Empirical and Analytical Correlations 289

control over the liquid layer thickness is of major importance to the control of the atomization quality.

4.3.2Water Atomization

A limited number of empirical correlations have been developed for metal droplet sizes generated by water atomization, as listed in Table 4.18. In these correlations,c is a system-specific constant,ϑ is the atomizing angle, i.e., angle between water nozzle axis and metal delivery nozzle axis, k is a proportional constant specific to atomizer type, melt type and melt temperature, n is a parameter depending on atomizer type, PW is the water pressure, UW is the water velocity, and m· W is the mass flow rate of water.

Some quantitative studies[498][501] on droplet size distribution in water atomization of melts showed that the mean droplet size increases with metal flow rate and reduces with water flow rate, water velocity, or water pressure. From detailed experimental studies on the water atomization of steel, Grandzol and Tallmadge[501] observed that water velocity is a fundamental variable influencing the mean droplet size, and further, it is the velocity component normal to the molten metal stream UW sin ϑ , rather than parallel to the metal stream, that governs the mean droplet size. This may be attributed to the hypothesis that water atomization is an impact and shattering process, while gas atomization is predominantly an aerodynamic shear process.[5]

In the empirical correlation proposed by Kato et al.,[503] the mean droplet size is inversely proportional to the water pressure, with a power index of ~0.5 for conical shaped annular-jet atomizers, and 0.7–1.0 for V-shaped flat-jet atomizers. This suggests a lower efficiency of the annular-jet atomizers in terms of spray fineness at high water pressures. The data of Kato et al.[503] were obtained for water pressures lower than 10 MPa. Seki et al.[502] observed the similar trend in the water atomization of nickel and various steels at higher water pressures (>10 MPa). Since k is dependent on both

290 Science and Engineering of Droplets

atomizer type and melt properties, no satisfactory general correlation has been developed and it must be determined by experimental measurements. Qualitatively speaking, the value of k is larger for Cu, Fe, and other high surface tension metals which are difficult to atomize.[5] The values of k and n proposed by Seki et al.[502] are given in Table 4.18.

Table 4.18. Empirical Correlations for Mean Droplet Size of Liquid Metals in Water Atomization via Jet Breakup

Correlations

Remarks

References

MMD = c /(U W sin ϑ)

Derived from

Grandzol &

m&

ϑ

spray data for

Tallmadge

c = 2750,

= 15°–30°

water-4620 steel

[501]

W

= 0.9–3.8 kg/s,

MMD = k P − n

;

MMD in µm,

P

in MPa

Derived from

Kato et al.

W

W

data for PW

[503]

For annular jet: n ≈ 0.5

[5]

; n = 0.58, k = 114

[502]

Seki et al.

For V-jets: n = 0.7–1.0 [5]; n = 0.56, k = 68 [502]

less than 10 MPa

[502]

æ d

0

ρ U

ö–0.57

æ d

0

ρ U

2 ö−0.22

MMD = kd

0

ç

L W

÷

ç

L

W

÷

ç

μL

÷

ç

σ

÷

è

ø

è

ø

For two-jet

Kishidaka

æ

m&

ö

0.043

water atomizers

L

[504]

´ ç

÷

ç m&

÷

è

W

ø

In the empirical correlation proposed by Kishidaka[504] for two-jet atomizers, melt nozzle diameter and physical properties, water velocity, and water to melt ratio are included. The constant k is again a function of atomizer geometry. The water velocity may be estimated with the following equation assuming loss-free water flow in the water nozzle(s):

Eq. (33)	U	W	≈ (2 P / ρ	W	)1 / 2
			W

Empirical and Analytical Correlations 291

Further studies are required to develop more comprehensive and general correlations for water-atomized metal droplets/particles.

Most water-atomized metal particles (powders) have been observed to follow the log-normal size distribution pattern. Relatively narrow size distributions of both fine and coarse particles may be generated by water atomization. A review of published data for droplet size distributions generated by gas and water atomization of a variety of liquid metals and alloys has been made by Lawley,[4] along with presentations of micrographs of surface morphology and internal microstructure of solidified particles.

4.3.3Centrifugal Atomization

Correlations have been developed for metal droplet sizes generated by centrifugal atomization, as listed in Tables 4.19 and 4.20. Similarly to normal liquids, centrifugal atomization of melts may occur in three regimes: (1) Direct Droplet Formation, (2) Ligament Disintegration, and (3) Film/Sheet Disintegration, as depicted in Fig. 4.3.[320] Champagne and Angers[320] studied the atomization of Al, Cu, Fe, Zn, and steel, and proposed a number group X:

Eq. (34)	X = X n / X d
with
Eq. (35)	&	ω	0.6	d	−0.68	,	X d = σ	0.88	−0.17	−0.71
	X n = VL								μ L	ρ L

This number group may be used to determine in which regime a centrifugal atomization takes place for given operation conditions and material properties. Since the group is not dimensionless and the plot was made using SI units, it should be used with caution. From Fig. 4.3,[320] it can be seen that for a given liquid metal/alloy at a

292 Science and Engineering of Droplets

given temperature (i.e., given material properties), a centrifugal atomization process may switch from Direct Droplet regime to Ligament regime, up to Film regime with increasing liquid flow rate, rotational speed, and/or decreasing electrode or disk diameter. For given operation conditions (liquid flow rate, rotational speed, and electrode or disk diameter), the reverse transition may occur, i.e., a centrifugal atomization process may switch from Film regime to Ligament regime, down to Direct Droplet regime with increasing surface tension and/or decreasing liquid viscosity and density. The transition from Direct Droplet regime to Ligament regime, or Ligament regime to Film/Sheet regime occurs at X = 0.07 or X = 1.33,[320] respectively.

Table 4.19. Correlations for Mean Droplet Sizes of Liquid Metals in Centrifugal Atomization

Process

Correlations

Characteristics &

References

Remarks

MMD =

K

æ 1

ö1 / 2

For atomization in

Yule &

ç

÷

both Direct Droplet

Dunkley

ω

è d

ø

and Ligament regimes

[5]

1

æ

3σ

ö1/ 2

Theoretical particle

Schmitt

MMD =

ç

÷

[489]

ç

÷

diameter

πωrpm è dρ L ø

Semi-empirical

correlation for rotating

æ

aσ

ö1 / 2

disk atomization or

REP; Fitted well with

MMD = ç

÷

data of Merl76 alloy

Tornberg

ρ

ω 2 d/2[1- (ωd/2) b μ

ç

L

L

] ÷

with a = 4.8, b = 0.93,

[486]

è

ø

d = 90 mm,

5000 ≤ ω ≤24000 rpm

Empirical and Analytical Correlations 293

Table 4.20. Correlations for Droplet Sizes of Liquid Metals in Rotating Electrode Atomization (REP)

Process

Correlations

Characteristics &

Refs.

Remarks

6

ω

−1.02

−

0.48

æ

σ

ö0.53

&−0.04

D10 = 1.77 ´10

ç

÷

d

ç

÷

VL

ρL

è

ø

Derived from data

−1.03

−0.51

æ

σ

ö

0.53

&− 0.03

for REP of Armco

6

iron, SAE 1090

ç

÷

D20

= 2.17 ´10

ω

d

ç

ρL

÷

VL

steel, Cu, Al and

è

ø

æ

σ

ö0.52

Zn.

6

ω

−1.03

−0.53

&−0.01

D30 = 2.50 ´10

ç

÷

d

ç

ρL

÷

VL

&

è

ø

V

are the

ω, d and L

Champagne

6

−1.05

−

0.55

æ

σ

ö0.53

&−0.01

angular velocity

D21 = 2.65 ´10

ω

ç

÷

&

d

ç

÷

VL

(rad/s), diameter

ρL

Angers

è

ø

(m) and melting

æ

σ

ö

0.52

rate (m3/s) of anode,

[320]

6

ω

−1.03

−0.56

& 0

respectively.

D

31

= 2.96 ´10

d

ç

÷

V

L

ç

÷

è

ρ L ø

Droplet sizes are in

σ

ö0.50

6

−1.03

−

0.58

æ

µm.

= 3.3´10

ω

ç

÷

& 0.02

D32

d

ç

÷

VL

è

ρ L ø

No effect of

σ

ö0.46

6

−1.0

−0.58

æ

viscosity

D43 = 3.65 ´10

ω

ç

÷

&0.06

d

ç

÷

VL

ρL

è

ø

æ

ö0.47

Dmin and Dmax

6

− 0.95

σ

correspond to

Dmin = 0.74 ´10

ω

− 0.44 ç

÷

&0.01

d

ç

÷

VL

values above and

Champagne

è

ρL ø

below which 99%

&

of the particles of a

0.38

Angers

æ

σ

given distribution

6

−0.85

−0.43

ö

&0.1

[320]

ç

÷

Dmax = 2.91´10

ω

d

ç

÷

VL

are found on a

è

ρL ø

weight basis. No

effect of viscosity

σ

0.48

0.02

& 0.03

Effect of viscosity

Champagne

D32 = 3.34 ´10 6

μ L

VL

considered within

&

ρ L 0.5ω 1.02 d 0.58

limited range of

Angers

viscosity values

[320]