properties and the total mass of the collector particles after collision. nl and ns
refer to the number of large and small particles inside a computational parcel.
If (ncoll nl) > ns, then there are physically not enough particles in the donor parcel to accommodate all of the collisions with the particles in the collector parcel. In this case, ncoll is reset to ns/nl and the mass of the particles associated with the particle parcel is set equal to zero and eventually removed from the calculation.
In a grazing collision event, only momentum exchange is considered since the time scales associated with heat exchange are much longer than the collision event. The particle velocities after collision are set as follows [43];
( )
Often, traveling particles collide with solid surrounding walls and the particle impact phenomena are investigated for various practical reasons: if particles do not deposit onto the surface (whether it is because of the liquid’s high surface tension, hydrophobic substrate, a thin air-film, or the Leidenfrost effect), painting/coating quality becomes rather poor. For spray cooling applications, the particle rebound phenomenon decreases the cooling efficiency. In automobile applications, excessive fuel spray impingement onto walls of the IC engine forms a liquid film whose fuel-rich burning leads to unwanted pollutants. At high impact speeds, particles break up (often referred to as “splashing”) into satellites, leaving scattered marks in the impacted region; this is detrimental for inkjet printing applications that requires a clean dot-printing onto a substrate. On the other hand, for spray drying applications, splashing should be promoted to produce powders by spraying mono-disperse drops onto various substrates. In these aforementioned applications, the particle impact phenomena should be controlled and understood properly to delineate their impact physics in the stochastic spray model.
2.8 Particle-wall impact
Traditionally, surface tension, viscosity, and substrate roughness (i.e., We, Re, and Ra,) are the primary parameters that describe drop impact phenomena [94]. Bouncing occurs when the surface tension force greatly exceeds the dynamic force. When the dynamic force overcomes the surface tension force, the particle spreads upon impact and can splash if the dynamic force is too excessive; otherwise it simply spreads. In this report, outlines for the (i) bouncing, (ii) sticking, (iii) splashing phenomena are briefly are reviewed and the relevant references for the description of their detailed modeling approaches are suggested (see Figure 13).
Figure 13 Configuration states during particle impact consisting of pre-impact (i.e., incoming stage), pre-impact (i.e., maximum diameter), and post-impact (i.e., sticking, bouncing, and splashing)
Bouncing
Bouncing is commonly observed during a low speed impact, where the surface tension effect is pronounced. This phenomenon is of low interest because most of the present applications are concerned with high-speed impacts. Furthermore, it is of relatively low interest to industries, as opposed to sticking and splashing phenomena, because it has few practical applications. However, bouncing does occur in drop or spray impact and, thus, its mechanism needs to be correctly understood for accurate impact modeling. Aziz and Chandra [95] noted: If the surface energy of a drop at its maximum spreading is greater than the dissipation energy, then the drop rebounds; otherwise, the drop will stick. Mao et al. [96] also suggested a rebound criterion with the dimensionless rebound energy, Er* as an implicit function of the maximum spread factor and the equilibrium contact angle. If Er*
>0, then the drop rebounds; otherwise, it sticks. More recently, Mukherjee and Abraham [97] presented a numerical simulation of rebounding using the lattice-Boltzmann model; but their study was limited to a case where the density ratios between the drop and surrounding gas were only 5 and 10.
Ideally, the density ratio should reach an order of ~103 for water-drop impact
in air. Park et al. [98] used the boundary integral method to model the rebounding phenomenon for a range of air pressures.
Sticking
Among various subjects on drop impact, sticking has received much attention because of its broad applications such as inkjet-printing, coating, painting, and spray cooling. The maximum spreading diameter often needs to be predicted to estimate the coating area-coverage or painting performance.
The accurate prediction includes the competition between the drop’s inertia and surface tension energy and viscous dissipation energy. Chandra and Avedisian [99] predicted the maximum spreading diameter using an energy conservation concept. Healy et al. [100] compared various suggested models for experimental data (Ford and Furmidge, [101], Stow and Hadfield, [102]) and concluded that the model by Yang [103] yielded the best result. Mao et al.
[96] also provided a correlation for the maximum spreading diameter as a function of the Reynolds, Weber number, and contact angle. Gong [104] also suggested another spreading model, but his results were not quantitatively compared with the results of other existing models. Sikalo et al. [105] utilized the VOF (Volume-of-Fluid) model for the maximum spreading diameter for highly viscous glycerin drops; their numerical results were compared with the experimental data of Stow and Hadfield [102]. For fire modeling applications, the sticking might be the source term for a fuel fire as a pool forms, and is, therefore, of interest. For solid propellant fires, the sticking can provide substantial deposition heat transfer to an object.
Splashing
Splash occurs for a relatively high speed impact, where inertia dominates over surface tension or/and viscosity. Various splash criteria were offered by Yarin and Weiss [106], Mundo et al. [107], and Cossali et al. [108]. Because of the dependence of splash on surface tension and viscosity, splashed droplets or satellites are normally scaled with the Weber, number and Reynolds or Ohnesorge (Oh) number (i.e., impingement characteristic parameter, Kcrit=aWebRec, where a, b, and c are constants; as for a drop onto film for example, Kcrit = Oh-0.4We = We0.8Re0.4 = 2100 + 5880 (δ*)1.44, where δ*= film thickness, non-dimensionalized by parent drop diameter, Dp), and so is the splashed liquid amount. At moderate impact speed, a corona is normally formed during spreading and the capillary waves along the rim of the corona in the azimuthal direction appear and eventually they snap-off, a phenomenon known as “corona splash.” When the substrate is wet (liquid film), a rising corona sheet or crown is of larger relative size due to the mass added by the film. The related work on this topic includes: Stow and Stainer [109], Marmanis and Thoroddsen [110], Rioboo et al. [111], Xu et al. [112], Xu [113], and Kalantari and Tropea [114]. The splashed droplets size appears to obey a log-normal distribution, according to Stow and Hadfield [102]. Post-impact characteristics of the splashed droplets, such as their speed and size, were also studied by Cossali et al. [108] and Roisman et al.
[115].
Surface roughness
Stow and Hadfield [102], Mundo et al. [107,116], Range and Feuillebois [117], Rioboo et al. [111] noted that the splashing criteria could change with surface roughness. Cossali et al. [108] provided an empirical fit for Kcrit for a dry surface, based on the previous data; Kcrit=649+3.76/R*, where R* is the surface roughness, non-dimensionalized by Dp. When the surface is sufficiently rough, “prompt splash” occurs, which is immediately followed by spreading. Breakthrough research was conducted by Xu et al. [112], who first showed that splash is a function of air pressure while conventional wisdom suggested that splash is dependent only on substrate roughness and the drop’s physical properties. Their finding provides a new degree of freedom in controlling splash.
A general description of the particle phase transport equation (i.e., conservation of mass, momentum, and energy) was reviewed on the basis of Eulerian-Lagrangian particle tracking method. In this modeling approach, suitable for the dilute two-phase flow system, the particle phase is modeled on a Lagrangian reference frame and the gas phase is modeled on an Eulerian reference frame. Explanation on the interaction between the particle phase turbulence model and the k-ε RANS (Reynolds-Averaged Navier-Stokes) turbulence for the Eulerian phase was also provided. A stochastic approach utilizing PDFs was adopted for several sub-models, such as drop size distribution, turbulence, and binary collision. Furthermore, various models for the particle-wall interaction were reviewed.
This research was supported by Research Center of Break-through Technology Program through the Korea Institute of Energy Technology Evaluation and Planning (KETEP) funded by the Ministry of Knowledge Economy (2009-3021010030-11-1). The author also acknowledges that a partial support was made for this project by the National Research Foundation of Korea NRF Grant (NRF-2010-D00013). The author is indebted to Prof. Paul DesJardin at University of Buffalo for his support in writing this review chapter.
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