Application of Prandtl’s Theory in the Design of an Experimental Chamber for Static Pressure Measurements

Abstract

Pumping in vacuum chambers is part of the field of environmental electron microscopy. These chambers are separated from each other by a small-diameter aperture that creates a critical flow in the supersonic flow regime. The distribution of pressure and shock waves in the path of the primary electron beam passing through the differentially pumped chamber has a large influence on the quality of the resulting microscope image. As part of this research, an experimental chamber was constructed to map supersonic flow at low pressures. The shape of this chamber was designed using mathematical–physical analyses, which served not only as a basis for the design of its geometry, but especially for the correct choice of absolute and differential pressure sensors with respect to the cryogenic temperature generated in the supersonic flow. The mathematical and physical analyses presented here map the nature of the supersonic flow with large gradients of state variables at low pressures at the continuum mechanics boundary near the region of free molecule motion in which the Environmental Electron Microscope and its differentially pumped chamber operate, which has a significant impact on the resulting sharpness of the final image obtained by the microscope. The results of this work map the flow in and behind the Laval nozzle in the experimental chamber and are the initial basis that enabled the optimization of the design of the chamber based on Prandtl’s theory for the possibility of fitting it with pressure probes in such a way that they can map the flow in and behind the Laval nozzle.

Mathematical and physical analysis of the character and position of shock waves during pumping of vacuum chambers

“…These equations are valid in the region of Knudsen number up to 10 -3 as seen in Figure 2. Previous analyses have been performed confirming that in the simulations we are dealing with we are still in the region where the Stock Navier equations are valid [20,21]. Comparative analyses have also been performed with published results of Dr. Danilatos, who used the Monte Carlo method for the case of differentially pumped chamber analysis, proving the agreement [3].…”

Mathematical-Physics Analyses of the Nozzle Shaping at the Aperture Gas Outlet into Free Space under ESEM Pressure Conditions

“…The same observation is made from speed, temperature, and surface pressure determinations [14,15]. His next work has established Sensors 2024, 24, 3436 2 of 26 that a thin-plate PLA assembly produces the minimum electron beam current loss at the high-vacuum-high-pressure boundary. This was expected because the thin PLA walls result in the most abrupt change in pressure, which generates the steepest gas density decrease along the axis.…”

Comparative Analysis of Supersonic Flow in Atmospheric and Low Pressure in the Region of Shock Waves Creation for Electron Microscopy

“…Subsequently, an analysis of this nozzle was performed on the previously tuned Ansys Fluent system [ 24 ], and the results were evaluated regarding the critical flow theory. This combination of theory and mathematical-physics analyses is one of the great advantages of modern research methodology [ 29 , 30 , 31 ].…”