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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Chamber

- Measurement of the flow velocity using a Pitot tube,
- Measurement of the pressure in the primary electron beam path,
- Temperature measurements in the supersonic flow region using a thermocouple,
- Analysis of the supersonic flow by the Schlieren optical method.

## 3. Flow Analyzes in the Experimental Chamber

## 4. Static Pressure Measurement According to Prandtl’s Theory

_{0}is the input pressure, p

_{v}is the output pressure, T

_{0}is the input temperature, T

_{v}is the output temperature, v

_{0}is the input velocity, v

_{v}is the output velocity, v

_{kr}is the critical velocity, ρ

_{0}is the input density, ρ

_{v}is the output density, M is the Mach number, $\varkappa $ is the gas constant = 1.14, A is the computational cross-section and A

_{kr}is the critical cross-section.

_{o}= 2000 Pa and p

_{v}= 100 Pa are based on the given relations and the results are shown in Table 1:

^{−1}:

_{0}= 297.15 K.

_{v}/ρ

_{0}= 0.0617. Then, it is possible to determine the value of output density ρ

_{v}= 0.00276 kg·m

^{−3}.

_{v}, ρ

_{0}, ρ

_{v}and T

_{v}were used as control values for the results obtained with Ansys Fluent (Table 2).

_{v}/ρ

_{kr}, which is shown in Table 1, the following applies:

_{kr}= 2 mm, the value A

_{kr}= 3.14 mm

^{2}. From relation 10, the calculated cross-section is equal to A

_{v}= 9.1 mm

^{2}, and therefore D

_{v}= 3.4 mm. The angle of 12° was determined according to [29].

## 5. Use of a Membrane Pressure Difference Sensor for Low Pressures Measurements

- For probe J: differential pressure value 4000 Pa.
- For probe I: differential pressure value 400 Pa.
- For probes A–H: differential pressure value 160 Pa.

- For the range below 600 Pa ± 0.5% of the range.
- For the range above 600 Pa ± 1% of the range.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Tihlaříková, E.; Neděla, V.; Dordevic, B. In-situ preparation of plant samples in ESEM for energy dispersive x-ray microanalysis and repetitive observation in SEM and ESEM. Sci. Rep.
**2019**, 9, 1–8. [Google Scholar] [CrossRef] [Green Version] - Neděla, V.; Konvalina, I.; Oral, M.; Hudec, J. The Simulation of Energy Distribution of Electrons Detected by Segmental Ionization Detector in High Pressure Conditions of ESEM. Microsc. Microanal.
**2015**, 21, 264–269. [Google Scholar] [CrossRef] [Green Version] - Jirák, J.; Neděla, V.; Černoch, P.; Čudek, P.; Runštuk, J. Scintillation SE detector for variable pressure scanning electron microscopes. J. Microsc.
**2010**, 239, 233–238. [Google Scholar] [CrossRef] [PubMed] - Neděla, V.; Tihlaříková, E.; Hřib, J. The Low-Temperature Method for Study of Coniferous Tissues in the Environmental Scanning Electron Microscope. Microsc. Res. Tech.
**2015**, 78, 13–21. [Google Scholar] [CrossRef] [PubMed] - Neděla, V.; Hřib, J.; Vooková, B. Imaging of early conifer embryogenic tissues with the environmental scanning electron microscope. Biol. Plant.
**2012**, 56, 595–598. [Google Scholar] [CrossRef] - Schenkmayerová, A.; Bučko, M.; Gemeiner, P.; Treľová, D.; Lacík, I.; Chorvát Jr., D.; Ačai, P.; Polakovič, M.; Lipták, L.; Rebroš, M.; et al. Physical and Bioengineering Properties of Polyvinyl Alcohol Lens-Shaped Particles Versus Spherical Polyelectrolyte Complex Microcapsules as Immobilisation Matrices for a Whole-Cell Baeyer–Villiger Monooxygenase. Appl. Biochem. Biotechnol.
**2014**, 174, 1834–1849. [Google Scholar] [CrossRef] - Neděla, V.; Tihlaříková, E.; Maxa, J.; Imrichová, K.; Bučko, M.; Gemeiner, P. Simulation-based optimisation of thermodynamic conditions in the ESEM for dynamical in-situ study of spherical polyelectrolyte complex particles in their native state. Ultramicroscopy
**2020**, 211, 112954. [Google Scholar] [CrossRef] - Maxa, J.; Neděla, V.; Jirák, J.; Vyroubal, P.; Hladká, K. Analysis of gas flow in a secondary electron scintillation detector for ESEM with a new system of pressure limiting apertures. Adv. Mil. Technol.
**2012**, 7, 111–116. [Google Scholar] - Maxa, J.; Bílek, M.; Hlavatá, P.; Vyroubal, P.; Lepltová, K. Comparisons Using Methods of Continuum Mechanics and Monte Carlo at Differentially Pumped Chamber. Adv. Mil. Technol.
**2016**, 11, 143–150. [Google Scholar] [CrossRef] [Green Version] - Danilatos, G.D. Velocity and ejector-jet assisted differential pumping: Novel design stages for environmental SEM. Micron
**2012**, 43, 600–611. [Google Scholar] [CrossRef] - Thevenin, D.; Janiga, D. Optimization and Computational Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Moran, M.; Shapiro, H. Fundamentals of Engineering Thermodynamics, 3rd ed.; John Wiley & Sons, Inc.: New York, NY, USA, 1996. [Google Scholar]
- Baehr, H. Thermodynamik, 14th ed.; Springer: Berlin, Germany, 2009. [Google Scholar]
- Bilek, M.; Maxa, J.; Hlavata, P.; Bayer, R. Modeling and simulation of a velocity field within supersonic flows in low-pressure areas. ECS Trans.
**2017**, 81, 311–316. [Google Scholar] [CrossRef] - Vyroubal, P.; Maxa, J.; Neděla, V.; Jirák, J.; Hladká, K. Apertures with Laval Nozzle and Circular Orifice in Secondary Electron Detector for Environmental Scanning Electron Microscope. Adv. Mil. Technol.
**2013**, 8, 59–69. [Google Scholar] - Neděla, V.; Tihlaříková, E.; Runštuk, J.; Hudec, J. High-efficiency detector of secondary and backscattered electrons for low-dose imaging in the ESEM. Ultramicroscopy
**2018**, 184, 1–11. [Google Scholar] [CrossRef] [PubMed] - Maxa, J.; Neděla, V. The Impact of Critical Flow on the Primary Electron Beam Passage through Differentially Pumped Chamber. Adv. Mil. Technol.
**2011**, 6, 39–46. [Google Scholar] - Dejč, M.J. Technická Dynamika Plynů; SNTL: Prague, Czechoslovak Republic, 1967. [Google Scholar]
- Uruba, V. Turbulence. Available online: http://www2.it.cas.cz/~uruba/docs/Aero/Turbulence_45.pdf (accessed on 12 October 2021).
- Roy, S.; Raju, R. Modeling gas flow through microchannels and nanopores. J. Appl. Phys.
**2003**, 93, 4870–4879. [Google Scholar] [CrossRef] - Van Eck, H.J.N.; Koppers, W.R.; van Rooij, G.; Goedheer, W.J.; Engeln, R.; Schram, D.C.; Cardozo, N.J.L.; Kleyn, A.W. Modeling and experiments on differential pumping in linear plasma generators operating at high gas flows. J. Appl. Phys.
**2009**, 105, 063307. [Google Scholar] [CrossRef] [Green Version] - Liu, Q.; Feng, X.-B. Numerical Modelling of Microchannel Gas Flows in the Transition Flow Regime Using the Cascaded Lattice Boltzmann Method. Entropy
**2020**, 22, 41. [Google Scholar] [CrossRef] [Green Version] - Danilatos, G. Optimum beam transfer in the environmental scanning electron microscope. J. Microsc.
**2009**, 234, 26–37. [Google Scholar] [CrossRef] - Danilatos, G.D. Figure of merit for environmental SEM and its implications. J. Microsc.
**2011**, 244, 159–169. [Google Scholar] [CrossRef] - Danilatos, G.; Rattenberger, J.; Dracopoulos, V. Beam transfer characteristics of a commercial environmental SEM and a low vacuum SEM. J. Microsc.
**2011**, 242, 166–180. [Google Scholar] [CrossRef] - Škorpík, J. Proudění Plynů a par Tryskami. Available online: https://www.transformacni-technologie.cz/ (accessed on 12 October 2021).
- Salga, J.; Hoření, B. Tabulky Proudění Plynu; UNOB: Brno, Czech Republic, 1997. [Google Scholar]
- Bayer, R.; Maxa, J.; Šabacká, P. Energy Harvesting Using Thermocouple and Compressed Air. Sensors
**2021**, 21, 6031. [Google Scholar] [CrossRef] [PubMed] - Daněk, M. Aerodynamika a Mechanika Letu; VVLŠ SNP: Košice, Slovakia, 1990; p. 83. [Google Scholar]
- Danilatos, G.D. Environmental scanning electron microscopy and microanalysis. Mikrochim. Acta
**1994**, 114, 143–155. [Google Scholar] [CrossRef] - Choi, E.; Kim, S.; Gong, J.; Sun, H.; Kwon, M.; Seo, H.; Sul, O.; Lee, S.-B. Tactile Interaction Sensor with Millimeter Sensing Acuity. Sensors
**2021**, 21, 4274. [Google Scholar] [CrossRef] [PubMed] - Kasai, M.; Sasaki, D.; Nagata, T.; Nonomura, T.; Asai, K. Frequency Response of Pressure-Sensitive Paints under Low-Pressure Conditions. Sensors
**2021**, 21, 3187. [Google Scholar] [CrossRef] [PubMed] - Drexler, P.; Čáp, M.; Fiala, P.; Steinbauer, M.; Kadlec, R.; Kaška, M.; Kočiš, L. A Sensor System for Detecting and Localizing Partial Discharges in Power Transformers with Improved Immunity to Interferences. Sensors
**2019**, 19, 923. [Google Scholar] [CrossRef] [Green Version]

**Figure 6.**Laval nozzle—characteristic method. Reprinted from ref. [26].

**Figure 7.**Laval nozzle—Bell nozzle. Reprinted from ref. [26].

**Figure 8.**Laval nozzle—Linear shape. Reprinted from ref. [26].

**Figure 10.**Pressure and Mach number on the path shown in Figure 3.

**Figure 12.**Comparison of static pressure on the axis and oblique surface of the nozzle according to Prandtl’s theory.

**Table 1.**Calculated values in critical input cross-section [28].

Mach Number | Output Velocity/Critical Velocity | Output Velocity/Input Velocity | Output Temperature/Input Temperature | Output Pressure/Input Pressure | Output Density/Input Density | Output Density/Critical Density |
---|---|---|---|---|---|---|

M_{v} | v_{v}/v_{kr} | v_{v}/v_{0} | T_{v}/T_{0} | p_{v}/p_{0} | ρ_{v}/ρ_{0} | ρ_{v}/ρ_{kr} |

2.6 | 1.8571 | 0.6521 | 0.4252 | 0.05 | 0.1179 | 0.3453 |

Theoretical Value | Ansys Fluent Value | |
---|---|---|

Mach number (-) | 2.6 | 2.598 |

Density (kg·m^{−3}) | 0.00276 | 0.00265 |

Velocity (m·s^{−1}) | 585.8 | 582 |

Temperature (°C) | 126.3 | 126.8 |

PROBE | A | B | C | D | E | F | G | H | I | J |
---|---|---|---|---|---|---|---|---|---|---|

EXPECTED PRESSURE DIFFERENCE (Pa) | 0–5 | 0–5 | 0–5 | 30 | 37 | 44 | 61 | 84 | 193 | 1451 |

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Šabacká, P.; Neděla, V.; Maxa, J.; Bayer, R.
Application of Prandtl’s Theory in the Design of an Experimental Chamber for Static Pressure Measurements. *Sensors* **2021**, *21*, 6849.
https://doi.org/10.3390/s21206849

**AMA Style**

Šabacká P, Neděla V, Maxa J, Bayer R.
Application of Prandtl’s Theory in the Design of an Experimental Chamber for Static Pressure Measurements. *Sensors*. 2021; 21(20):6849.
https://doi.org/10.3390/s21206849

**Chicago/Turabian Style**

Šabacká, Pavla, Vilém Neděla, Jiří Maxa, and Robert Bayer.
2021. "Application of Prandtl’s Theory in the Design of an Experimental Chamber for Static Pressure Measurements" *Sensors* 21, no. 20: 6849.
https://doi.org/10.3390/s21206849