Coagulation and breakup of droplets in turbulent couette flow: Theory, simulation, experiment.

Abstract

The problem of breakup and coalescence of droplets in fully turbulent Couette flow is investigated through theory, simulation and experiment. Solutions to the fragmentation equation are presented for three classes of the breakup product size distribution, two of which allow for particles to break preferentially into like-size products. In all three classes the overall rate of breakup is assumed to be a power of the particle size. All breakup product distribution functions with known analytical solutions are included as a subset of these three, general models. The analytic, sealing and moment solutions of each model are detailed along with a procedure for finding solutions when feed and withdrawal streams are present. For rates of breakup where smaller particles break up faster than larger ones, mass is lost to a phase of "zero" size particles and a power-law, small-size distribution forms. This process is identified as the "shattering" transition in analogy with gelation in coagulating systems. The analytical models of fragmentation are compared to experimental drop-size distributions obtained by digital imaging of two-phase, turbulent Couette flow. A new droplet sizing technique which uses the edges of the droplets along with a circularity measures is described. It is found that this technique performs as well as previously described region-growing algorithms while allowing irregularly shaped (overlaps) or poorly lit droplets to be recognized. Height, speed and time effects on the turbulent drop-size distribution in turbulent Couette flow are investigated. The time-dependant number distribution is found to equilibrate at a mean droplet size below the mixing length, suggesting that for Reynolds numbers above 15000 the turbulence stabilizes the droplets with respect to coalescence. This same effect has been observed in stirred tanks. Height above the base of the cylinder is found to have only marginal effect on the size distribution, except for short times $<$10 minutes. Comparisons between the reaction of the droplet number distribution to a step change in inner cylinder speed and theoretical size distributions show that, for short times, a model where all sizes of products are produced with equal probability follows the same trend as the experimental results.