Empirical and Analytical Correlations 313

Bennett and Poulikakos[380] also emphasized the important effect of droplet undercooling on solidification, although they indicated that solidification does not contribute significantly to terminating droplet spreading and the effect of solidification on arresting the spreading of a superheated droplet is likely to be secondary compared to the effects of viscous dissipation and surface tension. They further indicated that consideration of the undercooling encountered in rapid solidification dictates that the reduction in freezing time due to higher solidification speed is offset by the delay in nucleation, although rapid solidification that is typical of splat-quenching can produce much faster crystal growth kinetics than predicted by the Stefan solution.

Sobolev et al.[511] conducted a series of analytical studies on droplet flattening, and solidification on a surface in thermal spray processes, and recently extended the analytical formulas for the flattening of homogeneous (single-phase) droplets to composite powder particles. Under the condition Re >> 1, the flattening ratios on smooth and rough surfaces are formulated as:

Eq. (57)

ì0.8546 χ 0.5		Re0.25	[1 + 0.34 β Re0.5 ln(0.3 Re)]			smooth
ï	χ 0.5	Re0.25	[1 - 0.06ωα 0.5 Re 0.5
Ds / D0 = í0.8546
ï			+ 0.34 β Re	0.5	ln(0.3Re)]	rough
ï
î

where χ is the ratio of the remaining droplet mass after mass loss due to splashing to the initial droplet mass, α =ε /(D0/2), ε is the roughness height, β = Vs/u0, Vs is the solidification speed of the splat lower part due to the heat removal through the substrate, ω = Vs(D0/2)/(u0δ ), and δ is the thickness of the splat lower part.

4.4.4Partial Solidification Prior to Impact

Madejski’s solidification model did not account for partial solidification of a droplet prior to impact. San Marchi et al.[157]

314 Science and Engineering of Droplets

modified some of the assumptions in Madejski’s model and addressed the effects of different solid fractions of a droplet prior to impact on its flattening and solidification behavior. The modeling results showed that under the conditions typical of thermal spray processes, the impact kinetic energy of a droplet governs its spreading process, and increasing the solid fraction (or reducing the amount of liquid) reduces the extent of the droplet spreading. Partial solidification of a droplet prior to impact reduces the kinetic energy of the remaining liquid through reducing the volume or mass of the liquid. Thus, with increasing solid fraction, the kinetic energy of the liquid decreases, leading to a decrease in the spreading extent. The results also showed that the effect of the solid fraction on the decrease in the spreading extent is essentially independent of material systems considered. This further demonstrated that the partial solidification of a droplet prior to impact influences its spreading behavior primarily through reducing the liquid volume or mass. Overall, the partial solidification prior to impact does not affect the droplet deformation and solidification on substrate as much as might be expected. A 10% solid at impact results in a reduction in splat size of less than 4%. This is mainly because of the predominant effect of the impact kinetic energy on the flattening and solidification behavior. In addition, the solid fractions considered in this study were less than 0.4, limiting the generalization of the results. The effect of the solid fraction on the final splat morphology has been experimentally investigated and discussed in Ref. 409.

Further extensions of Madejski’s model[401] may include (a) turbulence effect, (b) Rayleigh instability or Taylor instability and droplet breakup, (c) vibrational energy, and (d) influence of solidification on flow.[514] Some issues related to the deformation and solidification of droplets on a flat substrate in splat quenching have been addressed in Refs. 380 and 514. To date, analytical models addressing droplet impingement on a semi-solid surface have not been found in available literature.

5

Theoretical Calculations

and Numerical Modeling

of Droplet Processes

Droplet generation and deformation processes involve complex physical phenomena, such as liquid-gas, or liquid-surface interactions, primary and/or secondary breakup of liquid, droplet dynamics, and in many applications, heat transfer and phase change. To date, no general theoretical treatment of droplet processes is available, except for few simple processes under restricted conditions. Therefore, numerical modeling and simulation have been increasingly employed for analysis of droplet processes and optimization of process designs. Computer modeling is usually preferred over experiments for several reasons. For example, computer modeling typically can be carried out more quickly and with less lead-time and expense than experiments. In addition, it can be much better controlled and used to explore a much wider range of conditions and systems, some of which are physically inaccessible.

In this chapter, basic theoretical calculations and numerical modeling of droplet generation and deformation processes of both normal liquids and melts will be discussed in detail. The review of modeling efforts will outline the current status and recent developments

315

316 Science and Engineering of Droplets

in numerical models and computational methods for the droplet processes. The information will also be useful for understanding various mechanisms governing droplet processes as well as effects of process parameters on droplet properties.

5.1.0ENERGY REQUIREMENTS AND EFFICIENCY

To generate droplets from a bulk liquid, a certain amount of energy is required to make some area of the liquid surface unstable. The surface then may rupture into fragments, which subsequently disintegrate into droplets. The energy requirement is specific to the technique used. For example, energy is needed to compress the gas used in gas atomization, and to melt and superheat a solid material in atomization of melts. In powder production process via atomization, the energy used for melting and superheating is wasted, because it is removed from droplets during flight in spray chamber through heat transfer to surrounding gas, and no heat energy recovery is usually made.

Theoretically, the energy, E, required to generate droplets from a liquid or melt can be roughly estimated by that needed to create the surface area of droplets, i.e., the product of the surface area of droplets and the surface tension of the liquid:

	i =N
Eq. (1)	E = σ åπ Di2
	i=1

where i represents the ith droplet, N is the total number of droplets generated from the liquid of mass mL in unit time, and Di is the diameter of the ith droplet. The mass of the liquid atomized in unit time can be formulated as:

				π	i =N
	Eq. (2)	mL	= ρL		åDi3
				6
					i=1

Theoretical Calculations and Numerical Modeling 317

In the above estimation of the energy requirement for atomization, the energy needed to overcome viscous force during liquid breakup is neglected. Under such assumption, the theoretical energy requirement for atomizing unit liquid mass can be calculated using the following equation:[5]

	Eq. (3)	E / mL	=	6σ
				ρL D32

The theoretical energy efficiency, η , is defined as:

	Eq. (4)	η =	theoretical input energy
			actual input energy to atomizer

Neglecting the energy for overcoming viscous force during liquid breakup, a simple equation for the theoretical energy efficiency has been derived by Yule and Dunkley[5] for a pressure-swirl atomizer:

Eq. (5) η = 600σ /( PD32 )

where the actual input energy is the product of the pressure drop P across the atomizer nozzle and the volume flow rate of liquid, which can be expressed as a sum of the volumes of N droplets created in unit time. For swirl jet sprays, the theoretical energy efficiencies are 0.22%, 0.22%, 0.49%, and 0.95% for water, oil, solder, and aluminum, at P of 10, 1, 6.7, and 2.4 MPa, and for D32 of 20, 59, 60, 150 μm, respectively.[5] The efficiencies are very small because only a very small portion of the kinetic energy supplied to the liquid is converted into surface energy on the droplets generated. A large percentage of the kinetic energy is consumed by accelerating the liquid and droplets.