Fundamental Phenomena and Principles 169

transient bubble activity. To obtain a near-monodisperse spray at high flow rates, the liquid film thickness must be maintained below the level at which bubbles can reach a resonant diameter. In practice, this is a matter of filming the liquid on the excited surface, and it appears that the hollow horn design may achieve this effect.[115]

3.1.2Secondary Atomization

Secondary atomization (secondary breakup, or secondary disintegration) is a process in which a droplet splits up into smaller droplets as a result of the interaction between the droplet (dispersed phase) and surrounding medium (continuous phase). Aerodynamic forces acting on a droplet during its motion may cause deformation of the droplet and under certain conditions may overcome internal cohesive forces, disintegrating the droplet into smaller droplets. Since the final droplet size distribution produced in an atomization process will be determined by the liquid properties in both the primary and secondary disintegration, it is of importance to review various mechanisms developed for the secondary breakup of a single droplet under the action of aerodynamic forces.

The fragmentation of liquid droplets due to sudden exposure to a high-velocity gas stream has many important applications in the fields of aerodynamics and propulsion. For example, the phenomenon of supersonic rain erosion caused by the impingement of rain droplets at high relative velocities on exterior missile or aircraft surfaces can be greatly alleviated through proper aerodynamic design. This can be achieved by designing a body whose detached shock is sufficiently far removed to allow for droplet shattering in the region separating the shock from the body surface. In propulsion field, the rate of mixing and combustion of liquid fuel droplets can be greatly enhanced by virtue of the droplet fragmentation process. The droplet breakup time may become a rate-controlling mechanism in combustion systems.

Theoretical Considerations. A droplet generated in the primary atomization may be unstable and may further disintegrate

170 Science and Engineering of Droplets

into smaller droplets, depending on the competition between external aerodynamic forces and internal forces due to surface tension and viscosity at any point on the droplet surface. Therefore, analyses of droplet deformation and secondary breakup require a detailed knowledge of the distribution of the forces on the droplet. Further, the distribution of the forces varies with time as the droplet shape changes as a result of the deformation caused by these forces. When the internal forces at any point on the droplet surface are sufficient to balance the external aerodynamic forces, an equilibrium state is reached. As long as a change in external forces at any point on the droplet surface is compensated by a corresponding change in internal forces, a droplet can remain stable. However, if external forces are very large and any change in them can not be balanced by a corresponding change in internal forces, the external forces may deform the droplet and subsequently disintegrate it into smaller droplets. After the disruption, the higher surface tension pressure of a smaller droplet may be large enough to accommodate the variations in the external forces so that a stable stage may be attained and no further breakup can occur. Otherwise, further deformation and breakup may follow until the surface tension pressure of a further smaller droplet is large enough to maintain a force balance at all points on the droplet surface. When this equilibrium state is attained, the droplet is stable. This droplet size is termed critical droplet size. Conceptually, a stable droplet of the size equal to or less than the critical droplet size has an infinite breakup time, while the breakup time of any droplet larger than the critical droplet size increases with decreasing droplet size. Even for a droplet larger than the critical droplet size, no breakup may occur if the external aerodynamic forces on the droplet decrease due to the variation in flow conditions or the internal forces increase due to the variation in surface tension and/or viscosity during the breakup time to the extent that an equilibrium state is reached, even if initially the external forces are sufficient to produce droplet breakup. Therefore, a rigorous analysis of droplet breakup demands information of dynamic distribution of various forces and the transient effects of flow conditions and physical properties.

Fundamental Phenomena and Principles 171

Basic Breakup Modes. Starting from Lenard’s investigation of large free-falling drops in still air,[267] drop/droplet breakup has been a subject of extensive theoretical and experimental studies[268]- [285] for a century. Various experimental methods have been developed and used to study droplet breakup, including free fall in towers and stairwells, suspension in vertical wind tunnels keeping droplets stationary, and in shock tubes with supersonic velocities, etc. These theoretical and experimental studies revealed that droplet breakup under the action of aerodynamic forces may occur in various modes, depending on the flow pattern around the droplet, and the physical properties of the gas and liquid involved, i.e., density, viscosity, and interfacial tension.

According to Hinze,[270] droplet breakup may occur in three basic modes:

(1)A droplet is initially flattened to an oblate, lenticular ellipsoid and then may be converted into a torus, depending on the magnitude of the internal forces causing the deformation. The torus subsequently becomes stretched and splits into smaller droplets.

(2)A droplet is initially elongated to a long cylindrical thread or ligament of cigar shape, and then breaks up into smaller droplets.

(3)Some protuberances may be created on a droplet surface due to local deformations. Under favorable conditions, these bulges detach from the droplet and disintegrate into smaller droplets.

The first mode may occur when a droplet is subjected to aerodynamic pressures or viscous stresses in a parallel or rotating flow. A droplet may experience the second type of breakup when exposed to a plane hyperbolic or Couette flow. The third type of breakup may occur when a droplet is in irregular flow patterns. In addition, the actual breakup modes also depend on whether a droplet is subjected to steady acceleration, or suddenly exposed to a highvelocity gas stream.[270][275]

Fundamental Phenomena and Principles 173

Suddenly exposed to a high-velocity gas stream, a droplet is deformed into a saucer shape with a convex surface to the gas flow. The edges of the saucer shape are drawn out into thin sheets and then fine filaments are sheared from the outer part of the sheets, which subsequently disintegrate into smaller droplets and are swept rapidly downstream by the high-velocity gas. Unstable growth of short wavelength surface waves appears to be involved in the breakup process.[210] This is known as shear breakup (Fig. 3.10).[246]

In both these modes of droplet breakup, the initial stage of the process involves a flattening of the droplet into a disk shape, normal to the flow direction. In addition, there are two distinct size ranges of droplets with smaller droplets moving downstream of the larger droplets. In the bag breakup mode, a droplet breaks upvia membrane formation with features similar to those seen in the co-flowing jet breakup cases. Structures present in the gas flow are responsible for the distorting and stretching of the liquid.[210] High-velocity droplet breakup mechanism is similar to the high-velocity jet breakup mechanism in the second wind-induced and possibly also the atomization regimes, which involve the unstable growth of short wavelength surface waves and the formation of fibers.[210] Generally, there seem to be similarities between the disintegration modes of round liquid jets in a coaxial air stream, and the breakup of droplets in an air stream. At similar aerodynamic Weber numbers, the disintegration modes are also similar (Table 3.2).

The experimental results of aerodynamic shattering of droplets behind shock waves revealed that the major variables affecting the high-speed breakup (Weber numbers much greater than 10) are the droplet diameter and the dynamic pressure of convective flow, with the liquid properties being less important. Ranger and Nicholls[246] extended previous experimental and analytical studies to a range of conditions typical of two-phase detonations, i.e., Mach numbers of 1.5–3.5 and droplet diameters of 750–4400 µm. Their shock-tube experiments revealed that droplet breakup can be temporally divided into two rather distinct stages.

174 Science and Engineering of Droplets

The first stage, called dynamic stage, is the period during which a spherical droplet is flattened and deformed into a planetary ellipsoid with its major axis perpendicular to the flow direction as a result of the external pressure distribution. The eccentricity of the elliptical profile changes with time.

The second stage, termed surface stripping stage, is characterized by boundary-layer stripping from the deformed droplet due to the shearing action of the convective flow that rapidly reduces the droplet to clouds of micromist. The stripping stage is well developed after the droplet is contacted by the shock. At the high Mach number (3.5), after a droplet is collided by the shock, a well-defined wake forms behind it.

Interestingly, the shape of the wake is similar to that developed behind a hypersonic blunt body where the flow converges to form a narrow recompression neck region several body diameters downstream of the rear stagnation point due to strong lateral pressure gradients. The liquid material, that is continuously stripped off from the droplet surface, is accelerated almost instantaneously to the particle velocity behind the wave front and follows the streamline pattern of the wake, suggesting that the droplet is reduced to a fine micromist.

The experimental observations also revealed that the droplet breakup is a continuous process of disintegration that begins shortly after the initial contact between a shock and the droplet, and proceeds until the droplet is completely transformed into a cloud of mist. The collision between the shock and the droplet has little effect on the shattering phenomenon. The main function of the shock is to produce a high-speed convective flow. The droplet breakup is mainly caused by the interaction between the droplet and the convective flow established by the shock. The shearing action exerted by the high-speed gas stream on the droplet periphery causes the formation of a boundary layer in the liquid surface. The layer is established very rapidly after the droplet is intercepted by the shock, and stripped away from the equator continuously. This boundary-layer stripping mechanism accounts for the droplet breakup in the shear breakup regime.

fluids,[288][289]

a polymer,[287]

Fundamental Phenomena and Principles 175

Recently, many researchers investigated shape fluctuations, oscillations, deformation, and/or breakup of a droplet containing during elongational flow in blends of viscoelastic in a gaseous medium,[290]-[295] in electric,[296]-[299]

magnetic,[300] or acoustical fields,[301][302] in shock waves,[303] during interaction with other droplets,[304] and by surfactant.[305] Various deformation theories,[288][289] and droplet breakup mecha- nisms[306]-[309] have been proposed.

Breakup Criteria. Generally, droplet breakup in a flowing stream is governed by its surface tension and viscous forces, and dynamic pressure. For liquids of low viscosities, droplet breakup is primarily controlled by the aerodynamic force and surface tension force, and may begin when a critical condition, i.e., an equilibrium between these two forces, is attained:

	π D2		ρ U 2
Eq. (27)		CD	A R	= π Dσ
			2
4

where CD is the drag coefficient, and it is assumed that the relative velocity between the droplet and surrounding gas is high. From this equation, the critical Weber number Wecrit, critical droplet size (maximum stable droplet size) Dcrit, and critical relative velocity UR crit may be derived as follows:

Eq. (28)

Wecrit

=

8

CD

Dcrit =

8σ

C

D

ρ U

2

A

R

æ

8σ

ö0.5

U

R crit

=

ç

÷

ç

÷

è CD ρ A D ø

176 Science and Engineering of Droplets

Similar criterion has been obtained by Taylor.[205] Hinze[270]

estimated Wecrit to be 22 for a free-fall droplet, and 13 for a lowviscosity liquid droplet exposed suddenly to a high-velocity air

stream. The latter value is comparable to those for water, methyl alcohol, mercury, and a low-viscosity silicone oil obtained by other investigators.[275]–[277]

For liquids of higher viscosities, the influence of liquid viscosity on droplet breakup needs to be considered. According to Hinze,[270] the critical Weber number may be modified to the following expression to account for the effect of liquid viscosity:

Eq. (29) We crit = We crit , 0 [1 + f (Oh )]

where Wecrit,0 is the critical Weber number for zero viscosity. Empirical relationship has been proposed such as:

Eq. (30)

Wecrit = Wecrit , 0 + 14Oh

In some practical processes, a high relative velocity may not exist and effects of turbulence on droplet breakup may become dominant. In such situations Kolmogorov,[280] and Hinze[270] hypothesized that the turbulent fluctuations are responsible for droplet breakup, and the dynamic pressure forces of the turbulent motion determine the maximum stable droplet size. Using Clay’s data,[281] and assuming isotropic turbulence, an expression was derived for the critical Weber number:[270]

Eq. (31)

Wecrit = 1.18

Sleicher[278] has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park[279] modified the hypothesis of Kolmogorov,[280] and Hinze,[270] and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an

Fundamental Phenomena and Principles 177

entrained fluid particle. On the basis of this hypothesis, a formulation was derived for the critical Weber number in which Hinze’s value of 1.18 is replaced by 1.04.

In highly viscous flows, the Reynolds numbers may be so small that dynamic forces may be no longer significant, and thus, droplet breakup may be dictated merely by viscous and surface tension forces. This may apply to the situation that, for example, some fluid globules are surrounded by a viscous fluid with a strong velocity gradient in the vicinity of the globules. A droplet is first elongated into a prolate ellipsoid by the viscous shear. When the viscous forces are sufficiently large compared to surface tension forces, droplet breakup occurs. This is often referred to as Taylor mechanism. This mechanism applies only when an undeformed or elongated droplet is small relative to local regions of a viscous flow, as indicated by Sevik and Park.[279] For large Reynolds numbers of external flows, the local regions may be very small in spatial dimensions compared to droplet sizes so that the dynamic pressure due to velocity gradient over the distance of the order of droplet size may become a dominant factor.

In practical situations, viscosities of both dispersed and continuous phases and their ratio may exert effects on droplet breakup. In particular, the viscosity of the dispersed phase may delay and impede breakup, as demonstrated by Meister and Scheele.[219] For a viscosity ratio of near unity, the perturbation wavelength has a minimum value corresponding to a maximum growth rate.[220][282] Either increasing or decreasing the value of the viscosity ratio reduces the tendency of droplet breakup.[220][270][282] Quantitative formulation encompassing the effect of the viscosity ratio has been proposed by Rumscheidt and Mason[283] such that:

	Eq. (32)	Wecrit	=		1 + (μ L / μ A )
				1 + (19 /16)(μ L / μ A )

where µA is the viscosity of air (continuous phase). This may explain the facts that dispersion by pure viscous flow is restricted to emulsification

178 Science and Engineering of Droplets

processes, and it is very difficult to disintegrate liquids of large viscosity ratios. In shock experiments, the critical Weber number for

breakup has been found to be Wecrit ≈ 12.[310]

Breakup Regimes. Hsiang and Faeth[285] made a comprehensive review of previous theoretical and experimental studies on secondary breakup of droplets. They extended the work of Hinze,[270] and Krzeczkowski, [311] and identified six regimes of droplet deformation and breakup, as depicted in Fig. 3.11,[270][285][311] and summarized in Table 3.3, where Wed and Oh d are defined as Wed = Dρ GUR2 / σ and Ohd = µL /(Dρ Lσ)1/2, respectively. This regime map can be used to determine if a droplet remains stable, or undergoes deformation or secondary breakup in addition to deformation. In this regime map, it can be seen that for a given Ohd, a droplet may remain stable and not deform at a small Wed. With increasing Wed, a droplet may undergo nonoscillatory or oscillatory deformation. The transition from bag breakup,[286] through multimode breakup to shear breakup,[246] regime occurs with further increase in Wed. At very high Wed, catastrophic breakup[312] involving complex breakup mechanisms may take place. For high viscosity liquids, i.e., with increasing Ohd, the breakup regime transitions are moved to higher Weber number values.

It is interesting to note that there are similarities in the breakup mechanisms, and even in the magnitudes of the Weber numbers between droplet breakup and jet breakup in various regimes. For example, the onset of the bag breakup occurs at the Weber number similar to that for jet breakup to enter the second windinduced regime. This is not surprising, because the Weber number measures the relative importance of gas inertia forces to surface tension forces, while these forces are influential in distortion and breakup processes of both droplets and jets. For the same reason, the breakup mechanisms of droplets and jets in a high velocity gas should also be similar as long as the instability wavelengths of surface waves are much less than the diameters of the parent droplets or jets, and thus, the identity of the bulk liquid would become irrelevant.[210]

180 Science and Engineering of Droplets

Hinze,[270] Hanson et al. [277] Gel’fand et al. [313] Krzeczkowski,[311] Reinecke and Waldman,[312] and Wierzba and Takayama[314] presented photographs of all the breakup regimes in the regime map. In high-pressure propulsion and combustion systems, acoustic and turbulent oscillations cause oscillatory deformation of droplets that may lead to secondary breakup. The breakup regime at the onset of secondary breakup has been termedbag breakup, also known as umbrella or hat-type breakup.[275] This regime involves deflection of a droplet into a thin disk normal to the flow direction, followed by deformation of the center of the disk into a thin bag (balloon-like) structure extending in the downstream direction. Both the thin center and thick rim subsequently disintegrate into droplets. The transition from bag breakup regime to shear breakup regime involves complex breakup processes, such as parachute breakup, chaotic breakup, bag-jet breakup, and transition breakup, etc.,[311] and therefore is referred to as multimode breakup. Shear breakup involves deflection of the periphery of the flattened droplet in the downstream direction and stripping of small droplets from the periphery. Shear breakup may occur when droplets are hit by shock waves.

Hwang and Reitz[315] studied the droplet breakup mechanisms in the catastrophic breakup regime. They found that the acceleration of a flattened droplet favors the development of the Rayleigh-Taylor instability.[206] The accelerating droplet breaks into large-scale fragments via the Rayleigh-Taylor instability. Ligaments and small droplets then form from the much shorter wavelength Kelvin-Helmholtz waves on the fragmented surfaces. Hwang and Reitz suggested that the droplet breakup mechanisms in the catastrophic breakup regime are similar to the high velocity jet breakup mechanisms in the second wind-induced regime, and possibly also the atomization regime, that involve the unstable growth of short wavelength surface waves.

The time tb for a droplet to undergo deformation prior to secondary breakup is a function of Ohd and a characteristic time

t*:[285]

	Fundamental Phenomena and Principles 181
Eq. (33)	tb / t * = 5 /(1 − Oh d / 7) ,	Wed < 103

where t* = D(ρL/ρG)1/2/UR. Similar formulations for tb were derived and used by Ranger and Nicholls,[246] O’Rourke and Amsden,[310] and Reitz.[316] For a constant Oh d, the breakup time is proportional to the droplet diameter and the square root of liquid-to-gas density ratio and inversely proportional to the relative velocity.[246] The distortion of a droplet prior to breakup has a significant effect on the droplet drag, and consequently on the droplet breakup process. During deformation, a droplet may flatten, leading to an increase in cross-stream diameter prior to breakup. The degree of droplet deformation at any given time t may be correlated to the maximum crossstream droplet diameter Dc,max and a characteristic time tc*:[285]

Eq. (34)	Dc − Dini	= 0.615	t
	Dc,max − Dini		tc*

where tc* = t*/(1 – Ohd / 7), Dc is the actual cross-stream droplet diameter, and Dini is the initial droplet diameter. The maximum cross-stream droplet diameter is a function of Wed, as reported by Hsiang and Faeth[285] based on their measurement data:

Eq. (35)	Dc,max	= 1+ 0.19We1/ 2	,	Oh	d	< 0.1, We < 102
		d				d
	Dini

Hsiang and Faeth[285] also presented measurement data for

Dc,max / Dini as a function of Wed in graphical form for Ohd=1.5 and 3.1. Thus, the maximum cross-stream droplet diameter over a range

of Ohd may be determined by interpolating these data.

In the breakup regimes, a droplet may undergo secondary breakup when the breakup time is reached. The droplet size distribution after bag or multimode breakup may follow the Simmons’ root-normal distribution pattern[264] with MMD/SMD equal to 1.1,