Empirical and Analytical Correlations 297

significantly different from those estimated with the equation of Nichiporenko and Naida.[488] Yule and Dunkley[5] warned that the published calculations of spheroidization time should be viewed with caution, and the use of the simplified form of the above equation under the assumption Dend/Dinit≈10 is not generally recommended. They further pointed out the confusion in the concept of spheroidization time, and suggested that the mechanisms for primary and secondary breakup should be considered separately in terms of the deformation and spheroidization times.

4.4.0CORRELATIONS FOR DROPLET DEFORMATION CHARACTERISTICS ON A SURFACE

One of the important deformation characteristics of droplets impinging on a surface is the degree of flattening, or flattening ratio, i.e., the ratio of final splat diameter (maximum spread diameter) Ds to the initial impact droplet diameter D0. This is also termed maximum spread factor, mainly used in characterizing normal-liquid droplet deformation.[411][507] Some analytical models[157][401][508]- [512] have been developed to describe the impact deformation, heat transfer, and/or phase change (solidification or evaporation) of droplets, assuming normal impact on a smooth surface. Although these models can not describe the detailed physical phenomena associated with deformation process, they are relatively simple in general and can correlate important dimensionless deformation parameters to initial impact parameters.

For the deformation of droplets of normal liquids at low impact velocities on a horizontal plane surface without phase change, Tan et al.[513] developed a physical-mathematical model with a droplet falling from a certain height under the influence of gravity. They derived quantitative relations for the dimensions of the deformed droplet, including the effects of initial droplet diameter, height of fall, and thermophysical properties of liquid. In this model, the behavior of droplet deformation was assumed to be governed by

298 Science and Engineering of Droplets

the gravitational force, surface tension force, viscous friction force, and interaction with the solid surface. Using a high-speed cine camera, they observed that when a droplet fell onto a horizontal surface from different heights, it flattened out at the initial instant of contact with the surface, reached its maximum size, and then turned to contract until an equilibrium state was reached. The extent of the contraction depends primarily on the height of fall and the surface condition at impact. For a given liquid such as water with D0 = 6 mm at room temperature, the maximum size of the deformed droplet, Ds , was expressed as a function of the height of fall, H, and the initial droplet size, D0:

		ìD			for H £ 0 and D < 4 mm
Eq. (41)	Ds =	ï 0				0
		í3.18D0 -0.009			for H £ 0 and D0 ³ 4 mm
		ï	b		for H > 0
		îexp(2.3aB )H		B
with
	ì158.75D0 - 2.33;				bB = 0.17	for 0 < H £ 2 mm
	aB = í		-1.82;		bB = 0.36	for 2 £ H £10 mm
	î158.75D0

where aB and bB are coefficients related to initial droplet diameter. It should be noted that this model predicts that when the droplet diameter decreases to below a certain value, the droplet would not deform, but instead preserve its spherical shape on a surface. Under the conditions of creep flows at very low or zero Reynolds numbers, the interaction with substrate surface, against the surface tension of liquid, may become the driving force for droplet spreading.[342] Under such conditions, the above model would be no longer valid. To determine the relative importance of gravitational force to surface tension force, one may use the Bond number, Bo = ρ LD20 g /σ , which is a ratio of the two forces.[514]

Empirical and Analytical Correlations 299

For the deformation of droplets of normal liquids on a heated, horizontal, flat surface with attendant phase change, i.e., evaporation, Chandra and Avedisian[411] derived the following equation based on a simplified model:

	9/2(D / D )4			3[(1− cosθ	LS	)(D / D )2	− 4]
Eq. (42)	s	0	+			s 0		≈ 1
	Re				We

with

θ LS ≈ arctan[kL (Tw −Tb )υL /( He A )]

where θ LS is the apparent advancing liquid-solid contact angle at the contact line, Tw is the substrate surface temperature, Tb is the boiling point of the liquid, υ L is the kinematic viscosity of the liquid, and A is the Hamaker constant. It was suggested that the contact angle θ LS be correlated to substrate surface temperature Tw to account for the effect of the surface temperature on droplet deformation. The influence of surface condition is carried entirely in the apparent contact angle.[385] It was found that this model predicts a maximum spread factor that is 20–40% higher than some experimentally measured data.[380]

Madejski’s model and correlations[401] are perhaps the most frequently quoted ones in literature related to surface impact and solidification of melt droplets. In his full model, Madejski considered the effects of viscosity, surface tension, and solidification on droplet flattening, but ignored the compressibility of liquid metal. The flattening ratio was formulated as a function of Re, We and Pe to quantify the viscosity dissipation of inertial force of an impact droplet, conversion of impact kinetic energy to surface energy, and solidification processes, respectively. The relative importance of these three processes can be then determined according to the relative magnitude of the three dimensionless numbers. Madejski’s full model consists of a complex integral-differential equation that has

300 Science and Engineering of Droplets

no analytical solution, and thus a general formula for the flattening ratio is difficult to find. However, the numerical fits made by Madejski resulted in some relatively simple correlations. In the general case of droplet spreading without solidification, the following equation was proposed for the calculation of the degree of flattening for We > 100 and Re > 100:

	3(Ds / D0 ) 2		1	æ Ds / D0		ö	5
Eq. (43)		+		ç		÷	= 1
	We				1.2941
			Re è			ø

Several other special cases have also been analyzed numerically, leading to simplified formulations for the flattening ratio, as detailed below in each model group.

Fukanuma and Ohmori[510] also presented an analytical model for the flattening ratio of a molten metal droplet on a surface as a function of time and compared the model prediction to experimental data for molten tin and zinc droplets. An expression for the flattening ratio was derived based on some simplified assumptions and approximations:

	6(Ds / D0 )2		1	æ Ds / D0		ö	6
Eq. (44)		+		ç		÷	=1
	We				1.06
			Re è			ø

Bennett and Poulikakos[380] made a detailed review of the analytical models for droplet flattening ratio. They pointed out that almost all treatments of surface tension effects in the analytical models were fundamentally the same, with differences arising from varying degrees of simplifying assumptions. The splat energy dissipated through surface tension effects was shown to be proportional to the spread factor squared plus the initial droplet surface tension and contact angle effects as considered in some models. The treatments of viscous energy dissipation, however, were dissimilar. The various models yielded viscous energy dissipation as proportional to the spread factor raised to the fourth, fifth, sixth, or eighth power. Madejski’s model provides the most theoretically

Empirical and Analytical Correlations 301

sound treatment of viscous energy dissipation and has the most published experimental support. However, his model effectively ignored the initial surface tension energy of a droplet prior to impact and the contact angle effect during spreading. Bennett and Poulikakos refined Madejski’s treatment of the surface tension effects using the treatment proposed by Collings et al.,[514] and Chandra and Avedisian,[411] and obtained the following general equation:

	3[(1– cosθ )(Ds / D0 )2 - 4]			1	æ Ds / D0		ö5
Eq. (45)			+		ç		÷	=1
	We					1.2941
				Re è			ø

where θ is the equilibrium liquid-solid contact angle. It was indicated that the dynamic effect of the contact angle on the spread factor is substantial and this effect increases with decreasing Weber number. For large contact angles, i.e., a droplet does not wet the surface, the influence of the Weber number on the spread factor becomes significant.

Recently, Pasandideh-Fard et al.[366] developed a simple analytical model to predict the maximum spread factor on a flat solid surface:

		æ	We +12		ö1/ 2
Eq. (46)	D / D = ç					÷
				1/2
	s 0	ç	3(1 - cosθ a ) + 4(We/Re		÷
		è			) ø

where θ a is the advancing contact angle. The advancing contact angle increases linearly with contact line velocity until an upper limit value is reached. Beyond that, the contact angle is independent of further increases in the velocity. A comparison of the model predictions with experimental measurements for a variety of drop- let-surface combinations and over a wide range of Weber number (26 < We < 641) and Reynolds number (213 < Re < 35339) showed that the discrepancy is less than 15% in most cases and is largest at low Re. For We >> Re1/2, the capillary effect becomes negligible so that Ds /D0 = 0.5Re0.25.

302 Science and Engineering of Droplets

In a subsequent study, Pasandideh-Fard et al.[367] derived a simple analytical model to estimate the maximum spread factor for a droplet with simultaneous deformation and solidification on a flat surface:

Eq. (47)

				æ	We +12			ö1/ 2
D	s	/ D	0	= ç				÷
						1/2	1/ 2
				ç	3(1 - cosθ ) + 4(We/Re			÷
				è			) + We[3St /(4Pe)]	ø

where the Stefan number and the Peclet number are defined as

St = cpL(Tm - Tw)/ Hm and Pe = u0D0/aL, respectively. In these equations, Tw is the initial substrate temperature and aL is the thermal

diffusivity of liquid droplet. In the above model, the following simplifying assumptions have been made: (a) The heat transfer is by one-dimensional heat conduction; (b) The substrate is isothermal; (c) The thermal contact resistance is negligible; (d) The Stefan number is small; and (e) The dimensionless deformation time is assumed to be 2.67. The magnitude of the term We[3St/(4Pe)]1/2 in the above model determines whether solidification influences droplet spreading. The kinetic energy loss due to solidification will be too small to affect the extent of droplet deformation if (St/Pr)0.5 << 1, where Pr = Pe/Re = vLaL. Discrepancies between the model predictions and experiments may arise due to the assumptions made in deriving the model, i.e., the negligible thermal contact resistance and the isothermal substrate. Analysis of splat cooling revealed that the effect of thermal contact resistance is negligible only if Bi = hD0/kL > 30. Thus, this model gives an upper bound on the thickness of solidified layer and the actual magnitude may be significantly lower. Hence, the model underestimates the maximum spread factor.

Overall, the available analytical models may be conveniently divided into four groups in terms of basic assumptions made and physical processes considered:

(1)Viscous Dissipation Domain: The decay of kinetic energy of an impacting droplet is due to viscous dissipation during flattening.

Empirical and Analytical Correlations 303

(2)Surface Tension Domain: The kinetic energy of an impacting droplet is converted to surface energy during spreading.

(3)Solidification Domain: Cooling and solidification of an impacting droplet occur simultaneously during flattening.

(4)Partial Solidification prior to Impact: Partial solidification of an impacting droplet prior to impact is considered.

For droplet spreading processes without attendant phase change, two primary domains have been identified, i.e., viscous dissipation domain and surface tension domain, which are discriminated by the principal mechanism responsible for arresting droplet spreading. In the viscous dissipation domain, the degree of flattening of a droplet is dominated by viscous energy dissipation, whereas in the surface tension domain, it is dominated by surface tension effects. According to Bennett and Poulikakos,[380] the condition under which surface tension effects dominate the termination of droplet spreading may be expressed as:

Eq. (48)

We < 2.80 Re0.457

If the dominating domain is selected correctly, the error induced by the simplification will be no more than about 20%. However, even well into the viscous dissipation domain, the effects of the surface tension are still significant, while in the surface tension domain, the effects of viscous dissipation disappear far more rapidly as one moves away from the borderline. In other words, the viscous energy dissipation contribution to the spread factor rapidly declines within the surface tension-dominated domain, while significant residual surface tension effects extend well into the viscous energy dissipation domain.

The correlations derived from the analytical models, numerical modeling, and experimental results are listed in Table 4.21. The dimensionless numbers used to describe the droplet deformation

304 Science and Engineering of Droplets

processes are summarized in Table 4.22. In these tables, TLeid is the Leidenfrost temperature, ρ S is the density of solidified material, U is the freezing constant, us is the speed of sound in liquid, aS is the thermal diffusivity of solidified material, T is the difference between droplet temperature and melting/liquidus temperature for solidification, or substrate surface temperature and vapor temperature for evaporation, T· is the splat cooling rate, and He is the latent heat of evaporation.

Table 4.21a. Correlations for Final Splat Diameter Created by a Droplet Impinging on a Solid Surface

			Correlations		Models	References
Group 1:
Decay of Kinetic Energy via Viscous Dissipation					Analytical	Jones[508]
D / D = 1.16 Re0.125
s	0
Ds / D0 = 1.2941(Re+ 0.9517)0.2 , for We-1=0					Analytical-	Madejski
Ds / D0 = 1.2941Re0.2 ,				for We-1=0, Re>100	Numerical	[401]
D / D = 1.0 Re0.2					Numerical	Trapaga &
s	0					Szekely [515]
Ds / D0 = 0.83Re0.2					Numerical	Hamatani
						et al. [516]
D / D = 0.82Re0.2					Numerical	Watanabe
s	0					et al. [517]
Ds / D0 = 1.04 Re0.2					Numerical	Liu et al. [18]
						Fukanuma
Ds / D0 = 1.06Re		0.17	,	for We-1=0	Analytical	& Ohmori
						[510]
Ds / D0 = 0.925 Re					Numerical	Bertagnolli
						et al. [518]
D / D = 0.5Re0.25					Analytical	Pasandideh-
s	0					Fard et al.
						[366]

Empirical and Analytical Correlations 305

Table 4.21b. Correlations for Final Splat Diameter Created by a Droplet Impinging on a Solid Surface

			Correlations						Models	References
Group 2:
Conversion of Kinetic Energy to Surface Energy									Analytical-	Madejski
									Numerical	[401]
Ds / D0	= (We / 3)0.5		-1	=0, We>100
		,	for Re
										Dykhuizen
									Analytical	[390]
We = 3(Ds / D0 )2 + 8 /(Ds / D0 ) − 11 ,
						no limit on We
Ds / D0	= 0.816We0.25								Analytical	Cheng
										[509]
Ds / D0	= (We/6 + 2)0.5								Analytical	Ueda et al.
										[519]
Ds / D0	= [(We / 3) /(1 − cosθLS )]0.5				for large Re					Collings
Ds / D0	= (We/6)	0.5		,					Analytical
										et al. [514]
		, for large Re and ? =180º
Ds / D0 ≈ [(We/3 + 4) /(1 − cosθ LS )]0.5							LS
					,		for Re-1=0			Chandra &
Ds / D0	≈ (We/3 + 4)		, for Re-1=0, T				>T	(θ =180º)	Analytical	Avedisian
						w				[411]
D / D = 1 + 0.463We0.345							Leid	LS
									Experi-	Senda
s 0			, on hot surface
									mental	et al. [422]

Table 4.21c. Correlations for Final Splat Diameter Created by a Droplet Impinging on a Solid Surface

Correlations

Models

References

Group 3:

Simultaneous Cooling and

Solidification

Analytical-

Madejski

Numerical

[401]

Ds / D0 = 1.4991( ρL / ρS )0.395(Pe/U 2 )0.1975

, for Re-1=We-1=0

Ds / D0 =

0.82(Pe /U

2

)

0.25

Analytical

Dykhuizen

[390]

ì0.8546 χ 0.5

Re0.25[1 + 0.34 βRe0.5 ln(0.3 Re)]

smooth

Sobolev

ï

0.5

0.25

0.5

0.5

Ds / D0 =

ï

Re

+

Analytical

et al.

í0.8546 χ

Re

[1 - 0.06ωα

ï

0.34 βRe0.5 ln(0.3 Re)]

rough

[511]

ï

î

æ

We +12

ö0.5

Pasandideh-

D

/ D = ç

÷

s

0

ç

3(1 - cos θ ) + 4(We/Re

0.5

0.5

÷

Analytical

Fard et al.

è

) + We[3St /( 4Pe)]

ø

[367]

Group 4:

Analytical-

San Marchi

Partial Solidification prior to Impact

Numerical

et al. [157]

306 Science and Engineering of Droplets

Table 4.22a. Dimensionless Numbers Used to Describe Droplet Deformation Processes with Simultaneous Heat Transfer and Phase Change (Solidification or Evaporation)

	Applications	Numbers	Definitions	Remarks	References
		Reynolds	Re = ρ L u0 D0 / μ L	Compare inertial force to	Madejski
		Number		viscous force	[401]
		Weber	2	Compare inertial force to	Madejski
		Number	We = ρ Lu 0 D0 / σ	surface tension force	[401]
	Deformation	Froude	Fr = u0 /( D0 g)0.5	Compare inertial force to	Tsurutani
	& Spreading	Number		gravitational force	et al. [334]
		Mach	Ma = u0 / us	Compare impact velocity	Hofmeister
		Number		to speed of sound in liquid	et al. [410]
		Bond	Bo = ρLD02 g /σ	Compare gravitational	Collings
		Number		force to surface tension	et al. [514]
				force
		Capillary	Ca = We/Re = u0μ L / σ	Compare inertial-viscous	Yarin &
		Number		force to surface tension	Weiss
	Splashing			force	[357]
		Ohnesorge	Oh = μL /( ρLσD0 )0.5	Compare internal viscosity	Walzel
		Number		force to surface tension	[398]
				force

Table 4.22b. Dimensionless Numbers Used to Describe Droplet Deformation Processes with Simultaneous Heat Transfer and Phase Change (Solidification or Evaporation)

	Applications	Numbers			Definitions				Remarks	References
		Peclet							Compare heat
				Pe = u 0 D0 / a S					removal by flow to	Madejski
		Number							heat extraction by	[401]
	Heat								conduction
	Transfer	Impact	I =		πρ L D02 u03				Compare kinetic	Matson
									energy transfer
										et al.
		Number			12 Ac h(TL – Tw )				rate to heat	[409]
									extraction rate
		Stefan		St = c pL T / H m					Compare sensible	Matson
									to latent heat in
										et al.
		Number							liquid-solid phase
										[409]
			F = [ Hm (1 − f s ) + c pL					T ]	change
	Solidification	Freezing							Compare	Matson
					πu0 ρ L D02
									solidification time
		Number								et al.[409]
				6 Ac h(TL – Tw )					to impact time
		Nucleation	N = 0.5 Hmu0 /(D0c pLT&)						Compare	Dykhuizen
									nucleation time to
		Number								[390]
									impact time
		Jakob		Ja = cpL T / H e					Compare sensible	Tio &
	Evaporation								to latent heat in	Sadhal
		Number							liquid-vapor phase
										[414]
									change