Solution of the general dynamic equation along approximate fluid trajectories generated by the method of moments

Abstract

We consider condensing flow with droplets that nucleate and grow, but do not slip with respect to the surrounding gas phase. To compute the local droplet size distribution, one could solve the general dynamic equation and the fluid dynamics equations simultaneously. To reduce the overall computational effort of this procedure by roughly an order of magnitude, we propose an alternative procedure, in which the general dynamic equation is initially replaced by moment equations complemented with a closure assumption. The key notion is that the flow field obtained from this so-called method of moments, i.e., solving the moment equations and the fluid dynamics equations simultaneously, approximately accommodates the thermodynamic effects of condensation. Instead of estimating the droplet size distribution from the obtained moments by making assumptions about its shape, we subsequently solve the exact general dynamic equation along a number of selected fluid trajectories, keeping the flow field fixed. This alternative procedure leads to fairly accurate size distribution estimates at low cost, and it eliminates the need for assumptions on the distribution shape. Furthermore, it leads to the exact size distribution whenever the closure of the moment equations is exact.

Comparison between solutions of the general dynamic equation and the kinetic equation for nucleation and droplet growth

“…͑22͒ is commonly used. In those cases the problem at r ‫ء‬ does not occur due to a time-dependent critical radius, 9 for example. If the critical radius is constant, the GDE therefore cannot be used with the above delta function.…”

“…Hagmeijer et al 9 applied the GDE to a condensing flow through a nozzle. In this study, the GDE was also used for droplets smaller than critical, while new droplets were still inserted at the critical size.…”

Efficient solution methods for N-component condensation

“…The algorithm is an extension of the Phase Path Analysis (PPA) algorithm derived for unary systems [20]. It is shown that the resulting cluster size distribution, which has a Dirac δ-function character, agrees well in integral sense with the solution of the NBD equations.…”

“…Applications can be found in the fields of condensing vapors [20], cavitation [54], crystallization [18,61], ferromagnetics [57], aerosol and atmospheric science [17], combustion science [41] and chemical processes [26]. In this thesis we will focus on the vapor to liquid phase transition.…”

“…[20] the so-called Phase Path Analysis (PPA) algorithm has been derived for an advected single-component vapor. In the present section we extend the method to a non-advected N-component vapor.…”