Theoretical Calculations and Numerical Modeling 331

collisions to follow the Poisson distribution, and proposed three possible outcomes of binary collisions for increasing collision Weber number: (a) permanent coalescence, (b) collision followed by breakup, and (c) shattering. It was indicated that the outcome of a binary droplet collision depends on forces acting on the coalesced pair of droplets. At low relative velocities or low Weber numbers, surface tension forces dominate over liquid inertia forces so that the droplets coalesce permanently. At high Weber numbers, the liquid inertia forces predominate so that the droplets separate, possibly forming satellite droplets, a phenomenon referred to asgrazing collision. At very high relative velocities, the droplets may shatter. Droplets of largely unequal sizes are more likely to coalesce than droplets of nearly equal sizes. The possible formation of satellite droplets due to collision and the shattering collision have not yet been included in the currently available models.

An attempt has been made by Tsouris and Tavlarides[561] to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mechanisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescence function was defined as a product of dropletdroplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions.

5.2.3Theories and Analyses of Spray Structures and Flow Regimes

The degree of development of liquid and air (gas) flows inside an atomizer governs the velocity profiles and turbulence levels at thenozzle

332 Science and Engineering of Droplets

exit, that in turn have a significant influence on the spray structure formed. Current theories and analyses of spray structures are based on the assumptions and/or measurements of droplet sizes and velocity distributions at a designated initial cross-section plane of a spray wherebreakup is assumed to be completed. A Lagrangian sectional approach for calculating droplet size distributions of vaporizing fuel sprays in turbulent jets has been developed on the basis of detailed and numerous local droplet size and velocity distributions in the sprays.[562] In this approach, the effects of dispersion, acceleration, coalescence and evaporation can be calculated at any locations downstream. Accordingly, a spray is divided into three regions along the downstream direction: near-field, far-field, and fully developed far-field regions.

The region from the end of the liquid breakup region down to the far-field region is defined as near-field region. In this region, droplet velocities are significantly higher or lower than local gas velocities, so that the droplets are decelerated or accelerated by the gas flow during momentum exchanges. Inter-droplet distances are small initially but increase subsequently as the droplets disperse with increasing distance downstream. Droplet size and velocity are not well correlated. Initially, droplet velocity varies dramatically and then the variance decreases as gas turbulence decays with increasing distance downstream. The probability of droplet collision and subsequent coalescence is high due to high droplet number density, significant transverse velocity components, impinging droplet trajectories, and turbulent fluctuation components of both the droplets and gas. Coalescence may reduce the length of the near-field region due to the momentum exchanges between droplets during collisions.

The region where droplets and gas are approaching equilibrium is defined as far-field region. This region may be characterized by the small velocity difference (less than 10%) between droplets and gas. The far-field region develops with distance downstream to fully-devel- oped far-field region. In the fully developed far-field region, the relative velocity between droplets and gas vanishes. Thus, momentum exchanges between the two phases also cease. The droplets are well dispersed in the gas with no collision or coalescence occurring, and

Theoretical Calculations and Numerical Modeling 333

spray becomes dilute with large inter-droplet distances. Droplet size is well correlated to its velocity. Droplet temperature is in equilibrium with local gas temperature so that no heat exchanges occur between the two phases.

Since a spray is a particle-laden turbulent jet, the fullydeveloped far-field region of a spray is analogous to the fullydeveloped region of a turbulent jet. In a turbulent jet, the profiles of mean velocity components, turbulent fluctuations, shear stress, and kinetic energy of turbulence tend towards self-similarity and thus can be described with a single self-similar profile independent of downstream distance by normalizing the respective quantities and the distance with characteristic quantities and length. Different quantities may achieve the self-similarity at different distances downstream from the nozzle exit. The fully-developed region of a turbulent jet is the region where all quantities have attained equilibrium and self-similar- ity of profiles. Similarly, the distance for different droplets in a spray to achieve the equilibrium with local gas velocity will vary, depending on the magnitude of droplet sizes. The fully-developed region represents asymptotic conditions that may not be achieved in a particular spray.

The denseness of sprays may be measured by the ratio of the inter-droplet distance to droplet diameter. For the ratios larger than 30, a spray is dilute, and evaporation rates are almost same as those of single isolated droplets evaporating in the same atmosphere. As the ratio decreases to 15, small departures from dilute conditions are encountered. For the ratio of 10, a spray is dense with substantial departures from dilute conditions. For the ratios less than 10, a spray becomes so dense that the gas phase may be saturated before evaporation is complete.

Sprays and other dispersed flows are normally divided into dense and dilute spray regimes. Faeth[563] described the characteristics of each of the regimes. A jet spray may be divided into four regimes: churning, thick, thin, and very thin, as defined by O’Rourke.[550] The churning regime is defined as the flow region where the volume fraction of the liquid equals or exceeds that of the gas, and the liquid is no longer dispersed in a continuous gas phase. This regime occurs very

334 Science and Engineering of Droplets

close to the nozzle exit where the intact injected liquid starts to break up into blobs and droplets. For this regime, the equations of mass, momentum and energy conservation for compressible two-phase flows have been derived by considering the phases to be co-existing continua.[564] However, it is still uncertain about the form and ranges of applicability of the coupling terms between the liquid and gas phases.

In the thick (dense) regime next to the churning regime, the volume fraction of droplets is significant in the two-phase flow. In spite of the significant volume fraction, droplets are still recognizable as discrete droplets or liquid blobs in a continuous phase. Droplet-droplet effects such as collisions and coalescences are extremely important. The rates of mass, momentum, and energy exchanges between the droplets and gas are influenced by neighboring droplets.

Downstream in the spray, droplet size is reduced due to evaporation and breakup. Far from the nozzle, droplets become isolated and have negligible mass and volume relative to the gas phase. This regime is termed very thin (dilute) regime. Although the mass, momentum, and energy exchanges between the droplets and gas continue in this regime, the state of the gas is not altered appreciably by the exchanges. Therefore, the gas flow can be computed ignoring the volume occupied by the droplets, and the droplets can be tracked using the spray equation.[541] Isolated droplet correlations can be used to calculate the exchanges, and collisions and other interactions between droplets can be ignored.

The intermediate regime between the thick and very thin regimes is referred to asthin regime. In this regime, the droplets occupy negligible volume but have significant mass compared to the gas due to the large liquid to gas density ratio. The droplets influence the state of the gas, that in turn influences the exchange rates between other droplets and the gas. For example, the wake produced by droplets at the tip of a transient spray leads to a reduced relative velocity between the subsequent droplets and the gas because these droplets experience drafting forces. The reduced drag on the subsequent droplets allows them to overtake the leading droplets at the tip of the spray. The process repeats until entering the very thin regime.

Theoretical Calculations and Numerical Modeling 335

5.2.4Modeling of Droplet-Gas and Droplet-Droplet Interactions

Droplet-gas and droplet-droplet interactions in a spray affect droplet size and velocity. For modeling of mass, momentum, and energy exchanges between droplets and gas in a spray, the knowledge of droplet-gas and droplet-droplet interactions is required. It is common practice to calculate the rates of mass, momentum, and energy exchanges by means of engineering calculations that, however, may not adequately describe the complex, unsteady characteristics of the exchange processes. Laminar convective models have been used, but are not applicable to turbulent flows. Very sophisticated single droplet models and highly developed codes by Raju and Sirignano[565] and Dandy and Dwyer[566] include the time-dependent equations for the exchanges at droplet-gas interfaces and are being extended to arrays of single droplets and clusters. Advances in measurement techniques and spray calculations will lead to correlations of probability density functions that will allow more comprehensive and realistic descriptions of the transfer processes.

Accurate representations of the drag coefficients of droplets over a wide range of flow conditions are a necessary prerequisite to the calculation of gas-droplet flows. Yung et al.[567] performed numerical modeling of unsteady motion of a spherical fluid droplet under the influence of gravity to examine the steady state droplet velocity and drag coefficient. Numerical results for these two quantities were presented in graphical form for a water droplet moving through air and an air bubble rising in water under conditions from creep flows to moderate Reynolds numbers.

Due to computer storage and run time limitations, it is not yet possible to accurately model the details of flows around each individual droplet in a spray. Thus, empirical or semi-empirical correlations are typically used to model the exchange processes between droplets and gas. Correlations for drag coefficients have been suggested by many researchers.[45][559][568]-[571] For thin sprays, the drag

336 Science and Engineering of Droplets

coefficient is often calculated as a function of the droplet Reynolds number using the single solid-sphere correlation.[568] For thick sprays, O’Rourke and Bracco[559] suggested a correlation that includes the effect of local void (air/gas) fraction on the drag coefficient.[550] The effects of droplet oscillation and distortion on the drag coefficient were investigated by Hwang and Reitz[315] and Liu et al.[572] who proposed a quantitative correlation to relate the magnitude of droplet deformation to its drag coefficient. Using this dynamic drag model, the calculated drag coefficient of a distorting droplet lies between that of a rigid sphere and that of a flat disk whose drag coefficient at high Reynolds numbers is about 3.6 times higher than that of a sphere. Renksizbulut and Yuen[573] suggested a correlation for drag coefficient to account for the effect of droplet vaporization. They showed that droplet vaporization reduces its heat transfer rate and drag coefficient, whereas Gonzalez et al.[574] found that the effect of droplet vaporization on its drag is small in diesel-type sprays.

Henderson[575] presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated.

Turbulence in a spray may have a significant effect on the drag coefficient.[576] The effect of turbulence is quantified by relative turbulence intensity. Torobin and Gauvin[577] found that for relative Reynolds numbers of the order of 103, relative turbulence intensities of the order of 20% can cause an order of magnitude decrease in the drag coefficient. Warnica et al.[578] presented data of drag coefficients for Reynolds numbers less than 100 and relative turbulence intensities as high as 60%. The data suggested that within the experimental accuracy achievable, the drag coefficient for a sphere can be used for a droplet.

Theoretical Calculations and Numerical Modeling 337

Table 5.1. Correlations for Drag Coefficient of a Single Sphere in Continuum and Rarefied Flows[575]

Valid Regime

Correlations

é

ì

æ

T

ö

ü

ù −1

ê

ï

ç 3.65 -1.53

w

÷

æ

ïú

T

ê

ï

ç

÷

Re öïú

C D

= 24 Re+ S í4.33 +

ç

÷

´ expç

- 0.247

÷ý

ú

T

S

ê

w

è

ø

ê

ï

ç 1 +

0.353

÷

ïú

ë

ï

è

T

ø

ï

Subsonic Flow

î

þû

Ma≤1

æ

0.5Ma

öé

4.5 + 0.38(0.03 Re+ 0.48

Re )

2

8

ù

+ expç-

÷

ê

+ 0.1Ma

+ 0.2Ma

ú

ç

Re

÷

ê

1 + 0.03 Re+ 0.48

Re

ú

è

ø

ë

û

é

æ

Ma öù

+ ê1

- exp ç-

÷

ú

0.6S

ë

è

Re ø

û

0.34

æ Ma

ö

1/ 2 é

2

1.058 æ T

ö

1/ 2

1

ù

0.9 +

+1.86ç

∞

÷

ê2 +

+

ç

w

÷

-

ú

2

2

4

Ma

ç

Re

÷

ê

S

S

ç

T

÷

S

ú

Supersonic Flow

∞

è

∞ ø

∞

∞

è

ø

∞

C D

=

ë

û

Ma≤1.75

æ

Ma

∞

ö

1 / 2

1 +1.86ç

÷

ç

÷

è

Re ∞ ø

Supersonic Flow

C (Ma

∞

, Re

∞

) = C

D

(1., Re)+

4

(Ma

∞

-1.)[C

D

(1.75, Re

∞

)-C

D

(1., Re)]

1<Ma<1.75

D

3

CD — Drag coefficient; CD(1.,Re) is the drag coefficient calculated using the correlation for subsonic flow with Ma=1.; CD(1.75, Re∞) is the drag coefficient calculated using the correlation for supersonic flow with Ma∞=1.75;

Ma — Mach number based on relative velocity between gas and sphere;

Re — Reynolds number based on sphere diameter, relative ve locity, and freestream density and viscosity;

S— Molecular speed ratio, S = Ma (γ/2)0.5, where γ is the ratio of specific heats;

T— Temperature of gas in freestream;

Tw — Temerature of sphere;

∞ — Refers to freestream conditions.

338 Science and Engineering of Droplets

Interactions of a shock wave with a single droplet and multiple droplets of various liquids have been investigated in many numerical modeling studies.[579]-[582] It was found[579] that for a flow across a shock, the Basset history force acting on a droplet can be many times larger than the viscous drag in the immediate shock region. It was demonstrated how this force influences the motion of particles of different densities or sizes across the shock. For the theoretical description of the motion of particles commonly used in flow measurement tracer techniques, however, it is justified to neglect the Basset integral in the

Basset-Boussinesq-Oseen equation.[568]

Resistance functions have been evaluated in numerical computations[583] for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases: (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,[584] and Ranz and Marshall,[505] among others. The correlation recommended by Whitaker is accurate to within ±30% for the range of parameter values listed. All properties except µs should be evaluated at T∞. For freely falling liquid droplets, the Ranz-Marshall correlation[505] is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account

Theoretical Calculations and Numerical Modeling 339

for the effects of droplet oscillation and distortion,[585] droplet vaporization and mass transfer,[555] as well as compressibility of gas.[154][156] The influence of superheating of diffusing vapor around a droplet on its heat transfer rate was examined by Priem et al.[586] As with the drag coefficient, turbulence also affects the heat transfer coefficient. The effect of turbulence is quantified by the relative turbulence intensity. A modified form of the Ranz-Marshall correlation has been proposed on the basis of experimental data for water, ethanol and methanol droplets at relative turbulence intensities up to 12%.[587]

Table 5.2. Correlations for Heat Transfer Coefficient of a Single Sphere in Gas

Correlations

Remarks

References

	Nu = 2 + (0.4 Re1 / 2 + 0.06 Re 2 / 3 ) Pr0.4 (				μ		)1/ 4	All	Whitaker
								properties
					μ s
					μ			(except µs)	[584]
	0.71 < Pr < 380, 3.5 < Re < 7.6 ×104 , 1 <					< 3.2		evaluated
					μs			at T∞
								Freely falling	Ranz &
	Nu = 2 + 0.6 Re1 / 2 Pr1/ 3							liquid	Marshall
								droplet	[505]
		x	) −0.7 ]					Effects of	Yao &
	Nu = 2 + 0.6 Re1/ 2 Pr1/ 3[25(							droplet
		D						oscillation &	Schrock
	3mm≤D≤6mm, 10≤x/D≤600							distortion	[585]
								included

Nu, Pr, Re —		Nusselt, Prandtl, and Reynolds numbers, respectively,
D	—	Droplet diameter,
T∞ , Ts	—	Gas temperature in freestream and sphere surface tem-
		perature, respectively,
x	—	Falling distance measured from rest,
μ, μs	—	Gas viscosity evaluated at T∞ and Ts, respectively.