Shatter of Drops in Streams of-Air W . R . LANE CHEMICAL DEFENCE E X P E R I M E N T A L E S T A B L I S H M E N T , P O R T O N , E N G L A N D
T
H E a t o m i z a t i o n of liquids has
T h i s paper summarizes t h e results of a study of t h e shatter of drops in streams
of air and discusses briefly some features of t h e phenomena which are of inmany important applications-for terest i n connection w i t h mechanisms involved i n t h e atomization of liquids. example, in internal combustion engine Using electronic flash and spark photography, stages of t h e shatter of inand gas turbine development, spray dividual drops exposed t o steady and transient air streams were identified and drying, agricultural and insecticidal interpreted in terms of fluid mechanics. T h e secondary droplets i n t o which spraying, and the production of aerosols adrop wasshattered were found t o be progressively smaller as t h e velocity of t h e for therapeutic and other purposes. air stream was increased, b u t a t t h e highest (supersonic) velocities used, they Many empirical studies have been were n o t so small as would be predicted by extrapolating t h e relationship estabmade of atomization in relation to lished for breakup i n a steady air flow. these and other applications, but comT h e results of this work help toward a clearer understanding of t h e action paratively little attention has been of aerodynamic forces in effecting t h e atomization of liquids, a process which given to the fundamental mechanics has many important applications. of the shatter of liquids into small droplets or to the physical conditions which favor fine atomization. This is not surprising, for atomization, as commonly observed, is unphenomenon is of interest, however, in connection wit,h atornimtion for two reasons. First, because it demonstrates in a striking doubtedly a complex process ( 2 , 10, 13), and a better understanding of the basic physics of the subject is not likely to result fashion a feature which is characteristic of sprays produced from further studies of atomizing devices. There seems to be by most atomizing devices-namely, the distribution of only a comparatively small fraction of the liquid mass among the finest little doubt that irrespective of JThether the shatter of the liquid droplets. Secondly, because the fine droplets resuking from the is brought about by the emergence of liquid into still air a t high breakup of the hollow bag are generat,ed from a thin stretched speed from a nozzle or by the interaction of a stream of liquid film of liquid. Similar thin liquid films are also seen in high and a fast Rowing gas stream, three stages can be distinguished speed photographs of sprays formed from atomizing nozzles ( 1 2 ) in the shatter process: and of liquid atomized in an air stream (8). If the velocity of the 1. Initiation of small disturbances a t the surface of the atomizing air stream is gradually increased, a stage is reached liquid, in the form of local ripples or protuberances when part of the emergent stream of liquid is blown out into a 2. Action of air pressure and tangential forces on these disturbances, forming ligaments which may break up into drops thin film which, on bursting, forms some very small droplet,s. 3. Further breakup of these drops in movement through the If the velocity of the air is further increased, the film formation air is not observed; the spray, then appears to be formed by disThis paper briefly describes some work done in 1948-49 which ruption of the liquid issuing from the nozzle into fine filaments is believed to have a bearing on the last of these stages. A fuller which break up to form droplet's. Whether from individual account of the investigation is to be published elsewhere. The drops or from a liquid stream, the finest droplets seem to he topic was chosen for study because it seemed to provide a comproduced by the bursting film mechanism, but' of course they paratively simple system in which the aerodynamic forces inrepresent only a small fraction of the liquid exposed to the stream. ducing shatter could be controlled and measured. Some of the features of the bursting drop may he explained on the basis of w-ell-established results of fluid mechanics. MeasureBREAKUP O F DROPS IN STEADY STREAM OF A I R ments of the distribution of pressure over the surface of a rigid In the first series of esperments, comparatively large drops, sphere placed in a wind tunnel show that a positive pressure of known size, were allowed t o fall down the axis of a small vertical exists over the front of the sphere and a reduced pressure a t its nind tunnel in which a steady downward stream of air of meassides and rear (Figure 2 ) . It is understandable', t'herefore, that a ured velocity could be maintained. The drops were photoliquid drop introduced into a stream of air should become flatgraphed through the transparent walls of the tunnel by the light tened on the side subjeckd to the positive pressure and extcnded from a "microflash" tube triggered photoelectrically by the falling a t the sides and rear; this deformation will be opposed by the drop. As it came under the influence of the air stream, the drop force of surface tension tending to keep the drop spherical, so that was seen to become increasingly flattened, and a t a critical veloca depression might be expected to form a t the cent,er of the upper ity of the air it was blown out into the form of a hollow bag surface of the drop. attached to a roughly circular rim. Bursting of this bag proUsing water drops of diameter d varying from 5.0 to 0.5 nim., duced a shower of very fine droplets, and the rim, which conmeasurements were made of the critical velocity, u,of the air tained a t least 70% of the mass of the original spherical drop, stream required to break them and the velocity, v , of the enbroke up later into much larger drops (5). trained drop a t the instant of breaking. The results are expressed The phenomenon of the breakup of water drops in a steady by the equation (5) stream of air (Figure 1) was observed by Hochschwender (4,6 ) ( U - v)'d = 612 in 1919. He allowed water drops to fall into a free upward (1) flowing current of air and saw that some of them were blown A relationship of the form ( u - v)2d = constant, would be inside out. Hochschwender does not appear to have been much expected on the simple assumption that a liquid sphere placed in a interested in the mechanics of this phenomenon; he was measursteady stream of air will break when the force due to the variation ing the electric charge generated in the breaking of water drops, of aerodynamic pressure over the drop exceeds that due to surface seeking to account for the origin of thunderstorms in the electritension, so that fication produced by rain drops in falling through the air. The
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INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1951
(u
- v)
1313 a d T d
and this relationship was in fact confirmed in experiments with a number of liquids chosen to cover a wide range of surface tension (< 28 t o 475 dynes per em.) measuring in each case the velocity of the air flow in the wind tunnel required to shatter drops of known size (6). Anomalous results were obtained with drops of aqueous solutions of surface active substances, but these could be explained on the basis of the known variation of the surface tension of such solutions with time. Viscosity appeared t o influence the breakup process only when it was very great-for example drops of glycerol; i t then tended to retard the breakup of the drop. In considering the possible significance of these results in relation to air-blast atomization it must be borne in mind that the bursting bag mode of breakup was studied, for convenience, with large drops (diameter 0.5 to 5 mm.), and there is no experimental evidence that the same phenomenon occurs with much smaller drops. If the relationship (u - v)2d = 612 holds good over a wide range of drop sizes, water droplets 5 microns in diameter would just remain intact in air moving a t sonic velocity relative to the drops. The results of further experiments suggest that drops larger than this are, in fact, able t o survive such blasts. BREAKUP O F DROPS IN FAST A I R BLASTS
There was a possibility that the breakup of a drop by the bursting bag process, which had been shown t o take place when the relative velocity of the air and the drop reached a certain critical value depending on the drop size, might be superseded by a different mode of breakup if the relative velocity was much in excess of that necessary to initiate breakup by “bagging.” This point was not settled by allowing drops to fall in the wind tunnel into a stream of air of velocity greater than that known to cause them to form “bags” at the point of observation in the working section, for then breakup by the same process occurred a t some point inside the flared entrance of the tunnel.
1 Figure 1.
Breakup of Water Drops i n Steady Stream of Air
CD = 0.4, which is the appropriate value for the range of Reynolds number in question, gives (U
- v)’d
=
1200
(3)
which is about twice the value found from the experiments. An overestimate would be expected since the photographs show that the drop does not remain spherical but approximates a lens shape before it bursts, and the drag of a disk is considerably less than that of a sphere. A more complete treatment of the deformation of a drop due t o the distribution of aerodynamic pressure over its surface has been given by Hinze (3). Using data published by Merrington and Richardson ( 7 )for the critical sizes of falling drops, Hinze deduced an expression which yields values of (u- v)Sd agreeing well with Equation 3. His predicted values for water drops suddenly exposed t o an air flow of constant velocity are nearer those which follow from Equation 1. Since the distortion of the drop leading t o its ultimate breakup is resisted by the force of surface tension, i t was important t o study the effect of variation of this property of the liquid. Simple dimensional analysis suggests
. IiOO‘ Figure 2.
Distribution of Pressure over Surface of Rigid Sphere i n Wind Tunnel
Drops were subjected t o fast air blasts by using a blast gun, similar in principle t o the shock tube which has been used ( I , 9) in studying the propagation of shock waves along tubes. A diagram of the gun is shown in Figure 3.
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VoI. 43, No. 6
BURETTE.
FLASH
-
I
A"I
F i g u r e 3.
~
'DIAPHRAGM,
D i a g r a m of B l a s t Gun
Compressed air was admitted into the compression chamher until a chosen pressure was indicated by the gage. On operation of the blast gun the polystyrene diaphragm \+as punctured by the spike and the compressed air expanded into the expansion chamber. A transient blast of air then emerged from the open end of the gun, striking a drop falling past the orifice a t a distance of 5 cm. from it. The velocity of the blast of air could be varied bv working with different air pressures in the compression chambkr of the blast gun. Vclocities up to 100 meters per second weie attained n i t h this apparatus, but much higher velocities weie reached with a more powerful gun described later. It was important that the transient blast of air emerging from the open end of the gun strike the drop as it fell past the center of the muzzle of the gun. This was accomplished by controlling the solenoid-operated spike, which punctuied the diaphragm,
F i g u r e 4.
L A W HOUSING,
by a photocell. The falling drop, released from a tip just above the muzzle of the blast gun, passed through a narrow horizontal beam of light directed across the orifice, and the resulting voltage pulse was amplified and caused to trigger a Thyratron which energized the solenoid. The drop was illuminated, a t a chosen instant during its shatter, by light from a microflash tube which was triggered when the breaking drop interrupted a second light beam also directed across the mouth of the gun onto another photocell. In this way the photographs shown in Figure 4 were obtained.
This mode of breakup is strikingly different from that observed a t the critical velocity in a steady stream of air. Instead of the drop being blown out into a thin hollow bag anchored to a rim,
S h a t t e r of Drops in T r a n s i e n t Air S t r e a m
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1951
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Table I. Critical Velocities for Shatter of Water Drops i n Transient Blast and Steady Stream of Air Diameter of drop (a), Mm. 4 0 3 0 2.0
1.0
0.05
Critical Relative Velocity, M./Sec. Transient Steady (UT) ( US) 12.0 12.1 12.6 16.0 24.0
UT Us
Uid
12.5 14 4 17.5 24.7 35.0
625 622 612 610 612
Period,of Oscillation
of Drop ( T I ,
0.96
0.83
0.72 0.65 0.69,
Sec. 0.023 0.015 0.008 0.003 0.001
of the spring (analogous to the maximum distortion the drop can sustain without breaking)
FT
=
1/2
Fs
Since the force tending to distort the drop is proportional to
-U T-- velocity for bursting by suddenly applied blast US
velocity for bursting in steady stream =
PERFORATED BRASS PLATE. ______._____-__-------I----
\
SAMPLING SLIDE.
Figure 5.
High Pressure Blast G u n
i t is deformed in the opposite direction and presents a convex surface to the flow of air. The edges of the saucer shape are drawn out into a thin sheet and then into fine filaments which in turn break up into droplets, From theoretical considerations regarding the distribution of pressure over the surface of the drop, Taylor ( 1 2 ) has deduced the shape which the drop would be expected to assume when accelerated in the air stream, supposing that it did not disintegrate. H e finds t h a t it would be flattened into a plano-convex lenticular body of diameter about twice the diameter of the original spherical drop. Measurements from Figure 4 gave a ratio quite close to this.
dz
ff
U2
4%
= 0.71
Table I shows that the ratio 0.71 is closely approached for drops of diameter 2, 1, and 0.5 mm. but is exceeded for the larger drops. This would be true if, for the larger drops, the blast velocity decayed before the drop attained its maximum distortion. Provided the full velocity of the transient air stream is maintained for half the free period of the drop, the ratio U T I U S might be expected t o be close t o 0.71. The period of vibration of a drop of water is given by r =
2.884 X 10-3d3/2
and the duration of the blast from the gun was 0.004 second, approximately. EXPERIMENTS W I T H M O R E POWERFUL A I R BLASTS
In the experiments already described the degree of shatter of the drop subjected t o the air blast was not measured but only assessed in a qualitative fashion by examining the photographs. At the higher intensities of blast the degree of shatter could no longer be judged in this manner, and it was necessary t o devise
CRITICAL VELOCITY FOR SHATTER I N TRANSIENT BLAST
U
The relationship between the size of drops of a given liquid and the velocity of the slowest air blast which would shatter them was determined by allowing drops of accurately known size t o trigger the blast gun and increasing the pressure in the gun in successive shots until the drops were found to break, producing fine droplets. The corresponding velocities were measured by photographing the emergent air pulse rendered visible by smoke generated in the gun; later these mere found to agree well with values calculated from the pressure of the air in the blast gun, using the Rankine-Hugoniot equations. The velocities for breakup in transient blasts were lower than in the steady stream, the divergence increasing for small drops. Taylor ( 1 9 ) has pointed out that the reason for this may be seen by considering the drop as a vibrating systemfor example a loaded spring-and comparing the force, FT,suddenly applied and the force, F s , gradually increasing, required t o produce the same extension. For a given maximum extension
Figure 6.
Spark Photographs of Shatter of Drops in Supersonic Air Stream
INDUSTRIAL AND ENGINEERING CHEMISTRY
1316
some means of measuring it. At the same time it was desirable to extend the experiments to much more powerful blasts and for this purpose another blast gun was used (Figure 5 ) . A drop was supported on thin filaments so as to be located on the axis of the blast gun and about 1.5 cm. from the muzzle. Compressed air was admitted slowly into the gun until a stout metal diaphragm sheared around its clamped periphery. A blast of air then emerged from the orifice of the gun, striking the drop and shattering it into droplets. Spark photography was used t o study the mechanism of shatter of drops in these high speed air streams (Figure 6).
i
i o
I 4c
y
!
'VELOCITY
1
I
II
I
i.i
50°/o -MASS
I
1
0 AIR B L A 5 T Q GOOMETRES/SEC.
I
I
II
i
DIAMETER OF DROPLETS
Relationship between Fineness of Shatter of Drop and Bursting Pressure i n Blast G u n
Figure 7.
The photographs of the early stages of the shatter show that the sequence of events is similar to that revealed in the less powerful blasts, the original spherical drop being distorted into a saucershaped disk while a thin layer of liquid is stripped off the drop and rapidly breaks up into droplets. In the shadowgraphs (Figure 6 ) the partially atomized drop is soon obscured in the opaque cloud of droplets, but the existence of a well-marked detached shock s a v e in the air stream immediat,ely behind the dark opaque mass would seem t>oindieatre that a portion of the drop is, a t that stage, still sufficiently intact to offer obst,ruction t o the air flow. I n the last photograph of the seYies, shatt,er of the drop appears to be almost complete.
necessary, however, to bear in mind that for so short a gun, the corresponding air blast velocity cannot be estimated reliably from the pressure in the compression chamber. An attempt was made to measure the supersonic velocities by inserting a sharp probe a t the place normally occupied by the drop and measuring the Mach angle from a spark photograph of the conical wave formed a t the tip of the probe. A Mach number 2.1 was found from such measurements for the highest air pressure used in the gun. This figure is probably an overestimate of the true Xach number of the flow; it is hoped to obtain more reliable estimates of the velocity in experiments which are now planned. The photographs of the drops in the process of disintegration suggest that' a layer of liquid is stripped off the drop by the air blast. The thickness of such a layer, for a given constant air velocit.y, can be calculated fairly accurately for a liquid of given density and viscosity, by applying approximate boundary layer theory ( 1 2 ) . For low air velocities the predicted thickness was in fair agreement with the thickness estimated from the photographs, but a t high air velocities the calculat~edthickness was too great to account for the size of the droplets collected in t,he experiments. Actually, of course, the shattering process is extremely rapid but' not instantaneous] and because of the acceleration of the deformed drop in the air blast the velocity of the air relative to the drop must diminish very rapidly. The thickness of the layer stripped off the drop by the air flow of continually diminishing relative velocity viill therefore increase wit,h time. Thus the shatter of the drop int,o a cloud of droplets of a wide range of sizes can be accounted for, the largest droplets being formed a t the latest stages of the shatter. Another possible mechanism involved in' t>lieshatter a t high air velocities, which also receives support from some of the photographs, is the production of unstable waves on the surface of the drop. When air flows over the surface of a liquid, xaves are generated, and waves of certain lengths are found t'o be uristable; t.he lengths a t which instability esists depend, among other things, on the relative air velocity. Taylor ( 1 2 ) has calculated, for various air velocities, the wave lengths which should increase most rapidly, and on the assumption that the diamet'er of a droplet formed when the crest of a wave is detached from the main body of the liquid is equal t o the wave length, he has conipared these values with values for the mass median diameter of the droplet cloud measured in these experiments. The calculation gives values which are smaller than the measured droplet diameters. This could be due to the finite t'hickness of the air boundary layer preventing the very small wave lengths from being unstable, and if the air boundary layer could be taken into account in the analysis, the predicted droplet sizes should be closer to those observed.
DEGREE OF A T O M I Z A T I O N OF DROPS
To avoid difficult,ies arising from evaporation of droplets during exposure to the blast and subsequently, a comparatively involatile liquid, dibutyl phthalate, was used in these experiments. A sample of the droplet cloud into which the drop was shattered was received on a glass slide, and the droplets were counted and sized and size-distribution curves plotted on logarithmic probability paper. This procedure was repeated for each bursting pressure investigated. The results are plotted in Figure 7. The atomization of the original drop became progressively finer as the pressure in the blast gun was increased to about 10 atmospheres, but thereafter further increase in bursting pressure caused little improvement in atomization, and even a t a pressure where the velocity of the air blast was well above sonic velocity, one half of the mass of the resulting spray was in the form of droplets of diameter greater than 15 microns. This does not imply that the decrease in size stops when supersonic conditions are att,ained but rather that the rat,e of decrease in size appears t o be slower than might be rspected from Equation 1. It is
Vo!. 43, No. 6
ACKNOWLEDGMENT
The author is indebted to \IT. C. Prewett and J. Edn arrls who contributed to the work on which this paper ia based and t o the Chief Scientist, Ministry of Supply, for permission to publish the paper. The photographs aie Crown Copyright and are reproduced by permission of the Controller H.B.hI. Stationery Office. NOMENCLATURE
d
CD u
= diameter of drop, mm. = drag coefficient of drop = velocity of entrained drop a t instant of breaking, meters/
second surface tension of drop, dynes/cm. 'T = period of vibration of drop, seconds k = a constant u = critical velocity of air stream, meters/second U S = ( Z L - v) for steady air stream, meters/second U T = (u- v) for transient air stream, meters/second p = density of air of stream, grams/ml. F s = gradually increasing force applied to vibrating system I"T = impiiloive forcr applied to vibrating systrin u
=
June 1951
INDUSTRIAL A N D ENGINEERING CHEMISTRY LITERATURE C I T E D
(1) Bleakney, W., Weimer, D. K., and Fletcher, C. H., Reu. Sci. Instruments, 20, 807 (1949). (2) Castleman, R. A,, Jr., Bur. Standards J . , Research, 6, 396 (1931). (3) Hinze, J. O., Applied Sci. Research, A. 1, 273 (1948). (4) Hochschwender, E., dissertation, Heidelberg, 1919. (5) Lane, W. R., and Edwards, J., Unpublished Ministry of Supply
Reports. (6) Lenard, P., Ann. Physik, 65, 629 (1921). (7) . . Merrington, A. C., and Richardson, E. G., Proc. Phvs. SOC., 59, 1-(1947).
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(8) Nukivama. S.. and Tanasawa. Y.. Truns. SOC.Mech. Enars. ( J a p a n ) ,6, No. 22, 11-7 and 11-15 (1939). (9) Payman, W., and Shepherd, W. C. F., Proc. Roy. SOC.(London), A186, 293 (1946). (10) Schweitzer, P. H., J . Applied Phys., 8, 513 (1937). (11) Simons, A., and Goffe, G. R., Aero. Research Council K and M, No. 2343. (12) Taylor, Sir Geoffrey, Min. of Supply Paper AC 10047/Phys ‘269 (1949). (13) Weber, C., 2. anyew. Muth. u.Mmh., 2, 130 (1931). .
I
RECEIVED January 3, 1951.
Droplet Size Distribution in Sprays R. A. MUGELE A N D H. D. E V A N S S H E L L DEVELOPMENT ‘20.. E M E R Y V I L L E . C A L I F .
A
N ACCURATE knowledge of drop G e n e r a l features of size d i s t r i b u t i o n are reviewed f o r dispersed systems. T h e size distribution a s a function of concepts o f ‘:mean diameter” and “ d i s t r i b u t i o n parameter” are clarified the conditions of the system is a preand generalized. Previously applied d i s t r i b u t i o n equations (Rosin and requisite for fundamental analysis of the Rammler, N u k i y a m a and Tanasawa, log-probability) are examined c r i t i c a l l y transport of mass or heat or of the in regard t o theoretical soundness, and application t o spray data. separation of phases in a dispersed sysA new equation, called t h e u p p e r - l i m i t equation, i s formulated and tem. For example, the “drop” size disproposed as a standard f o r describing droplet size d i s t r i b u t i o n s in sprays. tribution in a fractionation column deterIt is based on t h e differential equation of t h e “normal” or Gaussian dismines the rate of heat and mass transfer t r i b u t i o n , t h e distributed q u a n t i t y being y = In ax/(x, - x) where a is a and also the amount of entrainment. dimensionless parameter, x is droplet diameter, and x , , ~is m a x i m u m stable The size range of droplets introduced diameter. into, or created within, a cyclone sepaT h e u p p e r - l i m i t equation is applied t o a wide variety of experimental data rator must be known in order to design on sprays and m o r e l i m i t e d results o n other dispersoids. It is concluded the unit for the desired separation t h a t t h e new equation fits t h e available spray d a t a accurately, calculates t h e efficiency. I n liquid-liquid systems the mean diameters accurately, applies also t o emulsions and aerosols when t h e size distribution helps to determine mechanism of f o r m a t i o n is n o t t o o different f r o m t h a t of sprays, and indicates settling rates (and thus holdup), mass t h e t y p e of d i s t r i b u t i o n f u n c t i o n t h a t m a y be derivable f r o m t h e basic mechand heat transfer rates, entrainment, a n i s m of dispersion, when t h i s mechanism is better understood. For a and possibly coalescence rate. Again, mechanical spray, t h e relation of t h e parameters of t h e d i s t r i b u t i o n equation in many combustion studies, rate of t o physical properties and design variables i s indicated. evaporation or burning of individual drops must be estimated; here also a lets are sufficiently small to prevent detection of phase boundknowledge of the drop sines is important aries. By specifying a “system” we imply that the droplets Despite the importance of drop size distribution in evaluating have a common origin, usually a body of liquid. most separation processes, little is known about the actual size Examples of natural sprays are rains, fogs, waterfall mist, range of droplets in these units. This is due to difficulties inherent in measuring drop sizes in these types of apparatus. Howocean spray, and sneeze spray. ever, many data are available for size distribution in more simple atomizers such as spray nozzles of various types. It is believed that analysis of these data will make it possible to estimate drop size distribution in more complex apparatus with reasonable accuracy. This paper presents a summary of the investigation of drop size distribution in sprays in the following order: fundamental definitions and concepts, analysis of previously developed distribution equations, development and application of a new distribution equation known a s the “upper-limit” law, and comparison of the various distribution functions with experimental data. For the case of a mechanical spray, a brief discussion is given concerning relation of the parameters required for the upperlimit law to the design conditions and physical properties of the spray. L
(1
-
D E F I N I T I O N O F SPRAY
In the field discussed here, a spray is considered as a system of liquid droplets in a fluid continuous phase. By specifying “droplets” we eliminate macroscopic cases wherein large individual drops, slugs, or columns predominate, and microscopic cases wherein dispersion is of molecular order, or a t least where drop-
Figure 1.
Droplet Size D i s t r i b u t i o n (Schematic)
Among artificial sprays are fountain sprays, atomizer sprays, entrained liquid in a fractionating column, and the disperse phase in a solvent extraction column. Sprays considered in this paper all have a resultant motion
