Numerical Study on Bouncing and Separation Collision Between Two Droplets Considering the Collision-Induced Breakup

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1 Jornal of Mechanical Science and Technology (007) 585~59 Jornal of Mechanical Science and Technology Nmerical Stdy on Boncing and Separation Collision Between Two Droplets Considering the Collision-Indced Breakp Gwon Hyn Ko b, Hong Sn Ryo a,*, Nahm Keon Hr c, Seng Woo Ko d, Myong O Yon e a Institte of Advanced Machinery and Design, Seol National University, San 56-, Shillim-Dong, Ganak-G, Seol, 5-744, Korea b Corresponding Athor, School of Mechanical ngineering, Chng-Ang University, Hekseok-Dong, Dongjak-G, Seo, , Korea c Department of Mechanical ngineering, Sogang University, Sinsoo-Dong, Mapo-G, Seol, -74, Korea d Department of Mechanical ngineering, Chng-Ang University, Hekseok-Dong, Dongjak-G, Seo, , Korea e Department of Architectre, University of Seol, Jeon Nong-Dong, Dong Daemoon G, Seol, 0-74, Korea (Manscript Received Jne 9, 006; Revised December, 006; Accepted Janary 8, 007) Abstract The main prpose of the present stdy is to perform nmerical stdy on boncing and separation collision between two droplets considering the collision-indced breakp. In this stdy, the collision model proposed in or previos stdy is sed for simlation of collision-indced breakp, and we modify this model to consider the effect of liqid property on the behavior of droplet-droplet collision. This collision model is based on the conservation laws for mass, momentm, and energy between before and after collision and provides several formlae for post-collision characteristics of colliding droplets and tellite droplets. Improving the accracy of the model, in this stdy, appreciate criterion for boncing collision is added and dissipation energy dring collision process is newly modeled. To validate the new model, nmerical calclations are performed and their reslts are compared with experimental data pblished earlier for binary collisions of water, propanol, and tetradecane droplets. It is fond from the reslts that the new model shows good agreement with experimental data for the nmber of tellite droplets. It can be also shown that the predicted mean diameter by the new model decrease with increasing the Weber nmber becase of the collisionindced breakp, whereas the O Rorke model fails to predict the size redction via the binary droplet collision. Keywords: Collision; Break p; Boncing; Separation Introdction Collision dynamics of liqid droplets is important in evoltion of sprays sed in varios indstrial applications. In dense spray system and inter-spray impingement system (O Rorke, 98; Arai and Saito, 999), droplet collision can significantly affect the spray characteristics, sch as drop size and velocity * Corresponding athor. Tel.: ; Fax.: mail address: cfdmec@ca.ac.kr distribtion. The otcomes from binary droplet collision are classified for different types: bonce, escence, reflexive separation, and stretching separation, as seen in Fig.. Many researchers (Ashigriz and Poo, 990; Qian and Law, 997; Brenn et al., 00) observed that the separations prodce the tellite droplets from the interacting parts between two colliding droplets, and in trn reslts in the size redction in droplets see Fig. (c) & (d). This procedre is called a collision-indced breakp process whose effects are more prononced as Weber

2 586 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~59 θ B D rel D 005b) is sed for simlation of collision-indced breakp, and this model is modified to consider the effect of liqid property on the behavior of dropletdroplet collision. The comparisons of nmerical predictions with experimental data (Ashigriz and Poo, 990; Qian and Law, 997) are condcted for postcharacteristics of binary droplet collision process. Fig.. Kinetic and geometric parameters of the droplet collision. nmber increases. In addition, it is well known that the droplet collision phenomena highly rely pon the droplet property. By contrast with the collision of water droplets, the boncing phenomena occr over the wide range of Weber nmber, and also the srface tension energy loss becomes more significant in the collision of hydrocarbon fel droplets. In most of nmerical simlation on spray dynamics, the model of O Rorke (98) has been commonly sed for the droplet collision process. However, there are some problems that the O Rorke model incldes only escence and grazing separation. It means that the O Rorke model cannot mimic other important processes related to the binary collision, e.g., reflexive and stretching separations, and corresponding formation of tellite droplets. Moreover, becase the O Rorke model was derived on the basis of binary collision of water droplets, it may not be sitable for calclation of collision processes of fel droplets. Recently, we (Ko and Ryo, 005a, 005b) proposed the new droplet collision model considering droplet collision-indced breakp process with the formation of tellite droplets. This model consists of several eqations to determine the post-collision characteristics of colliding droplets and tellite droplets. These eqations were derived from the conservations of droplet mass, momentm, and energy between before and after collision, and made it possible to predict the nmber of tellite droplets, and the droplet size and velocity in the analytical way. In this model, however, the liqid property effects inclding boncing phenomena and srface tension energy loss of fel droplets are hardly considered. The main prposes of the present stdy are ths to perform nmerical stdy on the binary droplet collision considering the collision-indced breakp and the effects of liqid property. The collision model proposed in or previos stdy (Ko and Ryo, 005a,. Droplet collision model. Binary droplet collision phenomena Generally, the droplet collision process is described by three non-dimensional parameters as follows (Ashigriz and Poo, 990, Qian and Law, 997, Brenn et al., 00); We ρd rel / σ () D / D, () ( ) b B/ D + D, () where We is the Weber nmber based on droplet diameter, the droplet size ratio, and b is the impact parameter. ρ and σ are the density and the srface tension of liqid phase, and the sbscripts and represent smaller and larger droplets, respectively. B is calclated by taking the distance from the center of one droplet to the relative velocity vector, rel, placed on the center of the other droplets, as illstrated in Fig.. The otcomes from collision are classified for different types: bonce, escence, reflexive separation, and stretching separation, as seen in Fig.. As two droplets impinge each other, the gas between them is trapped and the pressre increases inside this gap. If relative velocity of droplets is not enogh to overcome the pressre force, two droplets do not impinge and go away from each other. This is referred to as boncing. For higher relative velocity, on the other hand, one droplet contacts another directly and in trn escence process takes place between them. At high Weber nmbers, the droplets have excess kinetic energy, which leads to the separation of droplets from droplet esced tentatively. The temporarily esced droplets tend to ndergo a reflexive separation and a stretching separation at low and high impact parameters, respectively. Many researchers (Ashigriz and Poo, 990, Qian and Law, 997, Brenn et al., 00) observed that the separations prodce the tellite droplets from the interacting parts between two colliding droplets, and in trn

3 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~ (a) (b) Fig.. Bondaries between collision regimes adopted in the O Rorke model [] model for. (c) (d) Fig.. Diagram of collision regimes: (a) boncing; (b) escence; (c) reflexive separation; (d) stretching separation. reslts in the size redction in droplets. This procedre is called a collision-indced breakp process whose effects are more prononced as Weber nmber increases.. O Rorke model Among otcomes of droplet collision, the O Rorke (98) model considers two regimes of permanent escence and separation, bt ignores the formation of the tellite droplets. Figre shows the bondaries between the regimes adopted in the O Rorke model for the collision of eqal-sized droplets. Followings explain the relationships in O Rorke model for the criteria and the post-collision characteristics of three regimes. The transition criterion from escence to separation is given in terms of escence collision efficiency as follows (Brazier-Smith et al., 97). s.4 f ( γ ) min,, (4) Wes where, f ( γ) γ.4γ +.7γ, γ D/ D and We ρ D + D σ. In addition, the sbscripts rel ( ) and represent smaller and larger droplets, respectively. The criterion of regime between escence and separation is determined by b. In other words, escence occrs if impact parameter b is less than and otherwise the separation occrs. The post-collision properties of the collision regimes sch as droplet diameters and velocities are determined from the conservation eqations of mass, linear momentm, and anglar momentm (O Rorke, 98). In escence regime, the droplet mass and the velocity are expressed as / ρd + nρd D a ρ, (5) a ( ρ D + nρd ) a, (6) ρ d a a where the sbscript a denotes a vale after collision and n is the nmber of escence. In the separation regimes, no mass is assmed to exchange between two colliding droplet parcels. The velocities are determined as ρ D + ρ D + ρ D ( ) ρ D + ρ D a ρ D + ρ D + ρ D ( ) ρ D + ρ D a b b, (7), For more details, good smmary is well docmented in references (O Rorke, 98; Bai, 996). (8)

4 588 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~59. The present model The present model is a modified version of the collision model, which is recently proposed by the athors (Ko and Ryo, 005a, 005b). Contrary to the O Rorke model, the present model takes accont for the stretching and reflexive separation regimes inclding the formation of tellite droplets. In addition, a criterion for boncing regime is introdced and the dissipation energy dring collision process is newly modeled to calclate the collision process of fel droplets more accrately. Figre 4 shows the bondaries between the regimes adopted in the present model for the binary collision of eqal-sized droplets. In the present model, the criterion proposed by strade et al. (999) is sed to distingish the boncing collision from other regimes. When the following condition is tisfied, the boncing occrs; + We <, (9) φ where φ ( )(4φ ) (cos(arcsin b)) + > λ ( λ) / 4 for λ <.0 ( λ) ( λ)/4 for λ.0 and λ ( b)( + ). The shape factor φ is given a vale of.5 by strade et al. (999). As seen in Fig. 5, the reflexive separation takes place for the head-on or near-centre collisions of two droplets, and the stretching separation occrs at the high impact parameter. Ashgriz and Poo (990) explained two separation processes sing the balance between the effective kinetic energy and the srface energy of the temporarily esced droplet. They proposed the bondaries from escence to reflexive and stretching separations as follows; / ( + ) We > 7( + ) 4( + ) 6 η + η (for reflexive separation), (0) / 4( + ) [( + )( b)( φ + φ)] We > [( + ) ( b )( φ + φ)] (for stretching separation), () / where the parameters η ( ξ) ( ξ ) and / η ( ξ) ( ξ ), and ξ (/ ) b( + ). Fig. 4. Bondaries between collision regimes adopted in the new model for. In addition, φ and φ denote the portions of interaction interaction region (the hatched parts in Fig. 4(b)) of two colliding droplets for stretching separation, i.e., V i φv, Vi φv () D ( λ) ( + λ) for h > 4 φ, () λ D ( λ ) for h < 4 D ( λ) ( + λ) for h > 4 φ, (4) λ D ( λ ) for h < 4 where λ ( b)( + ) and the interaction height, h 0.5( D + D)( b). V is the volme of droplet, the size ratio of two droplets, b is the impact parameter, sbscript i the interaction portion of droplets. qations (0) and () showed good agreements with their experimental data (Ashigriz and Poo, 990) for the regime bondaries between escence and separations. However, Ashgriz and Poo (990) did not offer any empirical or theoretical relationship on the droplet velocities and sizes, which are essential for describing the separation processes nmerically. On the basis of Ashgriz and Poo (990) s theory, the present stdy newly derives the mathematical formlae for the post-collision characteristics in the reflexive and stretching separations. Both reflexive and stretching separation processes accompany with the formation of tellite droplet as shown in Fig. 5. In order to determine the volme separated from the colliding droplets, the present stdy introdces the new parameter C V, called the separation volme

5 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~ coefficient that is defined as the ratio of separating volme to interaction volme (the hatched parts in Fig. 4) between two colliding droplets. It is assmed that this coefficient C V is proportional to the ratio of energy reqired for separation to total energy of two colliding droplets in the following manner. stretching energy h K CV K sep sep S + S, (5) where K sep represents the effective kinetic energy indcing the separation of the temporarily combined droplets, and S is the effective srface tension energy retaining the escence between two droplets. The present stdy determines these effective energies for two different separation processes sing the eqations of Ashgriz and Poo (990) as follows: stretching flow conteractive & excess srfaceindced flows srface energy Interaction volme (V i+v i) K S sep 6 / We( η+ η) σπ D ( + ) ( + ) + ( + ) (for reflexive separation) [( + ) ( b )( φ+ φ)] ρrelv ( + ) (for stretching separtion) (6) / 0.75 σπ ( D + D) (for reflexive separation) (7) / σ[ πvd λ( φ + φ)] (for stretching separation) Using above eqations, the separating volme and the size of colliding droplets after collision are calclated as follows: V C V + V ), (8) s D ja ( v i i ( C φ ) / D, (9) V j j where sbscript a means a vale after collision, and j,. φ is determined by qs. () and (4) for stretching separation, and is sed the vale of nity for reflexive separation becase the interaction between two droplets occr over the whole volme of temporarily combined droplets [see Fig. 4 (a)]. For simplicity, all tellite droplets are assmed to have me properties. In trn, conservation eqations for mass, momentm and energy are rewritten as tellite droplets (a) (b) tellite droplets Fig. 5. Schematic of the (a) reflexive and (b) stretching separations after collision between two droplets. Dj Dja + N D j j, (0) D j j Djaja + N D j j π ρ Dj j + σπ Dj j π ρ Dja ja + σπ Dja, (), () j + K + S + L where N is the nmber of droplets and sbscript means the tellite droplet. Additionally, and are the velocities of the smaller and larger droplets in the mass-centre coordinates, respectively, as given by rel ( + ) and rel ( + ), where is the relative velocity of two droplets. The total rel kinetic energy of tellite droplets can be represented by K N D V π ρ ρ s. () Also, L denotes the energy loss consisting of

6 590 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~59 dissipation energy and srface tension energy loss as follow: colliding droplets can be assmed to retain their velocities, i.e., i j L µ + dvdt + S xj x σ. (4) i where S is the additional srface area associated with the deformation dring collision process (Qian and Law, 997) In the or previos stdies (Ko and Ryo, 005a; 005b), the following eqation is derived on the basis of the relationship sggested by Jiang et al. (Jiang et al., 99) who expressed the viscos dissipation in terms of the kinetic energy of interaction volme: L, (5) α ρ φjv j j j where α denotes the energy loss coefficient. For water droplets, α has been empirically determined to be arond 0.5 by Jiang et al. (99). In this formla, the viscosity is exclded and the srface tension energy loss is also neglected. Qian and Law (997), however, reported that the critical Weber nmber for the reflexive separation is dependent on the Ohnesorge nmber, representing the ratio of the viscos to srface energies. This means that the viscos dissipation and srface tension energy loss are significantly affected by the liqid properties in the reflexive separation regime. In this stdy, ths, a new formla is derived on the Qian and Law (997) s relationship for the energy loss dring the reflexive separation process as follow: π σ α D L We + 6 ( α) β Z + ( α) γ 4 (6) where β and γ are the empirical constants and adopted as 0 and 5 from Qian and Law (997), respectively, / and Z µ /( ρrσ) is the Ohnesorge nmber. Considering the kinetic energy loss for the reflexive separation, the velocities of colliding droplets can be written as follows: / ja ( α). (7) j For the stretching separation, becase the loss of kinetic energy is involved within the interaction portion which becomes the tellite droplets, the. (8) ja j By solving the qs. (0) ~ () simltaneosly, the characteristics of tellite droplets can be finally obtained as follows: / CV ( φ + φ) D D. N (9) φ+ φ, ( φ + φ ) (0) N S / /, () σπ CV ( φ + φ) D Now, the post-characteristics of the separation collisions can be calclated sing qs. (9), and (7) ~ ().. Reslts and discssion This section examines the predictability of new model by comparing the predicted reslts with earlier pblished experimental data (Ashigriz and Poo, 990; Qian and Law, 997) for the binary collision of two droplets. Figre 6 shows the predicted We-b maps for the nmber of tellite droplets in case that the binary collision of two eqal-sized water droplets takes place. In this calclation, the energy loss coefficient α is 0.5 sggested by Jiang et al. (99). In the stretching separation regime, the nmber of tellite droplets is of the highest vale in the range from 0.4 to 0.6 in impact parameter. Until the impact parameter reaches to 0.4, the nmber of tellite droplets increases becase of the increase in the stretching energy. When the impact parameter ranges from 0.6 to.0, on the other hand, the probability of tellite droplet formation is redced de to the decrease in interaction region between two colliding droplets. In the reflexive separation regime, in addition, the tellite droplets are formed most actively in the case of the head-on collision. As mentioned previosly, it can be spported by the fact that the stretching effects indced by off-centre collision redce the reflexive separation. In Figs. 7 and 8, comparisons are made between the predictions and the experimental data of Ashgriz and Poo (990) for the nmber of tellite

7 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~59 59 Nmber of tellite droplets 4 xperiment (Ashgriz & Poo, 990) Prediction Fig. 6. Calclated nmber distribtion of tellite droplets for binary collision of two water droplets for Weber nmber Fig. 8. Comparison of the calclated tellite droplets nmber with the experimental data (Ashigriz and Poo, 990) for the head-on collision of two eqal-sized water droplets. Nmber of tellite droplets 9 6 We60, We8, We9, 0.47 xperiments Predictions (Ashgriz & Poo, 990) Impact parameter Fig. 7. Comparison of the calclated tellite droplet nmber with experimental data (Ashigriz and Poo, 990) for stretching separation collision of two water droplets. droplets. As seen in Figs. 7 and 8, the present model predicts well compared to the experimental data for the stretching separation and for the head-on collision at different conditions. For the head-on collision, it is fond that the nmber of tellite droplets increase with the Weber nmber. Figre 9 compares the predicted We-b map for the collision of eqal-sized tetradecane droplets with experimental data of Qian and Law (997). Althogh some discrepancy between the predicted bondary of boncing regime and experiments are shown at low Weber nmber, the predicted We-b map represents well the overall distribtion of collision regimes observed from experiment (997). In Fig. 0, the normalized mean diameter is presented for different Weber nmbers and it is averaged over the whole range of impact parameter at a given Weber nmber. Here, D 0 and SMD denote Fig. 9. Comparison of the calclated nmber distribtion of tellite droplets with experimental data (Qian and Law, 997) for the collision of eqal-sized tetradecane droplets. D 0 /D 0,SMD/D SMD (new model) D 0 (new model) SMD (O'Rorke model) D 0 (O'Rorke model) We Fig. 0. Comparison between the predictions by both models for the D 0 and SMD normalized by the initial droplet diameter D 0 for the binary collision of eqal-sized water droplets.

8 59 Hong Sn Ryo et al. / Jornal of Mechanical Science and Technology (007) 585~59 the arithmetic and Sater mean diameters, respectively. They are often sed to analyze the characteristics of droplet size distribtions in applications of liqid spray. Over the range of low Weber nmber, both models predict the increase of mean diameter after collisions becase of droplet escence. As the Weber nmber increases, however, both D 0 and SMD compted by the new model decrease gradally, indicating that the droplet breakp via collision increases with Weber nmber. Since the O Rorke model ignores the reflexive separation process at low impact parameters as well as the change in droplet size even in stretching separation process, the predictions of the O Rorke model are rarely varied over the whole range of Weber nmber. From these reslts, it is conclded that the new model is more reasonable than the O Rorke model for simlation in the droplet collision process leading to the droplet breakp. 4. Conclsions The present stdy has developed the binary droplet collision considering the collision-indced breakp and the effects of liqid property. In the present model, the post-collision characteristics of droplets were determined by the formlae based on the conservation eqations between before and after collision. For the validation of the new model, the calclated reslts are first compared with experimental data (Ashigriz and Poo, 990; Qian and Law, 997) on the binary droplet collision. In the stretching separation regime, the nmber of tellite droplets is of the highest vale in the range from 0.4 to 0.6 in impact parameter. In the reflexive separation regime, in addition, the tellite droplets are formed most actively in the case of the head-on collision. The predicted mean diameter by the new model decreased with increasing the Weber nmber becase of the collision-indced breakp. The O Rorke model, on the other hand, failed to predict the size redction via the binary droplet collision. From these reslts, it can be conclded that the new model for droplet collision is more reasonable than the O Rorke model for simlation in the collision-indced breakp process accompanied with formation of tellite droplets. Acknologement The athors wold like to acknowledge the financial spport from Ministry of Commerce, Indstry and nergy (005509). References Arai, M., Saito, M., 999, Atomization Characteristics of Jet-to-Jet and Spray-to-Spray Impingement Systems, Atomization and Sprays, Vol. 9, pp. 99~47. Ashgriz, N., Poo, J. Y., 990, Coalescence and Separation in Binary Collisions of Liqid Drops, Jornal of Flid Mechanics, Vol., pp. 8~04. Bai, C., 996, Modeling of Spray Impingement Processes, Ph.D. thesis, Imperial College of Science and Technology and Medicine, Department of Mechanical ngineering, University of London. Brazier-Smith, P. R., Jennings, S. G., Latham, J., 97, The Interaction of Falling Water Droplets: Coalescence, Proc. of the Royal Society of London A, Vol. 6, pp. 9~408. Brenn, G., Valkovska, D., Danov, K. D., 00, The Formation of Satellite Droplets by Unstable Binary Drop Collisions, Physics of Flids, Vol., pp. 46~477. strade, J. P., Carentz, H., Lavergne, G. and Biscos, Y., 999, xperimental Investigation of Dynamic Binary Collision of thanol Droplets-a Model for Droplet Coalescence and Boncing, International Jornal of Heat and Flid Flow, Vol. 0, pp. 486~49. Ko, G. H. and Ryo, H. S., 005b, Modeling of Droplet Collision-Indced Breakp Process, International Jornal of Mltiphase Flow, Vol., pp. 7~ 78. Ko, G. H., Ryo, H. S., 005a, Droplet Collision Process in an Inter-Spray Impingement System, Jornal of Aerosol Science, Vol. 6, pp. 00~. O Rorke, P. J., 98, Collective Drop ffects on Vaporizing Liqid Sprays, Ph.D. thesis, Mechanical and Aerospace ngineering, Princeton University, USA. Qian, J., Law, C. K., 997, Regimes of Coalescence and Separation in Droplet Collision, Jornal of Flid Mechanics, Vol., pp. 59~80.