Grantee Research Project Results
Final Report: Modeling Collision Efficiencies for Coalescence of Small Drops and Particles
EPA Grant Number: R827115Title: Modeling Collision Efficiencies for Coalescence of Small Drops and Particles
Investigators: Koch, Donald L.
Institution: Cornell University
EPA Project Officer: Hahn, Intaek
Project Period: October 1, 1998 through September 30, 2001
Project Amount: $305,155
RFA: Exploratory Research - Physics (1998) RFA Text | Recipients Lists
Research Category: Air Quality and Air Toxics , Land and Waste Management , Air , Safer Chemicals
Objective:
The objective of this research project was to determine the conditions under which aerosol droplet collisions lead to coalescence so as to derive collision kernels that can be used in drop population balances. Particular attention was given to the influence of non-continuum gas flow between colliding particles on the collision efficiency.
Summary/Accomplishments (Outputs/Outcomes):
An important component needed for modeling collision efficiencies in aerosol systems is the resistance produced by the non-continuum gas to the relative motion of the aerosol particles. In previous works, we predicted the resistance for the case of two rigid spheres when the mean free path is either much greater or much less than the sphere dimension. In each case, the drag force acting on the spheres was derived and the criterion for contact to occur due to non-continuum effects was obtained.
When droplets collide, the interfaces deform due to the pressure induced by the flow of the gas in the intervening gap. The drops may coalescence or bounce depending on whether the gas is forced out of the gap between the drops before the kinetic energy of their relative motion is transformed into energy of surface deformation. We have formulated a general theory that models the flow inside the drop and takes into account the evolution of the gap between the drops. The drop deformation in the near-contact inner region was determined by solving the lubrication equations and matching them to an outer solution. The resulting equations were solved numerically using a direct, semi-implicit, matrix inversion technique. By simultaneously including both strong retarding viscous forces and surface deformation, our work provided a rational basis for understanding and predicting collision outcomes. The general program developed during the course of our investigation allowed us to simulate the collision of two deformable drops, or the collision of a deformable drop with a deformable liquid surface in a non-continuum isothermal gas.
To complement this theoretical analysis, we developed an experimental apparatus to observe droplet collisions with a gas-liquid surface. Drops were fired from a piezo-electric drop generator toward the interface and the impact velocity was controlled by varying the distance from the generator to the interface. A pressure chamber encloses the apparatus, allowing variation in gas pressure and composition. The droplet trajectories were observed using a high-speed video camera and a long-range microscope. The trajectories were fit with a solution for the drop motion based on a balance of drop inertia, gravity, and nonlinear drag to determine the drop radius and the impact and rebound velocity.
The dynamics of droplet bouncing revealed by our numerical calculations were as follows. As two drops approached to within a gap thickness of aCa1/2, the droplet interfaces began to flatten as a result of the lubrication pressure in the gas. Here, Ca=µgU/σ is the capillary number, µg is the viscosity of the gas, U is the impact velocity, and σ is the surface tension. As the radial extent of the thin gas film region between the drops extends outward, the droplet shape in this region takes on a concave dimple shape. This reflects the presence of a pressure gradient driving the gas out of the gap. Eventually, the normal component of the drops' relative velocity was arrested when the kinetic energy of the drops motion is stored as surface deformation energy. The radial extent of the dimple region retracts inward and the dimple vanishes. As the dimple retracted, a low pressure region developed outside the dimple where the two surfaces were peeling away from one another. This pressure valley grew and became quite sharp when the dimple vanished. As the drops receded from one another, each formed a tail for a very short period of time.
The dynamics of a drop-interface bounce is quite similar to a drop-drop collision. However, the interface deforms much more than the drop. In the close-contact region, the interface and drop both have almost the same radius of curvature, which is twice the radius of the undeformed drop. This result arises because there are two interfaces to accomplish the pressure drop 2σ/a between the interior of the drop and the fluid sublayer.
A plot of the minimum gap separating the drops as a function of time revealed two local minima. The first occured at the rim of the dimple shortly after the dimple region reached its largest radial extent. A very thin gap was required at the rim to accomplish a pressure drop of 2σ/a (for a drop-drop collision) or σ/a (for a drop-interface) from the gas in the dimple to the gas just outside the dimple. The second minimum occured due to the approach of the tails to one another as the droplet(s) receded. The smaller of these two minima most likely will be the site for coalescence. The relative size of these minima depends on the detailed conditions of the collision. However, for all the cases of drop-interface collision encountered in our experimental program, the theory predicts coalescence due to the tail.
A convenient set of non-dimensional parameters with which to correlate the outcome of drop-drop and drop-interface collisions were the Weber number We=ρlU2a/σ, Ohnesorge number Oh=µg/(ρlaσ)1/2, and Knudsen number Kn=λ/a, where rl is the liquid density, a is the drop radius, and λ is the mean-free path of the gas. Both the experiments and theory indicated that the outcome was only weakly dependent on the Weber number. This helped simplify the specification of coalescence rate. Typically, drops collide with a distribution of relative velocities, but the transition from coalescence to bouncing is only a weak function of the velocity. A unique aspect of our work was the investigation of the role of the mean-free path of the gas on coalescence-bounce transition. It was found that a small decrease in pressure could greatly increase the critical Weber number above which drops bounce. We also varied the gas viscosity by using different gas compositions. A decrease in the gas viscosity led to less interfacial deformation and less opportunity for drop bouncing. The drop radius appeared in all the dimensionless groups mentioned above. However, the theory indicated that the effect of drop radius on Knudsen number is dominant. Thus, larger drops bounce more readily because the gap must thin to a smaller fraction of the drop radius before the continuum lubrication forces break down.
Although the primary goal of our study was to identify the conditions under which drops bounce or coalesce, we also compared other aspects of the drop dynamics with experimental measurements. For example, both theory and experiment indicated that the coefficient of restitution (the ratio of the drop velocity after and before collision) for drop-interface collisions was much smaller than that for drop-drop collisions. This occurs because the initial planar fluid interface deforms more easily, and a greater fraction of the kinetic energy is carried away by surface waves. The coefficient of restitution generally decreases with the increasing Weber number, and becomes independent of the gas viscosity (Ohnesorge number) at Weber numbers of about 1 or larger. At these large Weber numbers, the predominant mechanism dissipating the kinetic energy of the drops is the formation of surface oscillations. The amplitude of the interfacial deformation and the energy lost to surface waves increases with increasing Weber number. For smaller Weber numbers, the viscous dissipation of energy in the gas film becomes more important, while drops lose less energy when they bounce in a less viscous gas. Larger drops also have higher coefficients of restitution for the same We because of their smaller Ohnesorge number (which can be viewed as a dimensionless gas viscosity).
Following the coalescence of a droplet at an interface, we observed that a smaller daughter drop was emitted from the surface and rose at a velocity that could be larger than the impact velocity of the mother drop. The formation of such small drops could play a role in the formation of fine mists and the salt particles that serve as nuclei for drop formation in the atmosphere. Previous studies have identified a transition from daughter drop formation at We > 32 to no daughter drop formation for We < 32. For these large drops, the mechanism leading to the formation of a new drop involves the crater and jet formed in the liquid by the inertia associated with the impact of the drop. Our study involved much smaller drops for which the Weber number was of order one or smaller. The daughter drops formed in our experiments were caused by a surface tension driven flow. This was indicated by the fact that for We < < 1, the ratio of daughter to mother drop radius, ad/am = 0.55, and the daughter drop velocity Ud, scaled with a surface tension scaling i.e., Ud(ρla/σ)1/2= 0.38 were independent of We.
Journal Articles on this Report : 4 Displayed | Download in RIS Format
Other project views: | All 7 publications | 4 publications in selected types | All 4 journal articles |
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Bach GA, Koch DL, Gopinath A. Coalescence and bouncing of small aerosol droplets. Journal of Fluid Mechanics 2004;518:157-185. |
R827115 (Final) |
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Gopinath A, Koch DL. Hydrodynamic interactions between two equal spheres in a highly rarefied gas. Physics of Fluids 1999;11(9):2772-2787. |
R827115 (1999) R827115 (Final) |
Exit |
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Gopinath A, Koch DL. Collision and rebound of small droplets in an incompressible continuum gas. Journal of Fluid Mechanics 2002;454:145-201. |
R827115 (2000) R827115 (Final) |
Exit |
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Gopinath A, Koch DL. Dynamics of droplet rebound from a weakly deformable gas-liquid interface. Physics of Fluids 2001;13(12):3526-3532. |
R827115 (Final) |
Exit |
Supplemental Keywords:
air, atmosphere, precipitation, particulates, physics, mathematics., RFA, Scientific Discipline, Air, Waste, particulate matter, Environmental Chemistry, Physics, Atmospheric Sciences, Ecology and Ecosystems, Engineering, Chemistry, & Physics, Incineration/Combustion, collision efficiency models, pollution control technologies, air pollution modeling system, air pollution, Brownian motion, pollutant transport, differential sedimentation, emission controls, atmospheric transport, coalescence, particle surface interactions, incineration, pollution dispersion models, particle collision models, atmospheric models, particle collision, particle transportProgress and Final Reports:
Original AbstractThe perspectives, information and conclusions conveyed in research project abstracts, progress reports, final reports, journal abstracts and journal publications convey the viewpoints of the principal investigator and may not represent the views and policies of ORD and EPA. Conclusions drawn by the principal investigators have not been reviewed by the Agency.