1. Introduction

2. Materials and Methods

3. Results and Discussion

3.1. Analytical and numerical modeling

a) 

Finite element mathematical modeling

The developed mathematical model applies to dynamic systems with a finite number of degrees of freedom through the analytical formulation of structure dynamics with added damping. The analytical model of this formulation (the matrix differential equation of vibrational motion) is the one from classical dynamic analysis. In this context, loads or loading conditions vary over time and are applied instantaneously. Dynamic loads involve oscillating weights, impacts, collisions, and unpredictable amounts, and therefore, this case it supposes involve [20]:

➢

transient dynamic evaluation, employed to calculate the feedback of a structure to external loads that fluctuate unpredictably over the period.

In dynamic analysis, the matrix equations for force equilibrium are applied to a dynamic system [20], [21]:

- for a structure without foreign load:

$$\left[\mathrm{M}\right]\mathrm{x}\left[\ddot{X}\right(\mathrm{t}\left)\right]+\left[\mathrm{C}\right]\mathrm{x}\left[\dot{X}\right(\mathrm{t}\left)\right]+\left[\mathrm{K}\right]\mathrm{x}\left[\mathrm{X}\right(\mathrm{t}\left)\right]=0$$

(1)

- for a system with an external load:

$$\left[\mathrm{M}\right]\text{·}\left[\ddot{X}\left(t\right)\right]+\left[\mathrm{C}\right]\text{·}\left[\dot{X}\left(t\right)\right]+\left[\mathrm{K}\right]\text{·}\left[X\left(t\right)\right]=\left[F\left(t\right)\right]$$

(2)

where: M - mass; $\ddot{X}$ - acceleration; C - Rayleigh damping; $\dot{X}$- velocity; K - stiffness; X – displacement and F – load (all variabels are in matrical form), t - time.

By solving the equations (1) and (2), we can obtain the natural frequencies of a structure. The types of loads used in a static analysis are the same as those of a dynamic. The expected results from the software include natural frequencies, displacements, deformations, and stresses. All these outcomes can also be acquired in the total deformation, where δ_t is a scalar quantity, and:

$${\delta }_t=\sqrt{{\delta }_x^2+{\delta }_y^2+{\delta }_z^2},$$

(3)

where δ_x,y,z - the components of deformation along coordinate axes can be obtained in either global or local coordinates.

The associated differential equation has the form [15]:

$${\text{𝐌}}^{+}·\ddot{X}\left(\mathrm{t}\right)+{\mathbf{C}}^{+}·\dot{X}\left(\mathrm{t}\right)+{\mathbf{K}}^{+}·X\left(\mathrm{t}\right)={\mathbf{F}}^{+}\left(\mathrm{t}\right),$$

(4)

in which the matrices and vectors are specific to the dynamic model with added mass (M⁺) connected to the primary system through elastic connections (stiffness coefficient K⁺), damping connections (damping coefficient C⁺) and F⁺ - appropriate load.

b) 

Numerical modeling

Here is presented an approach to mathematical modeling established on the utilization of computational instruments for numerical simulations [22]. The process involves transitioning from problem formulation and equation establishment to the implementation of computational algorithms and analysis of results.

Due to the factors depicted beyond an evaluation of the pumping time should fundamentally differ for evacuating a vessel in the rough vacuum region compared to evacuating in the regions of medium and high vacuum.

In the case of gas evacuation of a vessel in gross vacuum mode (excluding supplementary quantities of gas or vapor), the effective pumping speed, s_ef of the pump-vacuum chamber assembly, depends only on the required pressure, p (after time t), on the volume, V of the chamber and the pumping time, t.

Accompanied by an invariable pumping speed, s_ef and supposing that the maximum pressure reached, p_f (the final/desired pressure) by the chosen pump model, is such that p_f << p, the pressure drop over time, p(t) in a vessel of vacuum is provided by the differential equation of the first order [2]:

$$- \frac{dp}{dt}=\frac{s_{ef}}{V}\text{·}p\mathrm{or}\frac{dp}{p}=- \frac{s_{ef}}{V}dt,$$

(5)

which by integration, considering that the pressure varies from the initial pressure, p₀ = 1,003 mbar at the time t = 0 to a minimum value, p (after the time, t), then the effective pumping speed, s_ef could be estimated as a function of the pumping time, t from equation/relation (5) as follows:

$${{\displaystyle \int }}_{1003}^p\frac{dp}{p}=- \frac{s_{ef}}{V}·t=>ln\frac{1003}{p}=-\frac{s_{ef}}{V}·t,$$

(6)

where from result:

$$s_{ef}=\frac{V}{t}·ln\frac{1003}{p}.$$

(7)

Noting $ln\frac{1003}{p}= \sigma$ - the dimensionless pressure factor and substituting in equation/relation (7) we obtain that the association amongst the effective speed, s_ef and the pumping duration, t, becomes:

$$s_{ef}=\frac{V}{t }·\sigma .$$

(8)

The ratio (V/s_ef) = τ, is commonly defined as a time invariant. Consequently, the pumping duration, t of a vacuum vessel from atmospheric load to a value of pressure p, will be:

(9)

The dependency of the σ parameter on the wished-for pressure is presented in Figure 5. It must be mentioned that the pumping speed of ordinary pumps drops less than 10 mbar with gas ballast and less than 1 mbar without gas heft. This elementary attitude varies for pumps of different capacities, but it is recommended to not be overlooked in determining the pumping time relying on the pump size. It is emphasized that equations (6 to 9) and also Figure 5 apply only when the final pressure reached alongside the pump applied is several orders of amplitude lower than the wanted pressure.

In the gross vacuum regime, the capacity of the chamber is determined for the duration engaged in the pumping routine. In the high and ultrahigh vacuum areas, the release of gases from the walls (of the corn seeds and the vessel) exhibits a significant function, in the medium vacuum region, the evacuating process is affected by both quantities. Additionally, in the medium vacuum zone, especially in the situation of rotary pumps, the maximum pressure that can be reached is no insignificant.

If the amount of gas accessing the chamber is established to be Q (in mbar·l/s) from the gas release from the seed enclosures, the chamber, and the leaks, the differential equation (5) for the pumping operation changes into [2]:

$$\frac{dp}{dt}=-\frac{s_{ef}\left(p-p_f\right)-Q}{V},$$

(10)

and through the integration of this equation (11), we obtain:

$$t=\frac{V}{s_{ef}}ln\frac{\left(p_0-p_f\right)-Q/s_{ef}}{\left(p-p_f\right)-Q/s_{ef}}$$

(11)

Unlike equation/relation (7), equation/relation (11) does not allow for a definition of the solution for the effective pumping velocity, s_ef; therefore, the s_ef for a well-known gas release cannot be obtained derived from the pressure drop curve over time without additional knowledge, Figure 6.

Therefore, in usage, the strategy will necessitate a pump with enough upward pumping speed, calculated from equation/relation (7) as an outcome of the volume of the gas-free chamber and the wanted pumping duration. Instead, the ratio Q/s_ef between the gas release velocity and this pumping speed is established. This ratio must be lesser than the needed pressure; for protection, it should be approximately ten times smaller. If the corresponding situation is not met, a pump with a comparable elevated pumping speed must be selected. In a situation where the pumping process is dominated by residual gas, pumping in a high vacuum region can be described by the relation [2]:

$$p= p_0·exp\left[-\frac{\left(s_{ef}/V_t\right)}{t}\right],$$

(12)

where: V_t - is the total volume of the system.

However, by far, the most significant uncertainty associated with pump performance, pressure, flow, and external leaks is due to gas release. Gas release rates can differ by several orders of amplitude, according to the component of a surface, its external intervention, humidity, temperature, and the exposure duration to vacuum. As it usually approaches asymptotically to the final pressure of a system, even small changes in gas loads result in significant differences in evacuation times (see Figure 6). Vacuum analysis, which aided in selecting a roughing pump, relied solely on the relationship between pressure and the time to reach this pressure, assuming the following relevant hypotheses for such an analysis: the system has no leaks, the pump is 100% productive, and nothing will vaporize in the vacuum vessels.

The calculation application was developed using Microsoft Excel, but any equivalent software to Microsoft Office that includes a spreadsheet with similar features to Excel can be used, making the adaptation of the application relatively easy. The choice of an Excel spreadsheet to simulate the variation of the vacuum system pressure over time is due to its capability to perform numerical description (in a table) applying symbolic statement as thoroughly as visual description using tables constructed for this purpose. The skill to activate and explore numerical, symbolic, and visual description dynamically makes the spreadsheet an essential instrument for promoting conceptualization and algebraic reflection [22]. Excel uses the VBA (Visual Basic for Applications) language, which is now widely used. It should be emphasized that VBA is a complex programming language. With its help, data can be manipulated, complex tasks automated, interactions with other Office applications can be performed, and much more [23], [24].

Using a spreadsheet, we can generate the graph of the precedent response (12) that reveals the relationship between the initial time, t, and the gas load pressure in the vacuum chamber, p(t), Figure 7. At the aforementioned, it is also feasible to see the so-called state curves, which represent the parametric error of estimation based on experimental data at each moment of the system's operation.

Since the particular solution (12) is a function that depends on parameters, s_ef (pumping speed in steady-state), V (volume of the vacuum chamber), and implicitly by σ (the dimensionless pressure factor), as well as the initial pressure of the gas load, p₀, at the initial moment t₀ = 0, we will observe how graphic elements are constructed that can be interactively modified using sliders for the constructive parameters of the chamber, diameter (D), and height (H), Figure 7. The design of the spreadsheet itself is indispensable to the development of the calculation section.

It should be emphasized that VBA is a complex programming language. With its help, data can be manipulated, complex tasks automated, interactions with other Office applications performed, and much more [24], [25]. Consequently, the relevant VBA editor was used and adapted accordingly for processing and analyzing the results obtained from the simulation. The interface of the "Vacuum Chamber Pressure Variation Simulation.xls" program, on the "Pressure" page, is presented in Figure 8.

To identify the real or correct values, the absolute error (EA) was calculated. The real values were considered to be the mean values of the data resulting from multiple measurements in physical experiments with corn seeds. The measured average real value for the gas load pressure in the vacuum chamber was 44.273 mbar. The graph of the data obtained from measurements for the pressure variation over time is presented in Figure 9a. The estimated values of pressure over time for three different values of σ (dimensionless pressure factor), σ = 23.73; 27.05; 32.18, were calculated using the mathematical model adopted in the simulation program, and the data can be observed in Figure 9b. The average values of the estimated pressures for each of the three data sets were 51.031, 45.920, and 40.140 mbar, respectively, with the estimated average value for all three data sets being 45.698 mbar. For each measurement, the following formula is applied to obtain EA:

EA = | Estimated value - Real value |

(13)

The variation of EA for the three sets of estimated pressure data with the created simulation program is presented in Figure 9c. The trend of the error variation is similar for the three curves shown, with a peak at the moment t = 0.33 s. The highest peak value of 132.472 mbar was recorded in the dataset with σ = 23.73, while the lowest peak value of 21.269 mbar is encountered in the curve with σ = 32.18, and for σ = 27.05, the peak has a value of 85.035 mbar.

It is observed for all three curves that after approximately 2 minutes, the error value significantly decreases, reaching a steady-state value of about 5.161 mbar.

Then, if there are multiple measurements, errors can be summed to obtain the total EA or the average of absolute errors can be calculated by dividing the sum of absolute errors by the number of measurements. The calculation of the average EA for the estimated pressure with the simulation model was performed using the relation:

Average: EA = (EA₁ + EA₂ + EA₃)/3 = (6.758+1.651+4.133)/3 = 1.425 mbar,

(14)

Therefore, a very small value justifies the correlation between the results obtained through simulation and the real ones.

5. Conclusions

References

1. Jousten K., Handbook of Vacuum Technology, Second Edition, Wiley-VCH Verlag GmbH&Co.KGaA, 2016, Weinheim, Germany.
2. Umrath W., Fundamentals of Vacuum Technology, Oerlikon Leybold Vacuum, 2007, Cologne, Germany.
3. *** The Vacuum Technology Book & Know how Book, Volume II, Pfeiffer Vacuum GmbH, 2013, www.pfeiffer-vacuum.com.
4. Silverio G-L., Chuck-Hernandez C., Serna-Saldivar S.O., Development and Structure of the Corn Kernel, Corn (Third Edition)- Chemistry and Technology 2019, 147–163. [CrossRef]
5. Govindaraj M, Masilamani P, Albert VA, Bhaskaran M. Effect of physcal seed treatment on yield and quality of crops: A review. Agriculture Reviews 2017, 38(1), 1-14. [CrossRef]
6. Farooq M.A, Ma W., Shen S., Gu A., Underlying Biochemical and Molecular Mechanisms for Seed Germination, Int. J. Mol. Sci. 2022, 23, 8502. [CrossRef]
7. Ramdan E.P., Perkasa A.Y., Azmi T.K.K., Aisyah, Kurniasih R., Kanny P.I., Risnawati, Asnur P., Effects of physical and chemical treatments on seed germination and soybean seed-borne fungi, IOP Conf. Series: Earth and Environmental Science 2021, 883,012022. [CrossRef]
8. Sonhaji YS, Surahman M, Ilyas S, Giyanto, Seed treatment improved seed quality, seed production, and controlled downey mildew disease on sweet corn, J. Agron. Indonesia 2013, 41(3), 242-248, http://repository.ipb.ac.id/handle/123456789/67027.
9. Deepa G.T., Chetti M., Khetagoudar M., Adavirao G., Influence of vacuum packaging on seed quality and mineralcontents in chilli (Capsicum annuum L.), J Food Sci Technol. 2013, 50(1),153–158. [CrossRef]
10. Ramdan E.P., Arti I.M., Risnawati, Evaluation of viability and corn seed-borne pathogens in physical and chemical treatment, J. Berkala Penelitian Agronomi 2020, 8(2),16-24.
11. Lawrence B., Bicksler A., Duncan K., Local treatments and vacuum sealing as novel control strategiesfor stored seed pests in the tropics, Agron. Sustain. Dev. 2017, 37, Article number 6. [CrossRef]
12. Arbaugh B., Rezaei F., Mohiti-Asli M., Pena S., Scher H., Jeoh T., A Strategy for Stable, On-Seed Application of a Nitrogen-Fixing Microbial Inoculant by Microencapsulation in Spray-Dried Cross-linked Alginates, ACS Agric. Sci. Technol. 2022, 2, 5, 950–959,. [CrossRef]
13. Li L., Li J., Shao H., Dong Y., Effects of low-vacuum helium cold plasma treatment on seed germination, plant growth and yield of oilseed rape. Plasma Sci. Technol. 2018, 20, 095502. [CrossRef]
14. Kouki A., Theory of vacuum, 2022. [CrossRef]
15. Marquardt N., Introduction to the principles of vacuum physics, Inst for Accelerator Phys & Synchrotron Radiat, CAS - CERN Accelerator School: Vacuum Technology 1999, 1-24, e-proceedings. [CrossRef]
16. Lateş M-T., Finite element method. Applications, Transilvania University Publishing House Braşov - 2008 (in Romanian).
17. Butnariu S., Mogan Gh., Finite element analysis in mechanical engineering. Practical applications in ANSYS, Transilvania University Publishing House in Braşov - 2014 (in Romanian).
18. Jamil T., Azher K., Tahir M.A, Ali Z.,, Hameed W., Javed H.H., Experimental and numerical dynamic analysis of plexiglass acrylic for impact energy using indigenously developed testing equipment, U.P.B. Sci. Bull., Series D 2022, 84(3), 191-206.
19. Ipate G., Ilie F., Cristescu A.C., Finite element 3D numerical simulation study of car braking systems and brake disc/drum–pad/shoe friction couple materials, TE-RE-RD, E3S Web of Conferences 2020, 180, 03003. [CrossRef]
20. Tickoo S., Ansys Workbench 14.0: A Tutorial Approach, Cadcim Technologies 2014.
21. Florin Gabriel Blaga F.G., The dynamic action of the wind on multi-storey metal structures. PhD Thesis, Technical University of Cluj Napoca – 2022 (in Romanian).
22. Oliveira M.C., Nápoles, S. Using a Spreadsheet to Study the Oscillatory Movement of a Mass-Spring System, Spreadsheets in Education (eJSiE) 2010, 3(3), Art.2, http://epublications.bond.edu.au/ejsie/vol3/iss3/2.
23. Ed Bott Ed., Woody Leonhard W., Special Edition Using Microsoft Office 2003, Student-Teacher Edition 2002, Google Books.
24. Roman S., (2002). Writing Excel Macros with VBA, 2nd Edition. O'Reilly Media, Inc. 2002.
25. Cooksey R., Descriptive Statistics for Summarising Data, Illustrating Statistical Procedures: Finding Meaning in Quantitative Data 2020 PMCID: PMC7221239, 61–139. [CrossRef]