J. Y. (989) Coalescence and separation in binary collisions of liquid drops, Journal of Fluid Mechanics, Vol. 22, pp Jiang, Y. J., Umemura, A., and Law, C. K., (992) An experimental investigation on the collision behaviour of hydrocarbon droplets, Journal of Fluid Mechanics, Vol. 234, pp Park, R. W. (97) Behavior of water drops colliding in humid nitrogen, Ph.D. thesis, Department of Chemical Engineering, The University of Wiconsin, p.577 Arkhipov, V.A, Vasenin, I.M & Trofimov, V.F. (983) Stability of colliding drops of ideal liquid, Tomsk. Translated from Zh.Prikl. Mekh. Tekh. Fiz. 3, Brenn, G., & Kolobaric (26) Satellite droplet formation by unstable binary drop collisions Physics of Fluids, Vol. 8, Art Ochs III, H. T., Beard, K. V., Laird N. F., Holdridge, D. J, and Schaufelbergert, D. E., (995) Effects of Relative Humidity on the Coalescence of Small Precipitation Drops in Free Fall,Journal of the Atmospheric Sciences, Vol. 52, No.2 4 Gotaas, C., Havelka, P., Jakobsen, H-A., Svendsen, H.F., Hase, M., Roth, N. and Weigand, B., (27) Effect of viscosity on droplet-droplet collision outcome: Experimental study and numerical comparison, Physics of Fluids, Vol. 9, Payr, M. Vanaparthy, S.H. and Meiburg E. (25) Influence of variable viscosity on density-driven instabilities in capillary tubes, Journal of Fluid Mechanic, Vol. 525, pp Bayer, S.I and Megaridis C.M. (26) Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics, Journal of Fluid Mechanics, Vol. 558, pp Czys, R. R. and Ochs, H. T. (988) "The influence of charge on the coalescence of water drops in free fall," Journal of the Atmospheric Sciences, November, pp Brenn, G., Valkovska, D. and Danov, K.D. (2) The formation of satellite droplets by unstable binary drop, Physics of Fluids Vol. 3, pp Willis, K. D. and Orme, M. E., (2) "Experiments on the dynamics of droplet collisions in a vacuum," Experiments in Fluids, Vol. 29, pp Neitzel, G.P & Dell Aversana, P. (22) Noncoalescence and Nonwetting Behavior of Liquids, Annual Rev. Fluid Mech. Vol. 3, pp Tsouris, C.,and Tavlarides, L.L (994) AIChE Journal, Vol. 4, No