In Sect. 4, we derive the first five moments of the radius from the analytic equilibrium PDFs, including moments for truncated DSDs (those with positive lower limits). In Sect. 3, we derive the analytic equilibrium solutions for the distributions and probability density functions (PDFs) of radius and of squared radius and from these obtain expressions for the median and mode radii. In Sect. 2, we show how the equilibrium radius distribution is realizedīy using a Monte Carlo method and compare the results to those that are obtained analytically in later sections. Including the loss rate due to sedimentation. The evolution of the droplet radius and squared radius distributions, In Sect. 1, we derive the equations which govern (2) the effects of droplet curvature and solute on the droplet growth rate can be neglected,Īnd (3) droplets fall relative to the turbulent flow at their Stokes' fall speed (for example, they are not affected by turbophoresis or thermophoresis). In this study, we assume that (1) droplets grow subject to a uniform mean supersaturation, Nor do we have a quantitative understanding of droplet fallout. In particular, we do not know the relative importance of mean supersaturation and supersaturation fluctuations, The reasons for this include the difficulty of accurately measuring supersaturation in a cloud chamberĪs well as uncertainties in our knowledge of the physical processes that determine the DSD. Obtaining a complete quantitative theory for the equilibrium DSDs has been elusive. When aerosols (cloud condensation nuclei) are injected at a constant rate,Īn equilibrium state is achieved in which the rate of droplet activation is balanced by the rate of droplet loss.Īfter a droplet is activated, it continues to grow by condensation until it falls out (i.e., contacts the bottom surface).Īlthough the resulting equilibrium droplet size distributions (DSDs) have been extensively measured in the Π chamber, and theoretical models proposed for some aspects of the DSDs Supersaturation is produced by isobaric mixing within the turbulent flow. It is possible to produce Rayleigh–Bénard convection by applying an unstable temperature gradient between the top and bottom water-saturated surfaces of the chamber. In a laboratory cloud chamber, such as the Π chamber at Michigan Technological University We also included some additional quantities derived from the analytic DSD: We found that accounting for the truncation radius of the measured DSDs is particularly important when comparing the theoretical and measured relative dispersions of the droplet radius. We found consistency between the theoretical and measured moments, but only when the truncation radius of the measured DSDs was taken into account.Īllows us to infer the mean supersaturations that would produce the measured PDFs in the absence of supersaturation fluctuations. We used statistics from a set of measured DSDs to check for consistency with the analytic PDF. Given the chamber height, the analytic PDF is determined by the mean supersaturation alone.įrom the expression for the PDF of the radius, we obtained analytic expressions for the first five moments of the radius, including moments for truncated DSDs. (3) a droplet's terminal velocity follows Stokes' law (so it is proportional to its radius squared). (2) when a droplet becomes sufficiently close to the lower boundary, the droplet's terminal velocity determines its probability of fallout per unit time, and (1) the droplets are well-mixed by turbulence, The loss rate due to fallout that we used assumes that We neglected the effects of droplet curvature and solute on the droplet growth rate. We derived analytic equilibrium DSDs and probability density functions (PDFs) of droplet radius and squared radius for conditions that could occur in such a turbulent cloud chamber when there is uniform supersaturation. When aerosols (cloud condensation nuclei) are injected into the chamber at a constant rate,Īnd the rate of droplet activation is balanced by the rate of droplet loss,Īn equilibrium droplet size distribution (DSD) can be achieved. Supersaturation is produced by isobaric mixing. In a laboratory cloud chamber that is undergoing Rayleigh–Bénard convection,
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