Equivalence of two different integral representations of droplet distribution moments in condensing flow

Hagmeijer, Rob

doi:10.1063/1.1630052

Cited by 12 publications

(17 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been demonstrated in Ref. 24, however, that both formulations are equivalent and, moreover, consistent with the GDE. The remaining integral can be further evaluated:…”

Section: Moment Equationsmentioning

confidence: 58%

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

2005

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…It has been demonstrated in Ref. 24, however, that both formulations are equivalent and, moreover, consistent with the GDE. The remaining integral can be further evaluated:…”

Section: Moment Equationsmentioning

confidence: 58%

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

2005

Self Cite

View full text Add to dashboard Cite

show abstract

“…To include nucleation in the GDE, usually a source term is added to Eq. ͑19͒ in the form of a delta function, 2,17 so that the GDE becomes…”

Section: General Dynamic Equationmentioning

confidence: 99%

“…Hagmeijer 17 presented the general solution of such a GDE in which J, ṙ, and r ‫ء‬ may be time dependent. If these parameters are constant in time, as they are here, a solution can be obtained in a more straightforward way, as we will now show.…”

Section: General Dynamic Equationmentioning

confidence: 99%

See 1 more Smart Citation

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

Holten

Dongen²

2009

The Journal of Chemical Physics

View full text Add to dashboard Cite

A comparison is made between two models of homogeneous nucleation and droplet growth. The first is a kinetic model yielding the master equations for the concentrations of molecular clusters. Such a model does not make an explicit distinction between nucleation and droplet growth. The second model treats nucleation and growth separately, fully ignoring stochastic effects, and leads to the continuous general dynamic equation (GDE). Problems in applying the GDE model are discussed. A numerical solution of the kinetic equation is compared with an analytic solution of the GDE for two different cases: (1) the onset of nucleation and (2) the nucleation pulse. The kinetic model yields the thickness of the condensation front in size space as a function of supersaturation and dimensionless surface tension. If the GDE is applied properly, solutions of the GDE and the kinetic equation agree, with the exception of very small clusters, near-critical clusters, and the condensation front.

show abstract

“…The approximation of g(n), equation (20), is denoted bỹ gðnÞ, and by introducing a similar construction as in equation (31), we define the error 2 as 2 eg À e g eg þ e g ð34Þ Equation (34) needs to be evaluated for the problem at hand, whereas equation (32) is problem independent. Both errors are depicted in Figure 2 and a root-mean-square (RMS) averaged error, "…”

Section: General Dynamic Equationmentioning

confidence: 99%

A comparison of cluster size distribution models in isothermal single component condensation

Putten

Sidin

Hagmeijer

2013

Proceedings of the Institution of Mechanical Engineers, Part A:

Self Cite

View full text Add to dashboard Cite

A review of single component condensation models is presented. The most exact model for condensation is the set of Becker-Dö ring equations. The first step toward reduced models is obtained by performing a Taylor series expansion on the discrete Becker-Dö ring equations, resulting in the Fokker-Planck equation. Further simplification of the FokkerPlanck equation leads to the stationary diffusion flux model, which is based on a steady state assumption in the region of small clusters. Discarding the small cluster size region results in the general dynamic equation, which is computationally the most attractive model. All condensation models are aimed at solving the cluster size distribution and differ in their ability to capture the physical process and their computational complexity. The assumptions needed to construct these reduced models are explained and the influences on the physical outcome are addressed. The models are applied to a nucleation pulse experiment and the computational times are compared.

show abstract

Equivalence of two different integral representations of droplet distribution moments in condensing flow

Cited by 12 publications

References 13 publications

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

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

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

A comparison of cluster size distribution models in isothermal single component condensation

Contact Info

Product

Resources

About