6 The Investigation of the Evolution of Cluster Beam …
71
CNT uses the macroscopic and equilibrium approaches to describe the nonequilibrium process that occurs in most cases on a nanometer scale. Important CNT results
are models of nucleation rate, droplet size growth rate, and critical nucleus size.
Recent experimental and computational works suggest that CNT does not
adequately describe a nucleation process. Potential sources of errors include: (a)
the use of the so-called “capillary approximation” that is the use of the concept of
“surface tension” and sphericity for small clusters, (b) the incorrect determination of
some macroscopic quantities (densities) in small clusters, (c) not taking into account
nonisothermal processes (droplet overheating) during condensation [19–21].
Recently, many approaches to modernizing the classical theory of nucleation
have been suggested [22–28]. For example, in [22, 23], the modified relations were
proposed for the nucleation rate and droplet size growth rate. These models contain
some coefficients that must be selected for each case under consideration. As a
reference model for the selection of these coefficients, the authors considered a
semi-analytical approximation proposed by Hagena and Obert [29, 30].
Using the method of moments, the chapter investigates the processes of condensation–evaporation during the flow of an argon jet from a micro-nozzle into a vacuum
chamber and the argon flow in the nozzle-jet-skimmer system.
Hereinafter, in Sect. 6.2, we consider the mathematical formulation of the model
problem. Section 6.3 extends the possibilities of MM due to the Hagena’s theory.
Section 6.4 contains the results of the numerical modeling. Section 6.5 concludes
the chapter.
6.2 Mathematical Model
The propagation of viscous heat-conducting condensing argon vapor in a nozzleskimmer system is considered. In continuum models, the nucleation function is used
to describe the process of cluster formation and further growth of droplets, their
growth rate determined from the ratio of partial pressure to vapor pressure. To simulate evaporation, the denucleation function is used in MM instead of the nucleation
function [31].
For high concentrations of condensing gas, we can assume that the droplet temperature and medium temperature are the same [32]. In this case, two parameters will
have the main effect on condensation: accommodation coefficient α and droplet
growth rate β. Each of these parameters can be selected for a specific gas. Previously, the authors considered the case of condensation of water vapor [14, 33] in the
experiment [22]. It was shown that the acceptable values of the coefficients are α =
1, β = 0.1.
Section 6.2.1 describes the complete system of equations. Section 6.2.2 realizes
method of moments. Section 6.2.3 contains the thermodynamic relations for closing
of the equations.
71
CNT uses the macroscopic and equilibrium approaches to describe the nonequilibrium process that occurs in most cases on a nanometer scale. Important CNT results
are models of nucleation rate, droplet size growth rate, and critical nucleus size.
Recent experimental and computational works suggest that CNT does not
adequately describe a nucleation process. Potential sources of errors include: (a)
the use of the so-called “capillary approximation” that is the use of the concept of
“surface tension” and sphericity for small clusters, (b) the incorrect determination of
some macroscopic quantities (densities) in small clusters, (c) not taking into account
nonisothermal processes (droplet overheating) during condensation [19–21].
Recently, many approaches to modernizing the classical theory of nucleation
have been suggested [22–28]. For example, in [22, 23], the modified relations were
proposed for the nucleation rate and droplet size growth rate. These models contain
some coefficients that must be selected for each case under consideration. As a
reference model for the selection of these coefficients, the authors considered a
semi-analytical approximation proposed by Hagena and Obert [29, 30].
Using the method of moments, the chapter investigates the processes of condensation–evaporation during the flow of an argon jet from a micro-nozzle into a vacuum
chamber and the argon flow in the nozzle-jet-skimmer system.
Hereinafter, in Sect. 6.2, we consider the mathematical formulation of the model
problem. Section 6.3 extends the possibilities of MM due to the Hagena’s theory.
Section 6.4 contains the results of the numerical modeling. Section 6.5 concludes
the chapter.
6.2 Mathematical Model
The propagation of viscous heat-conducting condensing argon vapor in a nozzleskimmer system is considered. In continuum models, the nucleation function is used
to describe the process of cluster formation and further growth of droplets, their
growth rate determined from the ratio of partial pressure to vapor pressure. To simulate evaporation, the denucleation function is used in MM instead of the nucleation
function [31].
For high concentrations of condensing gas, we can assume that the droplet temperature and medium temperature are the same [32]. In this case, two parameters will
have the main effect on condensation: accommodation coefficient α and droplet
growth rate β. Each of these parameters can be selected for a specific gas. Previously, the authors considered the case of condensation of water vapor [14, 33] in the
experiment [22]. It was shown that the acceptable values of the coefficients are α =
1, β = 0.1.
Section 6.2.1 describes the complete system of equations. Section 6.2.2 realizes
method of moments. Section 6.2.3 contains the thermodynamic relations for closing
of the equations.