A stochastic droplet collision model with consideration of impact efficiency

A stochastic droplet collision model (Sommerfeld [9]), based on the creation of a fictitious collision partner is described, taking into account impact efficiencies. The model of O’Rourke [5] is considered to predict the outcome of water-droplet collisions, being grazing or coalescing, and to predict post-collision velocities. The relevance of impact efficiencies is discussed for water droplet collisions on the basis of the inertial parameter. As a result, regions of importance are defined, in which the impact efficiency has to be taken into account. The influence of impact efficiency on coalescence rates of water droplets is discussed by comparing normalised critical displacements and collision frequencies. The assumption of a step function for modelling impact efficiencies, as valid for laminar flows, is not applicable to turbulent flows if the integral scale of turbulence is in the order of the large droplet diameter. Introduction Droplet collisions are a phenomenon occurring in all dispersed liquid-gas systems. By it’s simple existence this phenomenon has to be taken into account in industrial processes like spray drying, atomisation in engines or Ion launders. The change of the droplet size distribution due to merging or splashing of colliding droplets may significantly influence the process performance. The occurrence of collisions assumes that the collision partners come into touch with each other. That is not a matter of course, small droplets may follow the flow field around a larger one, therefore preventing a collision and reducing the impact efficiency of the system. The impact efficiency not only depends on the diameter ratio of the approaching partners, but also on the surrounding flowfield, being laminar or turbulent. Schuch and Löffler [8] performed calculations and experiments for small solid particles and fixed droplets in a laminar flow, formulating relations for Reynolds number dependent impact efficiencies. Pinsky and Khain [7] showed the random character of hydrodynamic droplet interactions in turbulent flows and indicated the possible increase of impact efficiencies due to turbulence effects. If collisions between droplets finally happen, one can observe different phenomena in the post-collision stage, such as permanent coalescence, bouncing or separation with production of small satellite droplets. Numerous investigators dealt with the object of defining boundaries between this collision modes. The majority of them concentrated their research on water-air systems (Brazier-Smith et al. [2]), others on organic liquids like Heptane, Propanol or glycerin (Jiang et al. [4]). The main achievement of this work was the description of boundaries of the collision scenario expressed as functions of critical Weber numbers and collision angles. Only few researchers investigated other phenomena related with droplet collisions, like mass exchange of merging or separating droplets (Potvysotsky et al. [6]). Almost no information is available about post-collision velocities of colliding droplet pairs. Such knowledge is needed for a detailed modelling these phenomena. The present study concentrates on the modelling of droplet collisions in turbulent flows, considering impact efficiency. The relevance of the impact efficiency and it’s influence on coalescence rates is discussed, concentrating on water-air systems. Stochastic Droplet collision model The core of the stochastic collision model is the creation of a fictitious collision partner, which is done with the help of local size and velocity distributions of the droplet phase. Hence, the fictitious droplet is a representative of the local droplet population. This way of deciding whether a collision takes place or not does not require the knowledge of locations of neighbouring droplets and the time consuming search for collision partners. Only the droplet sizeand velocity distribution functions must be stored for each computational cell (Sommerfeld [9]), usually expressed by size classes or parameters of standard distribution functions. In order to describe the collision process in detail, the following physical effects have to be considered: • The instantaneous velocity of small droplets moving in a turbulent flow are not independent. If the droplets are able to respond to the turbulent fluctuations they are moving in the same turbulent eddy upon collision. Hence, their velocity is correclated through their response to turbulence. • With the correlated velocities a collision probability can be obtained based on kinetic theory of gases if the size of the colliding droplets is not too different. • In case of a collision between small and large droplets, which is most likely to occur (Ho and Sommerfeld [3]), the impact probability is reduced, since the small droplet may move with the relative flow around the larger droplet, also called collector. • Once a collision occurs it has to be decided whether this collision results in coalescence or rebound (note, that also other collision scenario are possible) and the post-collision velocities have to be determined. For clarity of the results, the correlation of the velocities of colliding droplets (Sommerfeld [9]) was not considered in the present study. In order to decide whether a collision between the real and fictitious droplet takes place during a time step or not, two steps are necessary: the calculation of the collision probability and the calculation of the impact probability. In order to simplify this calculation a co-ordinate transformation into a cylindrical frame of reference is carried out, in a way that fictitious droplet is set stationary. In order to find the point of imp act, a collision cylinder is defined, with an axis aligned with the relative velocity vector (Fig. 1 a). Fig. 1: Inter-droplet collision configuration (a) and the modelling of the collision efficiency (b) The collision probability Pcoll is calculated in this cylindrical frame of reference according to kinetic theory: ( ) t n u d d t f P d rel fict real coll coll ∆ ⋅ ⋅ ⋅ + ⋅ = ∆ ⋅ = r 2