182 Science and Engineering of Droplets

1.2 or 1.5.[285] A bimodal distribution pattern has been observed by other researchers[313] for different experimental conditions. For shear breakup, a bimodal behavior has also been found.[285] This is deemed to be caused by the core droplet that remains after the stripping of smaller droplets from its periphery ceases. In bag or multimode breakup, however, several large droplets form from the ring at the base of the bag, as compared to the single core droplet in the shear breakup process. A correlation for SMD has been derived by Hsiang and Faeth[285] based on their experimental measurements:

Eq. (36) ρg SMDU R2 /σ = 6.2(ρL / ρG )1/ 4 [μL /(ρL DiniU R )]1/ 2 Wed

with a correlation coefficient of the fit being 0.91. Thus, the entire droplet size distribution after bag or multimode breakup may be determined once the SMD is calculated from Eq. (36).

Formulations for SMD of secondary droplets have also been derived by other researchers, for example, O’Rourke and Amsden,[310] and Reitz.[316] O’Rourke and Amsden[310] used the χ -square distribution[317] for determining size distribution of the secondary droplets. They speculated that a breakup process may result in a distribution of droplet sizes because many modes are excited by aerodynamic interactions with the surrounding gas. Each mode may produce droplets of different sizes. In addition, during the breakup process, there might be collisions and coalescences of the secondary droplets, giving rise to collisional broadening of the size distribution.

3.1.3Droplet Formation in Atomization of Melts

Most commercial and near-commercial atomization processes for liquid metals/alloys involve two-fluid atomization or centrifugal atomization. As suggested by many experimental observations, twofluid atomization of liquid metals is typically a three-stage process,[318][319] whereas centrifugal atomization may occur in three different regimes.[5][320] Many atomization modes and mechanisms for normal liquids may be adopted or directly employed to account

Fundamental Phenomena and Principles 183

for these atomization processes for liquid metals. For example, the Rayleigh mechanism may govern the disintegration of liquid ligaments formed during the breakup of a liquid jet in two-fluid atomization or generated in the Ligament regime of centrifugal atomization, whereas the mechanisms proposed by Fraser et al.,[116][207] and by Mansour and Chigier,[209] as well as the Taylor instability theory may be relevant to the disintegration of liquid sheets or films formed in two-fluid atomization, or generated in Sheet regime of centrifugal atomization. Simultaneous occurrence of two or more modes in a single process is possible, depending on the physical properties of melt (and gas) and atomizer geometry and configuration.

Droplet Formation in Gas Atomization. Experimental and

modeling studies[160][161][169][318][319][321]–[325] have shown that gas

atomization of liquid metals in spray forming and powder metallurgy processes may take place in two primary modes, i.e., liquid jetligament breakup and liquid film-sheet breakup.

Liquid Jet-Ligament Breakup (Fig. 3.12) may be an operating mechanism in gas atomization of melts with a free-fall atomizer. See et al.[318][319] made experimental and analytical studies on droplet formation during gas atomization of liquid metals with a free-fall atomizer using photographic techniques. These studies revealed that high velocity gas atomization of liquid metals is a three-stage process: (1) primary disintegration, (2) secondary disintegration, and (3) spheroidization and solidification. In the first stage, when a liquid metal stream issuing from a delivery nozzle enters into the influence region of discrete gas jets, it expands laterally and forms an expanding hollow cone above the geometric impingement point of the gas jets. The lateral spreading is caused by (a) the liquid being squeezed into the interstices of the dynamic pressure contours generated by the gas jets as they approach each other, and (b) an upward thrust generated by the reverse (recirculating) gas flow along the metal stream axis due to the negative pressure gradient in the upward direction. The magnitude of these effects depends upon factors such as metal specific gravity, metal stream diameter, nozzle design features (notably impingement angle), and gas jet velocity, among others. The periphery of the cone is presumably in the form of

184 Science and Engineering of Droplets

ligaments connected to the base of the cone. The cone spreads laterally until viscous and surface tension forces can no longer hold the liquid together as the cone falls under the action of aerodynamic forces and gravity into the region of higher gas velocities. The changes in dynamic pressure initiate and enhance the growth of disturbances on the liquid surfaces. As the cone thins out, ligaments and/or other irregular shapes are torn off or stripped from its periphery in a random manner, and subsequently break up into droplets following the Rayleigh mechanism. At high velocities, this tearing action may be due to the aerodynamic forces caused by the pressure variations in the gas flow around the waves and bulges on the liquid surfaces.[5] This primary disintegration occurs in time intervals much less than 5 × 10-4 s.[319] After the disintegration of each cone, another is formed which then undergoes disintegration. The cycle of the formation and disintegration of the metal cone repeats as the atomization proceeds. The initial expansion of the liquid metal stream is an important step in the primary disintegration because it determines the effective surface area of the liquid metal in contact with the gas. The exact manner in which droplets are formed depends on the dynamic instability of the surface disturbances, turbulence in the gas flow, and the variation in the gas velocity field around the metal surface. Although a fundamental understanding of the detailed atomization mechanism is yet to be established, it is widely accepted that in most cases the Rayleigh instabilities growing on the surfaces of the torn liquid ligaments lead to the formation of droplets.[4][147][238] Subjected to pressure forces and shear forces, a droplet formed by primary disintegration may undergo secondary disintegration if the dynamic pressure due to the gas jet velocity exceeds the restoring force due to the surface tension of the liquid metal. Solidification of the liquid metal/droplets may occur before, during and/or after the primary disintegration and/or the secondary disintegration, depending on process parameters and material properties. The final shape of a semi-solid or solid metal particle is determined by the relative magnitudes of the time needed for solidification to complete, and the time required for surface tension force to restore the metal droplet into a sphere.

186 Science and Engineering of Droplets

Recently, detailed experimental observations using highspeed videography techniques[326] provided insightful information on the primary breakup stage in gas atomization of melts using a free-fall atomizer with two-level nozzle configuration. It is suggested that for the typical conditions prevailing in spray forming processes, the membrane-type breakup mode (or even fiber-type breakup mode at very high gas flow rates) proposed by Faragó and Chigier[210] may be the dominant operating mechanism(s) governing the liquid metal jet breakup. The interaction between the liquid metal jet and the gas jets from the top-level nozzles initiates small disturbances on the liquid surface. While the round liquid jet enters into the ‘influence zone’ of the gas jets from the second-level nozzles, it develops into a thin liquid membrane as a result of the distortion and thinning by the gas flow. Aerodynamic shear forces exerted on the membrane create the Kelvin-Helmholtz instability, leading to the formation of surface waves. Liquid accumulates at the edges of the thin membrane, forming a liquid frame while waviness increases in amplitude. The frame then breaks up into droplets via the nonaxisymmetric Rayleigh mechanism. It is speculated that the ‘liquid metal cone’ observed in the experiments of See et al.[318][319] may be a temporal image of the Kelvin-Helmholtz waves because the jet breakup in a co-flowing gas is a pulsating process, as suggested by Faragó and Chigier.[210] However, in the work of Faragó and Chigier[210] no recirculating or transverse flow was present under the co-flowing condition. Therefore, the liquid jet-ligament breakup mechanism derived from the work of See et al.[318][319] might be different from the membrane-type breakup mode.

Liquid Film Sheet Breakup (Fig 3.12) is often encountered in high pressure gas atomization with a close-coupled atomizer.[324][325][327][328] This atomization mode is also a three-stage process, i.e., primary disintegration, secondary disintegration, and spheroidization/solidification.In the first stage, a liquid metal flows downward through a delivery tube initially. At the exit of the tube, it changes its flow direction to radially outward along the tube base plane, forming a thin liquid film, due to the recirculating gas flow in

Fundamental Phenomena and Principles 187

this region. The liquid film flowing radially then turns into a conical liquid sheet at the edge of the tube base plane, owing to the shear forces and pressure forces exerted by the gas flow parallel and perpendicular to the sheet surface, respectively. The pressure forces cause the sheet to align with the gas flow while the shear forces cause the sheet to accelerate downstream within the gas flow field. The conical liquid sheet forms a hollow cone beneath the delivery nozzle. The relatively high surface tension of metal melts produces a longitudinal streakiness in the spray due to the instability in the conical sheet near the nozzle. The sheet extending parallelly to the gas flow is highly unsteady and can vary from a length of essentially zero to a length extending a significant distance downstream into the gas flow field. While the sheet thins out, it fragments into droplets via the mechanisms proposed by Fraser et al.[116][207] and by Mansour and Chigier.[209] The droplets generated by the primary breakup accelerate, deform, and cool in the gas flow field. Secondary breakup of the droplets may occur, depending on their deformation and cooling/ solidification rates. Similarly to the situation in the liquid jet-liga- ment breakup mode, solidification of the liquid metal may occur at any stage during the atomization. Hence, the relative magnitudes of the time needed for solidification to complete and the time required for surface tension force to spheroidize a droplet determine its final shape.

In both atomization modes, as thin unstable ligaments, and/ or sheets disintegrate into round droplets, atomization gas may plausibly be trapped into the droplets under certain conditions. For alloys with alloying elements which readily react with atomization gas, for example, oxidize to form refractory oxides, solidification may be delayed and spheroidization is prevented so that rough flakes may form. For such alloys, the atmosphere in the spray chamber must be inert and protective to avoid the formation of any refractory and to foster spheroidal shape of droplets.

While gas atomization of liquid metals is generally viewed as a three-step process, the Kohlswa ultrasonic gas atomization is suggested to be a single-step process.[172] When ‘particles’ of an atomization gas at extremely high velocities strike a liquid metal

188 Science and Engineering of Droplets

stream, the liquid responds like a rigid solid but with low shear resistance. It is assumed that each gas particle may be able to knock out one metal droplet. However, the actual efficiency is low due to energy losses through reflection of gas pulses and heat generation, etc. To date, no mathematical model is available for quantifying the atomization mechanism in USGA.

Generally, gas atomization of melts produces a wide range of droplet size distributions. This is plausibly due to the uncontrolled disintegration of liquid stream into sheets and/or ligaments by gas jet(s) with a velocity distribution and the random breakup of the sheets and ligaments into droplets downstream of the jet impingement zone. For external-mixing atomizers, maximizing energy utilization of the high velocity atomization gas is of critical importance. At the high velocities used in gas atomization, the gas jets rapidly become turbulent and spread so that their velocities decay. A gas jet may maintain a centerline velocity approximately equal to its exit velocity for about six nozzle diameters downstream,[5] depending generally on Mach number.[175] If liquid breakup is completed within this high velocity zone, the fineness of atomization is optimized and a relatively narrow, mono-modal size distribution may be produced. If liquid breakup occurs downstream of the high velocity zone, a mono-modal, yet coarser, spray may be generated. Atomization spanning the high and low velocity zones may produce bimodal size distributions.[5] Therefore, using high gas velocities with optimized atomizer designs, such as HPGA and USGA, may be a way to increase the yield of fine droplets with a relatively narrow size distribution. Along with the rapid cooling potential, high velocity methods appear to be very attractive for many applications.

In the high-velocity atomization processes, the effect of the compressibility of atomization gas could be quite appreciable. Extending Taylor’s analysis[245] for a gas flow over a liquid to a compressible gas flow, Bradley[329][330] developed an expression for the fastest growing wavelength that dominates the disruption at the interface. Supposing that surface tension nips off the crest of the wave into a filament whose diameter is some fraction of the

Fundamental Phenomena and Principles 189

fastest-growing wavelength, Bradley derived the following expression for the most frequently occurring droplet diameter:

	Eq. (37)	D =	2πε
			0.53kmax

where ε is a parameter taken as ¼, and 2π /kmax is the fastest-growing wavelength. In Bradley’s model, kmax is related to process param-

eters through a dimensionless parameter χ or χ m:

	ì	χρGU s	2	0 < Ma < 1
	ï
		σ
kmax	ï
	= í	2
Eq. (38)	ï	ρGU s		1 £ Ma < 10
	ï	χ mσ
	î

where Us is the sonic velocity (speed of sound in gas), and Ma is the Mach number. The speed of sound in gas can be calculated using the following expression:

Eq. (39)	U	s	= (γ P / ρ	G	)1/ 2 = (γ RT )1 / 2
			G		G

where PG is the gas pressure, and the values of the gas constant R and the isotropic factor of gas γ can be found in Table 2.7 for various gases. The droplet diameter is then formulated as:

	ì	2.96σ		0	< Ma <1
	ï
		2
Eq. (40)	ï	χρGU s
	D = í	2.96σχm
	ï			1	£ Ma <10
	ï	2
	î	ρGU s

The dimensionless parameter χ or χm is a function of Mach number and can be determined from a universal curve for the atomization of various liquid metals, i.e., a plot of χ or χm vs. Mach number, as presented by Bradley in Ref. 329 for subsonic gas flow, and in

190 Science and Engineering of Droplets

Ref. 330 for sonic and supersonic gas flow, respectively. Although the mathematical treatment of this model is sound and the derived equation for droplet diameter is simple for use, some important process parameters and material properties are not included, such as gas to liquid mass flow rate ratio, liquid viscosity, and nozzle geometry parameters. These parameters have been proven to affect the resultant droplet size in a significant manner. In addition, there seems to be no justification in assuming a constant value for the parameter ε . In fact, Rao and Mehrotra[331] have shown by analyzing atomization data for molten lead, tin, and lead-tin eutectic alloy that the parameter ε is a function of atomizing angle and nozzle diameter:

Eq. (41)	ε = −0.12 + 0.57ϑ + 0.38d 0	for tin
Eq. (42)	ε = −0.14 + 0.66ϑ + 0.48d0	for lead

where ϑ is the atomizing angle, i.e., the angle between gas nozzle axis and liquid nozzle axis.

Another limitation of Bradley’s model is that the liquid phase is assumed as an initially stationary flat layer of infinite depth over which a gas flows with uniform velocity. This is seldom the condition in practical atomization processes involving liquid metals. In Bradley’s model, there is no indication of the origin of the spread in droplet size generated in gas atomization. Moreover, as pointed out by Mehrotra,[321] Bradley’s model, and many previous models are pertinent to atomization process in which gas flows parallel to liquid surface. Thus, the models may not be able to adequately represent gas atomization processes of liquid metals involving ring-like atomizers. In such processes, gas streams typically impinge on a liquid stream at an angle. Thus, the velocity component perpendicular to the liquid metal surface is most likely to have a significant effect on the fluid dynamics of liquid metal breakup. The instability of liquid surfaces accelerated in a perpendicular direction has been analyzed by Taylor[206] and Lewis.[332]

Fundamental Phenomena and Principles 191

Droplet Formation in Water Atomization. In water atomization of melts, liquid metal stream may be shattered by impact of water droplets, rather than by shear mechanism. When water droplets at high velocities strike the liquid metal stream, some liquid metal fragments are knocked out by the exploding steam packets originated from the water droplets and subsequently contract into spheroidal droplets under the effect of surface tension if spheroidization time is less than solidification time. It is assumed that each water droplet may be able to knock out one or more metal droplet. However, the actual number of metal droplets produced by each water droplet may vary, depending on operation conditions, material properties, and atomizer designs.

Droplet Formation in Centrifugal Atomization. The mechanisms of centrifugal atomization of liquid metals are quite similar to those for normal liquids. Three atomization modes have been identified in rotating electrode atomization process, i.e., (1) Direct Droplet Formation, (2) Ligament Disintegration, and (3) Film/Sheet Disintegration.[189][320] These are also applicable to the centrifugal atomization of melts in general. The Direct Droplet Formation mode is an operating mechanism at lower liquid flow rate. In this mode, liquid flows toward the periphery of a disk/cup or an electrode, and accumulates at the edge in the form similar to a small torus that deforms into many protuberances. From these protuberances, droplets are directly thrown off. The phenomenon is attributed to the Taylor instability,[205] which in this case is caused by centrifugal force, and opposed by surface tension force. Droplets formed in this mode are almost free of satellites. Under certain conditions, however, a bimodal droplet size distribution may be generated, if two droplets are thrown off from each protuberance, a large one from its head, and a small one from its tail, at equal numbers. With increasing liquid flow rate, the droplet formation process may transit into the Ligament Disintegration mode. In this mode, the protuberances develop to larger amplitudes than in the Direct Droplet mode before the elongated ligaments break up due to the Rayleigh instability. Droplet size distribution is typically bimodal. With increasing liquid flow

192 Science and Engineering of Droplets

rate, droplet size increases and the weight fractions of small and large droplets become similar. Further increasing liquid flow rate to a critical value at which the ligaments can no longer accommodate the flow of liquid, a thin continuous film or sheet forms, i.e., the droplet formation process transits into Film/Sheet Disintegration mode. In this mode, the sheet extends from the lip to an equilibrium length at which the contraction energy at the sheet edge due to surface tension is equal to the kinetic energy of the extending sheet. The sheet then breaks up via the mechanisms proposed by Fraser et al.[116][207]

Droplet Formation in Roller Atomization. In roller atomization of melts, the atomization mechanism was proposed to be similar in nature to the phenomenon of cavitation occurring in oil lubricated bearings.[188] During roller atomization, a liquid metal sheet forms in the gap between the rolls. Holes appear in the sheet, and seem to propagate as the sheet moves from the rolls. The sheet cavitates a small distance downstream of the nip between the rolls, forming perforations and threads. At low roll speeds (< 3000 rpm), the perforations formed propagate in the liquid sheet that issues from the gap between the rolls. At high roll speeds, the liquid may flow around the points of cavitation to form thin threads. The number and size of the perforations and threads are dependent on the number of cavities formed. It was speculated that droplets originate from the ligaments formed by impact of the rims produced from adjacent perforations (the perforated-sheet disintegration mechanism proposed by Fraser et al.).[116][207] The liquid threads break up into droplets by the propagation of liquid jet instabilities following the Rayleigh mechanism.

A mathematical analysis of the process is extremely difficult and requires to solve the Reynolds equation of lubrication theory and apply the solution to the cavitation boundary conditions. A twodimensional analysis of the pressure distribution in the plane of the roll nip showed that the liquid pressure rises sharply to a large value near the nip, and drops equally sharply to a minimum just beyond the nip. Before large negative pressures are reached, the liquid may cavitate as a result of the expansion of entrained gases within the liquid.