278 Science and Engineering of Droplets

Generally, the mean droplet size is proportional to liquid surface tension, and inversely proportional to liquid density and vibration frequency. The proportional power index is ~1/3 for the surface tension, about -1/3 for the liquid density, and -2/3 for the vibration frequency. The mean droplet size may be influenced by two additional parameters, i.e., liquid viscosity and flow rate. As expected, increasing liquid viscosity, and/or flow rate leads to an increase in the mean droplet size,[130][482] while the spray becomes more polydisperse at high flow rates.[482] The spray angle is also affected by the liquid flow rate, vibration frequency and amplitude. Moreover, the spray shape is greatly influenced by the direction of liquid flow (upwards, downwards, or horizontally).[482]

In ultrasonic standing wave atomization, increasing the amplitude of the sonotrodes and/or the gas pressure reduces the mass median diameter of the droplets generated.[143]

4.3.0CORRELATIONS FOR DROPLET SIZES OF MELTS

As described above, a number of empirical and analytical correlations for droplet sizes have been established for normal liquids. These correlations are applicable mainly to atomizer designs, and operation conditions under which they were derived, and hold for fairly narrow variations of geometry and process parameters. In contrast, correlations for droplet sizes of liquid metals/alloys available in published literature[318][323][328]-[331][485]-[487] are relatively limited, and most of these correlations fail to provide quantitative information on mechanisms of droplet formation. Many of the empirical correlations for metal droplet sizes have been derived from off-line measurements of solidified particles (powders), mainly sieve analysis. In addition, the validity of the published correlations needs to be examined for a wide range of process conditions in different applications. Reviews of mathematical models and correlations for

Empirical and Analytical Correlations 279

droplet sizes generated by twin-fluid atomization have been made by Mehrotra,[321][322] with emphasis on gas atomization of liquid metals/alloys using free-fall type of ring atomizers. In a recent review, Aller and Losada[485] have surveyed the mathematical models developed for commercial and some near-commercial atomization processes for metal powder production.

4.3.1Gas Atomization

Analytical,[321][322][331][486] and experimental studies[162][164][318][319][323] on the atomization of molten metals have been performed in an effort to elucidate atomization mechanisms and control droplet size. These studies have led to the development of various correlations that may be used to predict the mean size and the size distribution of atomized droplets. Analytical, empirical, and semi-empirical correlations for droplet sizes of liquid metals/alloys are summarized in Tables 4.15 and 4.16, for gas atomization via jet breakup and film/sheet breakup, respectively. Among these, the empirical correlation developed by Lubanska[323] is probably the best known and most quoted one for free-fall type of ring-like atomizers due to its reasonable accuracy under various experimental conditions. Lubanska correlated a considerable amount of spray data for low melting-point alloys, Sn, Fe, Al, Cu, steel and wax, and proposed a modified form of Wigg’s correlation.[75] In Lubanska’s correlation, important process parameters and material properties are related to the mass median droplet size by the following equation:

MMD

Eq. (26)

d0

	éæ		m&	L
	= kD êçç1	+
			m& G
	ëè

	ö		ν L ù1 / 2
	÷
		ú
	÷
	ø	ν G We û

where d0 is the melt delivery nozzle diameter, kD is a constant whose value was found to be 40–50 for a wide variety of spray data (air-Sn, Fe, Al, Cu, steel, low melting-point alloys, and wax), and vL and vG

280 Science and Engineering of Droplets

Table 4.15. Empirical Correlations for Droplet Sizes of Liquid Metals in Gas Atomization Jet Breakup

Correlations

Process Characteristics

References

& Remarks

éæ

ù1/ 2

Derived from gas-Sn

MMD

&

ö

ν L

(Fe, low melting point

= k D

êç1

+

mL

÷

ú

alloys) spray data with

d 0

ç

&

÷

ν G Weú

multiple gas jets from a

Lubanska

ê

mG

ø

ëè

û

free-fall, ring-like

[323]

ρ LUG2 d 0

atomizer: d0 =1/4–7/8";

We =

, kD

»40–50

σ

Applicable to Cu, Al,

steels, wax, oil

Derived from air-Sn (Pb,

ù m

Pb-Sn) spray data using

MMD

é

&

ö

ν

free-fall atomizer with

Rao &

æ

m

L

L

nozzle diameters of 1.3,

= k

ç1+

÷

Mehrotra

ú

D ê

&

÷

1.8, 2.4 mm, atomizing

d0

ç

ν G Weú

ê

mG

ø

angles of 15,25, 35,45°,

[331]

ëè

û

and different melt

superheat

Semi-empirical

correlation for gas

atomization with free-

fall atomizer; Gas jet

é

aσ

m& L ù

1/ 2

impacts melt stream

MMD =

vertically;

Tornberg

ê

ú

(1 – bd0

Fitted well with data for

[486]

ë ρLU G0

μ L ) m&G û

Fe-,Ni-, Co-, Cu-based

alloys; the values of a

and b should be

determined

experimentally.

MMD » 43 /(SGC)1/ 2

For gas-atomized Al, Fe,

Dunkley

Cu-Al and Sn-Sb

[487]

(a) Dpeak = D10 (0.136 ×10-5

Re+ 0.294 )

For air-atomized bronze

at (a) constant metal

nozzle diameter but

Dpeak = 0.382d0 ξ −1We−1.77

different annular gas jet

velocities,

Nichiporenko

(b)

182000<Re<238000,

We=7.95;

[488]

ρ GUG 0 d G0

(b) varying metal nozzle

Re =

We =

ρ LU L20 d 0

diameters but constant

,

annular gas jet velocity,

μ G

σ

4.4<We<7.95, Re =

206000

Schmitt

Semi-empirical

correlation for a ring-

[489]

like atomizer

Empirical and Analytical Correlations 281

Table 4.16. Analytical and Experimental Correlations for Metal Droplet Sizes in Gas Atomization via Film or Sheet Breakup

Correlations

Process Characteristics &

References

Remarks

Subsonic gas flow:

Derived from an analytical

model for flat, infinitely

D =

11.85εσ

thick liquid layer; Effects of

,

gas compressibility

χρGU s2

Bradley

included; Effects of

Sonic/supersonic gas flow:

gas/liquid ratio, liquid

[329,330]

viscosity, and nozzle

Rao

11.85εσχ m

D =

,

ε = 0.25

geometry not included; χ

&

ρGU s2

and χm can be determined

Mehrotra

ε = −0.12 + 0.57ϑ +

0.38d 0

from the universal curves

[331]

for tin

for metals in [329] for

ε = −0.14 + 0.66ϑ +

0.48d 0

subsonic gas flow and in

for lead

[330] for sonic/supersonic

gas flow

D ρ

G

U 2

æ ρ

G

U

G

μ

L

ö2 / 3

0.5

G = ç

÷

σ

ç

ρL

σ

÷

Derived from spray data of a

Miller

è

ø

& Giles

æ

ρ σ d

ö1/ 6

æ m&

ö

variety of metals with close-

´

f

[4]

ç

L

0

÷

ç

G

÷

coupled atomizers

ç

2

÷

ç

÷

è

μL

ø

è m& L

ø

Film breakup:

æ

ρ

L

ö0.5 U Lt f

MMD »

2.9ç

÷

Derived from air-water

ç

÷

U G

è

ρG ø

(water-glycerol mixtures)

For 15≤ ρ Lσ 2 /(μ L2 ρGU G2 ) ≤5000,

spray data using an internal

mixing air-blast atomizer

ρ

U

t

/ μ

with prefilming effect (a

60≤

L

L

f

L

≤2200

coaxial liquid injector

placed at the throat on

Sheet breakup:

centerline of a converging-

Hespel

æ

2

ö

0.3

diverging air nozzle) similar

ts

to Ünal nozzle;

[169]

Liquid

MMD =

ç

ρLU L

÷

et al.

4.3ç

2

÷

Re

0.22

,

injector without or with a

[328]

è

ρGU G ø

small deflecting plate at its

exit plane creates conditions

Re =

ρLU L ts

for Film or Sheet breakup;

Observed by high-speed

μ L

photography; Measured by

For 720≤ ρ Lσ 2 /(μ L2

ρGU G2 ) ≤5100,

video image analyzer;

Calibrated by holography;

.

5

ρ L ts σ / μ L2

.

5

Applicable to liquid metals

≤

,

due to dynamic similarity

1.1 10

≤3 10

. -4

≤

μ LU L / σ

. -2

9.1 10

≤1.2 10

282 Science and Engineering of Droplets

are the kinematic viscosity of liquid metal (melt) and atomization gas, respectively. The Weber number is defined as We = PLUG2d0/σ , where UG is the gas velocity at the impact of the gas jets with the liquid metal stream, that was derived (in Lubanska’s work) from curves for velocity decay in sonic jets. Lubanska’s correlation fitted well with the experimental data of Tamura and Takeda for copper,[490] Thompson’s data for aluminum,[491] and Clare and Radcliffe’s data for wax.[461] However, it failed to fit the experimental data of Rao and Mehrotra[331] for tin, lead, and lead-tin alloy using nozzles of different diameters and atomizing angles. Large discrepancies between the experiments and the predictions based on this correlation have also been reported for aluminum alloys,[492] suggesting the inadequacy of this correlation for the application to close-coupled atomizers with sonic-supersonic-sonic transitions. In addition, no experimental data and correlations are available for the size and size distribution of droplets generated with a linear atomizer which is composed of a long rectangular melt delivery nozzle and gas jet nozzle.[178]

Yule and Dunkley[5] made a detailed analysis of advantages and drawbacks of Lubanska’s correlation. By comparing the predictions of Lubanska’s and Wigg’s equations to measured data, they showed that Wigg’s equation[75] is in a reasonable fit to the data for MMD > 500 µm, and gives better fit than Lubanska’s equation for MMD < 500 µm. Thus, they recommended to reconsider Wigg’s equation:

0.5

0.1

0.1

0.2

−0.3

−1.0

æ

m& L

ö0.5

Eq. (27)

MMD = 20ν

L

m&

L

d

0

σ

ρ

G

U

R

ç1+

÷

ç

÷

è

m& G ø

because it has some theoretical reasoning behind it and appears to offer a better accuracy than Lubanska’s modification to it. It should be noted that in the above equation, the length scale h defined in

Empirical and Analytical Correlations 283

Wigg’s original equation is substituted by liquid nozzle diameter d0. It is uncertain, however, if one could use the liquid nozzle diameter to substitute this length scale, although the prediction of this equation is not sensitive to the choice. In addition, there is evidence that the dependency of droplet size on surface tension is more pronounced in reality than that indicated by this equation. Thus, Yule and Dunkley pointed out that there is scope for exploring the use of other correlations which contain higher powers of surface tension.

To make Lubanska’s correlation applicable to more general conditions, Rao and Mehrotra[331] proposed to substitute the square root by an exponent m which is not a constant, but a function of atomizing angle and independent of nozzle diameter (Table 4.17). In addition, in the modified Lubanska’s correlation, the parameter kD is also no longer a constant but varies with nozzle diameter and insensitive to atomizing angle (Table 4.17).

Table 4.17. Experimental Values of Parameters					kD and m* in
Modified Lubanska’s Correlation[331]
	Nozzle Diameter (cm) →		0.24	0.18		0.13
	Atomizing Angle (°)	↓
	15		6.31	8.19		11.72
			(0.301)	(0.303)		(0.300)
	25		6.26	8.23		11.67
			(0.282)	(0.283)		(0.282)
	35		6.29	8.32		11.60
			(0.265)	(0.267)		(0.261)

*Values of m are in parenthesis

On the basis of a force balance model, Tornberg[486] derived a semi-empirical correlation for free-fall type of atomizers:

	é		aσ	& ù1/ 2
						m
Eq. (28)	MMD = ê					L	ú
		0	(1 – bd0			&
	ë	ρ LU G		μL ) mG û

284 Science and Engineering of Droplets

where a and b are constants related to atomizer design, material system, and operation conditions, and UG0 is the atomization gas velocity at nozzle exit. This model encompassed a force applied to a melt to break up the liquid into droplets, and an internal friction force within the melt counteracting the atomization force. The atomization force included that created by the dynamic pressure on the melt stream, and the drag force created by gas surrounding the melt. The dynamic pressure was neglected in the model. The internal friction force is due to the liquid viscosity. The pressure from all these forces exerted on a droplet was then balanced by the surface tension force, leading to the above correlation. Although the physical meaning of this correlation is clear, it is apparently also system-dependent. Nevertheless, this correlation has been found to generate much better agreements with experimental data for free-fall type of atomizers than other available correlations.[493]

Dunkley[487] analyzed droplet size data for gas-atomized Al, Fe, Cu-Al and Sn-Sb, and developed an empirical correlation between the specific gas consumption, SGC, and mass median diameter of droplets for various atomization gases, pressures, and atomizers:

Eq. (29)

MMD ≈ 43 /(SGC)1/ 2

where SGC is in m3/kg, MMD is in µm, and the standard deviation of the correlation is within ±10 µm. This empirical correlation may also be employed as a useful rule of thumb for estimating gas requirements.

Some other correlations that have been specifically developed for liquid metals include those proposed by Nichiporenko,[488] Schmitt,[489] Thompson,[491] and Date et al.[494] Nichiporenko[488] correlated the diameter of particles (powder) of predominant fraction, Dpeak, with the arithmetic mean particle diameter (linear average diameter), D10, and the Reynolds number as well as the Weber

Empirical and Analytical Correlations 285

number and metal nozzle diameter for air-atomized bronze under different experimental conditions. The definitions of the Reynolds and Weber numbers are given in Table 4.15, where UG0 and UL0 are the gas and liquid metal velocities at the gas and metal nozzle exit, respectively, dG0 is the equivalent diameter of the annular gas nozzle, and ξ is the predominant fraction corresponding to the peak of particle size frequency distribution curve. Schmitt[489] derived a correlation for the maximum droplet diameter on the basis of dimensional analysis and assuming that the disintegration process encompasses forces due to inertia, surface tension, and viscosity of melt as well as inertia of atomization gas. The correlation was used to explain his data for the atomization of a Ni-base alloy (IN 100) with argon using a ring-like atomizer. Thompson[491] derived an empirical equation to correlate atomization gas pressure, liquid metal temperature, nozzle diameter, and metal head to atomization rate. Date et al.[494] also related atomization gas pressure to the percent yield of each mesh fraction with orthogonal polynomials based on a statistical analysis of atomization data.

Many atomization experiments of liquid metals with free-fall type of atomizers via jet breakup[318][319][323][331][495][496] revealed that droplet/particle size distributions follow closely the logarithmicnormal distribution pattern. As discussed above, the logarithmicnormal distribution can be fully described by two parameters, i.e., the mean of the distribution and the standard deviation. The values of both these parameters can be readily determined from the plot of experimental data (cumulative weight percentage vs. droplet size) on a log-probability graph paper. Thus, the entire droplet size distribution can be characterized by two parameters, i.e., the mass median diameter D0.5, and the geometric standard deviation of the distribution, σ LN, which satisfies a relationship:

Eq. (30) σ LN = D0.84 / D0.5 = D0.5 / D0.16 = (D0.84 / D0.16 )1/ 2

286 Science and Engineering of Droplets

according to the mathematical characteristics of the log-normal distribution. In a sieve test, the mass median diameter D0.5 corresponds to the opening of a screen mesh which lets through 50 wt.% of the atomized powder. Similarly, droplet sizes D0.16 and D0.84 correspond to the openings of the screen meshes which let through 16 wt.% and 84 wt.% of the atomized powder, respectively. In the log-probability graph, the 50 wt.% value of droplet diameter gives the mass median diameter. In addition, the standard deviation of the distribution can also be empirically correlated to the mass median diameter, as suggested by Lubanska:[323]

Eq. (31)

D0.5 = 13σ LN3

or

σ LN = 0.425D01./53

Thus, both the mean droplet size and the size distribution may be predicted using these correlations [Eqs. (26), (27), (28), or (29) and Eqs. (30), (31)] for given process parameters and material properties. For a given atomizer design, the standard deviation of droplet size distribution has been found to increase with the melt flow rate, but appears to be less sensitive to the gas flow rate.[5] Moreover, the variation of the standard deviation is very atomizerand melt-specific. An empirical correlation which fits with a wide range of atomization data has the following form:

Eq. (32)

σ LN = aD0b.5

where a and b are system-specific constants. The values of the standard deviation of droplet size distributions are given in Tables 2.3,2.4 and 2.5 for different atomization processes. In spray forming processes, the mass median droplet diameter is typically in the range of 50 to 150 µm, and the log-normal standard deviation is generally in the range of 1.7 to 2.3, depending on process parameters and materials involved.[497]

Empirical and Analytical Correlations 287

Similarly to the trends in the atomization of normal liquids, droplet properties such as mean droplet size, and size distribution for liquid metals/alloys depend on a large number of parameters. These include thermophysical properties of melts and atomization gas, and operation variables such as liquid temperature and flow rate, gas pressure and flow rate, and atomizer geometry parameters and configurations. For a given atomizer design and under identical operation conditions, different atomization gases and/or metal/alloy systems may produce droplets of different mass median sizes and size distributions due to the difference in thermophysical properties of the gases (Tables 2.6 and 2.7) and/or metals/alloys (Tables 2.8, 2.9, 2.10 and 2.11). Gas velocity and gas to metal mass flow rate ratio are two important factors governing the resultant droplet sizes. As evident from the correlations (Tables 4.15 and 4.16), the mean droplet/ particle size increases with increasing liquid viscosity, surface tension and/or gas density, and it decreases with increasing gas viscosity, liquid density, and/or gas velocity. The mean droplet/particle size reduces with increasing gas to liquid mass flow rate ratio and approaches an asymptotic value at large gas to liquid ratios. The dynamic force of atomization gas, in addition to the gas to liquid mass flow rate ratio, is also an important factor affecting the resultant metal droplet sizes.[498] As confirmed by many atomization experiments for melts,[80][82][322][323] the mean droplet/particle size generally increases with increasing liquid nozzle diameter for plain-jet airblast type of atomizers, i.e., free-fall type of atomizers with an annular gas jet or discrete gas jets. However, the experiments of Nukiyama and Tanasawa[79] showed no effect under their experimental conditions and nozzle dimension. For other types of nozzle designs, such as those used by Gretzinger and Marshall,[102] the mean droplet size decreases with an increase in liquid nozzle dimension. Therefore, published data and correlations should be used with caution, and attention should be paid to the operation conditions, nozzle configurations and nozzle dimensions used for deriving the correlations. Moreover, the atomizing angle, or generally speaking,

288 Science and Engineering of Droplets

atomizer geometry parameters and configurations also have an important impact on the resultant droplet sizes and size distributions. Increasing the atomizing angle may reduce the mean droplet size (Table 4.17).[331]

In addition to the effects similar to those in normal liquid atomization, the thermophysical properties of melts and atomization gases also influence the metal droplet size by affecting the heat transfer between the two phases. Putimtsev[499] studied the atomization of Fe, Ni, Cu, Ag and Fe-Ni alloys in air, nitrogen, argon, and helium, and observed that the metal droplets atomized by gases of low density, low thermal conductivity, and/or high viscosity remain in liquid state for longer time, thus increasing the probability of secondary disintegration and generation of finer droplets. Metals with high thermal conductivity and/or low thermal capacity cool faster, inhibiting droplet deformation and disintegration and leading to large droplets/particles. For example, Ag and Cu have lower viscosity and surface tension than Fe and Ni. However, much smaller yield of finer droplets was obtained for Ag and Cu due to their high thermal conductivities. For the similar reason, increasing liquid temperature or melt superheat can reduce the mean droplet size.[331] Moreover, gas chemistry critically affects the size distribution, shape and thermal history of atomized metal droplets, as demonstrated by Liu et al.[154][174] and Zeng et al.[173][500] Retention of ligament shape due to the formation of oxide skin,[319] or change in droplet size distribution due to oxygen blended in inert atomization gas[154][174] are some examples of the effects of gas chemistry.

In gas atomization via film or sheet breakup (Table 4.16), the mean droplet size is proportional to liquid density, liquid viscosity, liquid velocity, and film or sheet thickness, and inversely proportional to gas density and gas velocity, with different proportional power indices denoting the significance of each factor. In recent experimental studies on liquid sheet and film atomization processes using a close-coupled atomizer, Hespel et al.[328] concluded that the