Empirical and Analytical Correlations 295
Eq. (36)	N teeth = 0.215(ρ Lω 2 d 3 /σ )0.417 ( ρ Lσd / μ L2 )0.167

In both the Direct Droplet and Ligament regimes, the mean droplet size is inversely proportional to the rotational speed ω and the square root of the electrode or disk diameter d:[5]

Eq. (37)

MMD = Kω –1d –0.5

This approximate relationship is similar to those for centrifugal atomization of normal liquids in both Direct Droplet and Ligament regimes. However, it is uncertain how accurately the model for K developed for normal liquid atomization could be applied to the estimation of droplet sizes of liquid metals.[5] Tornberg[486] derived a semi-empirical correlation for rotating disk atomization or REP of liquid metals with the proportionality between the mean droplet size, rotational speed, and electrode or disk diameter similar to the above equation. Tornberg also presented the values of the constants in the correlation for some given operation conditions and material properties.

From these correlations, it is clear that the mean droplet size may be controlled by changing rotational speed, and/or disk diameter for a given liquid metal. In addition, the droplet size increases with increasing liquid viscosity and/or surface tension, and/or decreasing liquid density. However, the effect of liquid viscosity is inconclusive. A large disk diameter and/or a high rotational speed are required to obtain a high metal velocity and fine droplets. The technical limitations to spin a large disk at a high rotational speed along with the additional effect of viscosity make it very difficult to produce fine powders via centrifugal atomization.[486]

4.3.4Solidification and Spheroidization

In atomization of melts, the final droplet size also depends on the relative magnitude of the time tb for a droplet to undergo deformation prior to secondary breakup, and the time tsol required for

296 Science and Engineering of Droplets

cooling a superheated metal droplet to its melting temperature and for droplet solidification to complete. The latter time can be estimated using the following expression[319] for a small Biot number and negligible radiative heat transfer:

	Eq. (38)	tsol	=	Dρ L
				6hc

é

æ T

D

- T

ö

DH

m

ù

ê

c

lnç

G

÷

+

ú

- T

T - T

pL

ç T

m

÷

ë

è

G ø

m

G û

where TD is the initial droplet temperature and hc is the heat transfer coefficient. Ranz-Marshall correlation[505] has been frequently used to determine the heat transfer coefficient:

Eq. (39)	hc	=	kG	(2 + 0.6 Re0.5 Pr 0.33 )
			D

where Pr is the Prandtl number. If the breakup time is less than the total solidification time, secondary breakup may occur.

The final droplet/particle shape is determined by the time required for a deformed droplet to convert to spherical shape under surface tension force. If a droplet solidifies before the surface tension force contracts it into a sphere, the final droplet shape will be irregular. Nichiporenko and Naida[488] proposed the following dimensionally correct expression for the estimation of the spheroidization time, tsph:

3 π 2

μL æ

1 ö4

4

4

Eq. (40)

tsph

=

ç

÷

(Dend

- Dinit )

4 VDσ

2

è

ø

where VD is the droplet volume, Dinit is the smallest dimension of the initial, non-spherical droplet, and Dend is the final droplet diameter. A more detailed and general model for the estimation of the spheroidization of a droplet has been proposed by Rao and Tallmadge.[506] The model-predicted spheroidization times are not