1 Travel Grouping of Evaporating Polydisperse Droplets in Oscillating Flow- Theoretical Analysis DAVID KATOSHEVSKI Department of Biotechnology and Environmental Engineering Ben-Gurion niversity of the Negev Beer-Sheva ISRAEL Astract: Basic criteria are estalished for the phenomenon of travel grouping, oth for the possiility of it to occur as well as for the associated characteristic time. These criteria are suggested to e incorporated in the rate of spray evaporation, thus enaling for the estimation of the effect of grouping due to periodic oscillation on the overall vapor production of a multi-size spray. It is shown that the relatively small droplets tend to form groups and thus decrease their evaporation rate. On the other hand, the lager droplets tend not to group, thus maximizing their rate of evaporation. Hence, such travel grouping rings closer the evaporation rates of different droplet size ranges. With respect to the opposite effect, that is, the influence of evaporation on grouping, it is shown that once a grouping or a non-grouping mode of ehavior has een estalished, evaporation does not change this ehavior as time passes or, in other words, droplet s history determines a single mode of time-ehavior. Key-Words: Spray Modeling, Sectional Method, Grouping 1 Introduction The configuration in which the motion of evaporating droplets is influenced y an oscillating flow occurs in many spray applications . One such an application involves fuel-droplet vaporization in an oscillating or pulsating comustion chamer. The spatial spreading of the fuel vapors is a key parameter regarding the characteristics of the comustion, while that spreading of the vapors is primarily influenced y droplet trajectories and y the local concentration of the liquid phase . Thus, if we deal with relatively fast moving multi-size spray droplets we may primarily picture the vapors of droplets to form a narrow cylindrical trail, as a function of their size, marking the droplet trajectory, which later diffuse according to the local vapor concentration gradients. With this picture in mind, for the purpose of the current study we consider a Lagrangian type of approach (y Tamour ) to follow the dynamics of multi-size evaporating spray droplets, with special emphasize on the phenomenon we denote here as travel grouping. By this notation we refer to cases in which droplets tend to gather into groups (see for example the work y Bellan and Harstad ) as a function of various operating conditions to e later addressed. Figure 1 shows a schematic description of the general configuration. Fig. 1. Schematic description of travel-grouping in oscillating flow field. The effect of grouping on the evaporation has een a suject of various studies (see a partial list in ). However, non of them deals with the opposite effect, that is, the effect of evaporation on grouping, an effect to e addressed here. We wish to emphasis that we will not make an attempt here to otain a new formulation for the evaporation of a group of droplets, rather, we will enale criteria to predict when the use of such availale rates should e turned on. Suggestion of modification for the evaporation rate itself will also e addressed, ut only in a qualitative manner.
2 In the route of the current work, the following section deals with the equation of motion of the droplets associated with the effect of the oscillations of the host-gas flow. This asic issue was discussed in a recent work y Katoshevski et al.,  and thus we will show only the major points with that respect, while a new criteria for grouping will e presented here. Then, we will address the dynamics of the droplets coupled with their evaporation. 2 Droplet motion and grouping/nongrouping criteria For a qualitative investigation of the prolem we consider droplet motion in a one-dimensional periodic fluid flow. As a model of such flow one can apply the following mathematical form : ut, x a sinkx t (1) where u t, x is a fluid velocity at a time t at a location x, a is a the mean flow velocity, is the amplitude of the velocity oscillation, k is the wave numer, and is the angular velocity which is linearly proportional to the oscillation frequency. Without less in generality one can assume that a, 0. The flow has a wave length of L 2 / k along the x-axis and a period T 2 / in time. The wave that is formed propagates along the x-axis in a velocity of w / k. Let us normalize the velocities y w, the distance y k -1 1 and the time y. This leads to the following equation of motion for the droplets (note that from now on we refer to normalized velocities only, and for convenience, omit any indication for designating a normalized character), dud 1 u ud, (2) dt St where St is the Stokes numer which is linear with the square of the droplet diameter. u d denotes the velocity of a droplet. After sustituting the explicit form of the host gas velocity u t, x from Eq. 1 and introducing a new variale x t, the droplet equation of motion ecomes: 2 d 1 d a 1 sin. (3) 2 dt St dt St St Now, replacing time t y ct, where c / St, leads to sin, (4) where the prime denotes a derivative with respect to. In the aove, the parameters and are defined as, 1/ St (5) and a 1 /. (6) In the previous study  the analysis of the aove equation has revealed that there are two major modes of droplet grouping ehavior in such a flow configuration that can e descried in the plane. The first mode of ehavior is pronounced y motion of droplets in a fixed travel-group, that is, moving constantly within the original family while the second major mode is the shifting of droplets from one family to the other in addition to reakup of groups. As and are associated with the operating conditions, such as the host gas velocity, the frequency of the oscillations as well as droplet size, we may say that in certain operating conditions we expect to oserve mode I, that is the family type of group motion and in other conditions mode II is expected, that is the non-family type, will e oserved. In addition to these modes of grouping, for certain operating conditions clear non-grouping situations are predicted. It is important to note that Eq. 2 is valid for low droplet Reynolds numers, ased on the velocitydifference etween the host gas velocity and that of the droplet. But, from our calculations we conclude that using a drag coefficient C D as a function of the Reynolds numer for Re>1 leads to grouping tendency which is even greater. In most spray cases the desire is to reakup groups of droplets in order to maximize evaporation. From further analysis of the plane we ring here new criteria for reaching non-grouping, or nearly nongrouping situations. The following relations should e maintained y the operating conditions, for low particle Reynolds numers 1 (7a), OR, 1 AND / 2 (7) and to ensure non-grouping, or nearly non-grouping situations even for a wider range of droplet Reynolds numers we have to maintain the following relation arising from Eqs. 5-7, a 1 NG St 1 (8) where NG stands for Non-Grouping. We note here, that according to our numerical investigation we conclude that grouping is not distinguished already when NG >2 (corresponding to the division y 2 in Eq. 7), and the aove relation is for the sake of generality.
3 Note that the aove relation is presented here for the first time and was not otained in the previous study y Katoshevski et al., . Keeping in mind that the velocities are normalized y w / k and that the Stokes numer is linear with the frequency, then, if one increases the frequency NG will also increase and the tendency will e toward non-grouping situations. The same ehavior of tendency to group is true for an increase in droplet size, or an increase in droplet liquid density, oth causing an increase in St. The aove criteria can e used for correcting the evaporation rates according to grouping/non-grouping situations. This will e addressed in the next section, while here we first would like to demonstrate some related features of the grouping ehavior as shown in Figs Figure 2 presents a clear grouping case, in which the droplets of 30 microns in diameter that are initially dispersed along a distance of one wave-length (L in length) along the x axis form two travel groups. For that case the value of NG is Then, if we increase the frequency, so that (which is 2 times the frequency) increases from 1000 Hz to 1800 Hz, NG ecomes 1.04, and one of the two groups splits up into two, that is, a secondary reakup occurs of the overall group, as shown in Fig. 3. Our calculations show that, a further increase of to 2000 Hz leads to an increase of NG to 1.11 and causes another stage of group reakup. Now, we jump a few steps ahead and increase the droplet size to 70 microns and to 6000 Hz in order to show a clear non-grouping case, as presented in Fig. 4. In that case NG is well over unity (equals 4.45) as in the criterion of Eq.8. In order to account for the grouping effect on evaporation, which is the topic of the next section, it will e useful to achieve an approximate analytical solution of the equation of droplet motion under the grouping criteria. For such an approximation, ased on Eqs. 7a-, we may consider the grouping criterion as, and for an asymptotic solution account for and define Equation 4 then ecomes sin (9) and the solution can e represented asymptotically as (0) (1)... (10) (0) The first approximation will e const., which implies from the definition of x-t) that the group velocity is aout the same as that of the wave propagation. Thus, we may conclude that for the sake of descriing the clear-grouping situation we can account for the dimensional velocity of the groups as equal to G / k, where the index G denotes group. This value of G is also deduced from the trajectories presented in Fig. 2. In order to satisfy the initial condition of the particle initial position we set (0) (0). For the correction term which is in the order of we allow ourselves to linearize the sinus term and otain the following approximation, ( 0) t sin t (11) St St From the aove, one can estimate the dimensional time which passes until grouping is clearly estalished, which is, St tg tˆ (12) where t G is the dimensional characteristic time for grouping to occur, where tˆ is a constant that equals 2. For example, with respect to the parameters of Fig. 2, this corresponds to the dimensional time of sec, which agrees well with the trajectories shown in that figure. Fig. 2. Trajectories of droplets in the grouping mode, for non-evaporating droplets of 30 microns in diameter, and 1000 Hz.
4 Fig. 3. Non-grouping mode, for non-evaporating droplets of 30 microns in diameter, and 1800 Hz. Fig. 4. Non-grouping mode, for non-evaporating droplets of 70 microns in diameter, 6000 Hz. 3 Travel grouping and evaporation coupling We now turn to suggest how to incorporate the effect of grouping on evaporation, and visa- versa, that is, the effect of evaporation on grouping. For the frame of our qualitative description we choose to employ the Lagrangian method [3, 7], for a mono-disperse spray, dq C Q (13) dt where Q is the total mass of droplets in a control volume. C is an evaporation coefficient. In case a multi-sized spray is addressed, the overall size range is divided into size sections, and the aove equation will then incorporate an additional term accounting for droplets that move from one size section to the other as their size is reduced due to evaporation . For the purpose of the current study we treat a mono-size spray, which can e used also for the case where similar-sized groups of droplets of a multi-size spray move closely together. The evaporation coefficient is general a function of the droplet size, the local temperature, the liquid specifications, and can also incorporate the effect of the velocity difference etween that of the droplet and that of the host gas y a Ranz-Marshal type of correlation . As mentioned aove, in the frame of our approximations we use a Lagrangian type of an approach , and for the estimation of overall production of the vapors we may write, dm C Q (14) dt When dealing with the production of vapors y specific range of droplet sizes that may travel within a group we will sum up the contriutions of the various groups to the overall vapor production. In general, we can estimate the overall vapor production y dividing the time frame into two periods which are (I) up to t G and (II) after t G. In the second period, the fact that grouping is accounted for will hinder evaporation comparing to the evaporation rate from isolated droplets. This effect on evaporation will e accounted for y introducing a theoretical correction to the evaporation coefficient. The correction for the evaporation coefficient due to the grouping is pronounced y multiplying these coefficients y a function E, where E( G) 1, if NG 1 and t tg E (15) 1 otherwise and G is the known correction term suggested y Chiu and co-workers (Sirignano, 1999, and references therein) for evaporation and comustion of non-isolate droplets. Based on studies on group comustion (see in Ref. ) G is primarily a function of the Lewis numer, Le, the total numer of droplets in the group N, the averaged droplet radius r and the spacing etween the centers of the droplets s, 2 / 3 G ~ Le N ( r / s) (16) Thus, in addition of eing a function of the sectional size range, the liquid properties, and the temperature , the sectional evaporation coefficients may also e
5 adjusted y the aove E function. Adding the Ranz- Marshal correlation  denoted here as R M, we may write C ~ RM (Re, Pr) E( G, NG, tg ) (17) where Re and Pr are the Reynolds and Prandtl numers, respectively. Hence, in a formal way of representation, the overall vapor production from a multi-sized spray undergoing grouping and is eing followed numerically, may e estimated y tg t m( t) C Q dt E C Q dt (18) o t G where the evaporation coefficient C is for nongrouping situation ut includes the effect of the Ranz- Marshal correlation . When a spray flame is to e considered (as for example, y Greenerg ), which is not the case in the present work, it will e important to add the effect of diffusion in order to descrie the time evolution of the spatial vapor distriution in the comustion chamer, along with the comustion instaility which is a result of flow oscillations. With respect to droplet coalescence, the same treatment can e applied, that is, as grouping will enhance the sectional rates of coalescence, these rates  should e adjusted during the time greater than t G. 4 Demonstration and conclusions For demonstration, let us consider the case of two mono-sized sprays of 10 m and 70 m in diameter. We have calculated the trajectories of these droplets, without evaporation, and then added the effect of droplet size reduction, employing the commonly used D-square law of evaporation (and we modify Eq. (1) so that the time derivative will account for droplet size reduction). For a qualitative demonstration we have the square of the droplet diameter reduced dramatically during the time of 0.08 seconds, to e consistent with the time scale of the former figures. Thus, in Fig. 5a the droplet size reduces from 10 microns to 1 micron. Comparing to our nonevaporating calculations we find that the reduction in size does not affect the grouping ehavior of these small droplets. This coincides with the aove mentioned conclusion, that smaller droplets have higher tendency to form groups. Thus, after grouping has occurred, a further reduction in size will not reak that group. With respect to the larger droplets (Fig. 5), their diameter reduces from 70 microns to 6 microns during the same period of time of 0.08 seconds. Although the size of these relatively large droplets reduces consideraly with time as they evaporate, our calculations show that the tendency of these droplets to group is negligile, and this is due to their history as non-grouping droplets. Thus, for large droplets, which initially do not form a travel-group, evaporation and coincidently reduction of size, will not lead to grouping ehavior. Now, we add to the aove the effect of the slow-down in evaporation as grouping occurs. This is done y employing the function E(G) of Eq. 15. Grouping is relevant to the small droplets as already mentioned. Thus, we do not apply the function E(G) on the large droplets of Fig. 5. As already concluded, from Fig. 5, the rate of evaporation does not change the grouping ehavior of the small droplets once grouping has een estalished. Hence, slow-down in evaporation rate will not change the trajectories of Fig. 5a. What is then left is to estimate the change in the vapor production, or the change in evaporation rate of the group of small droplets compared with the same numer of isolated ones. Laowsky  has shown that the D-square law essentially descries the evaporation rate of nonisolated droplets, ut with a decrease factor, represented here y the function E(G). That study, as well as others , have shown that depending on spacing etween droplets, and operating conditions, the reduction in the evaporation rate can e sustantially larger than 50%. As our simple mathematical modeling can represent only a qualitative picture, for that frame of work we can use such an estimation for the purpose of a qualitative demonstration. In addition, from the study y Tamour  regarding the aove mentioned sectional approach we may estimate that the averaged sectional evaporation coefficient C d (where the su-index d denotes droplet-size) of droplet initially with size of 10 microns is 3 times larger than the corresponding rate of a 70 micron droplet, when they are isolated. Hence grouping can ring the evaporation rates of these two classes of sizes to very close values. The amounts of vapors produced from two sizes-groups together as a function of time m(t) can e evaluated y Eqs. (14) and (18). As a first-order approximation, accounting for t>t G we otain, m( t) Q Q (0) 1 ~ (0)! 1 Exp( C70 t) " ~ ~ Exp( C t )! 2 Exp( EC ( t )" 10 G 10 t G (19)
6 where the tilde represents an averaged value of vaporization rate. The caliration of the coefficients in Eq. (19) can e done when experiments are involved. In conclusion, the smaller size droplets tend to form groups and thus decrease their evaporation rate, while the lager droplets do not tend to group, thus maximizing their rate of evaporation. This may ring closer the evaporation rates of different size ranges. The criterion for travel grouping introduced here, along with the characteristic time until grouping may occur can e of use in cases where multi-size spray droplets are sujected to an oscillatory flow field. The current work also points out that once grouping or non-grouping mode of ehavior has een estalished, nor evaporation nor time will cause a switch of that mode. a Acknowledgment This research was supported y the fund for promotion of research at the Ben-Gurion niversity of the Negev. References:  W. A. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays, Camridge niversity Press,  N. A. Chigier, Progress in. Energy and Comustion Science, Vol. 2, 1976, pp  Y. Tamour, A Lagrangian Sectional Approach for Simulating Droplet Size Distriution of Vaporizing Fuel Sprays in a Turulent Jet, Comustion and Flame, Vol. 60, 1985, pp  J. Bellan, and K. Harstad, The Dynamics of Dense and Dilute Clusters of Drops Evaporating in Large Coherent Vortices, Proceedings of the Comustion Institute, Vol. 23, 1990, pp  D. Katoshevski, Z. Dodin, and G. Ziskind, Aerosol Particle Clustering in Oscillating Flows- Mathematical Analysis, Atomization and Sprays, in press,  S. C. Crow, F. H. Champagne, Orderly Structure in Jet Turulence, Journal of Fluid Mechanics, Vol. 48, 1971, pp  J. B. Greenerg, The Burke-Schumann Diffusion Flame Revisited- with Fuel Spray Injection, Comustion and Flame, Vol. 77, 1989, pp  V. E. Ranz, and W. R. Marshall, Evaporation from Drops, Chemical Engineering Progress, Vol. 48, 1952, pp  M. Laowsky, Calculation of Burning Rates of Interacting Fuel Droplets, Comustion Science and Technology, Vol. 22, 1980, pp Fig. 5. Trajectories of evaporating droplets: (a) 10 microns, and () 70 microns Hz.