Semi-empirical Formula Research Articles

Frequency equation and semi-empirical mechanical coupling strength of microcantilevers in an array.

A characteristic equation for the frequencies of the T-shaped and overhang-shaped cantilevers is derived for the first time. We show that there are optimum values of the overhang lengths and widths that maximize the frequency and the number of maxima is corresponding to the mode number. The frequency of higher-order modes could be tuned by changing the overhang dimensions. Especially, a semi-empirical formula for the coupling strength between cantilevers in an array is proposed where the strength presents a cubic decrease with the overhang width and a linear increase with the overhang length , . There is a very good agreement between the proposed formula and the values obtained in recent experiments by other researchers.

The total and the partial charge-changing cross sections of 40Ar fragmentation on elemental targets at 500 A MeV

The total and the partial charge-changing cross sections of 40Ar fragmentation on CH2, C, Al, Cu and Pb targets at the highest energy of 495 A MeV were investigated using CR-39 plastic nuclear track detector. The total charge-changing cross-section for hydrogen target was calculated based on the results of polyethylene and carbon targets. It was found that the total charge-changing cross sections for fragmentation of 40Ar on H, C, Al, Cu and Pb targets were independent on the beam energy in our studied energies, and were consistent with the predictions of Bradt-Peters semi-empirical formula. Nilsen semi-empirical formula, NUCFRG2 and PHITS simulation codes given an underestimation of experimental data. The partial cross sections for projectile fragment production were independent on the beam energy in our studied energies for each target and do not show a significant odd-even effect. The partial cross sections for projectile fragment production were also compared with different empirical parametrization formulas, such as different version of Cummings, Nilsen, FRACS and EPAX3. Generally, EPAX3 and FRACS empirical parametrization can give a reasonable estimation of data within 25% relative errors.

Density, viscosity, surface tension, and spectral analysis of polyethylene glycol 300 + 1,2-Propylenediaminebinary liquid mixture

At T = 293.15 ∼ 318.15 K, the density (ρ), viscosity (η), and surface tension (γ) data of polyethylene glycol 300 (PEG300) and 1,2-propane diamine (1,2-PDA) binary liquid mixture in the entire composition range were systemically measured. Meanwhile, the influence of different temperatures on the physical properties of the binary liquid mixture was deeply discussed. The values of ρ and η of PEG300 were compared with the reported values of ρ and η of different linear alcohols under the same experimental conditions. It was found that the values of ρ and η of linear alcohols were related to the number of carbon chains. Under isothermal conditions, the values of ρ and η increased with the number of carbon chains. After that, excess molar volume (VmE), excess viscosity (ηE), surface tension deviation (Δγ), excess Gibbs energy of activation of viscous flow (ΔG∗E), and molar isobaric thermal expansion (Ep,m) of PEG (1) + 1,2-PDA (2) binary mixture were calculated. The η values were fitted with four semi-empirical formulas, and the VmE, ηE, and Δγ values were fitted with Redlich-Kister (R-K) equation. IR, UV, FLS, and Raman spectral technologies were used to further prove the interaction among different molecules of the liquid mixture.

Semi-empirical Formula for X-rays Interaction Cross Sections and Energies with Atoms

Semi-empirical Formula for X-rays Interaction Cross Sections and Energies with Atoms

An iterative transfer matrix approach for estimating the sound speed and attenuation constant of air in a standing wave tube.

In this work, an iterative method based on the four-microphone transfer matrix approach was developed for evaluating the sound speed and attenuation constant of air within a standing wave tube. When the air inside the standing wave tube is treated as the material under test, i.e., as if it were a sample of porous material, the transfer matrix approach can be used to identify the air's acoustic properties. The wavenumber within the tube is complex owing to the formation of a visco-thermal boundary layer on the inner circumference of the tube. Starting from an assumed knowledge of the air properties, an iterative method can be applied in the post-processing stage to estimate the complex wavenumber. Experimental results presented here show that although the results are sensitive to ambient temperature, a semi-empirical formula previously proposed by Temkin [(1981). Elements of Acoustics (John Wiley & Sons)] matches closely with the measured sound speed and attenuation constant, as does a theoretical formulation proposed by Lahiri et al. [(2014). J. Sound Vib. 333(15), 3440-3458]. Further, it is shown that the Temkin [(1981). Elements of Acoustics (John Wiley & Sons)] and Lahiri et al. [(2014). J. Sound Vib. 333(15), 3440-3458] predictions accurately represent the variation of sound speed with frequency, in contrast to the formula recommended in the ASTM E1050 standard [(2019). American Society for Testing and Materials], in which the sound speed is assumed to be independent of frequency.

Partially accreted crusts of neutron stars

Context. Neutron stars in low-mass binary systems are subject to accretion when material originating from the companion star accumulates on the surface. In most cases, the justified and common assumption in studying the properties of the neutron star crust is the fully accreted crust approximation. However, observations of some X-ray transient sources indicate that the original crust has not been completely replaced by accreted material, but is partly composed of the compressed original crust. Aims. The crust of an accreting neutron star beyond the fully accreted crust approximation was studied; a two-part (or hybrid) crust made of the original crust that is compressed and of the accreted material crashing onto it was reconstructed as a function of the accretion stage. The differences in the composition and energy sources for the fully accreted and hybrid crusts influence the cooling and transport properties. Methods. A simple semi-empirical formula of a compressible liquid drop was used to compute the equation of state and composition of the hybrid crust. Calculations were based on the single-nucleus model, with a more accurate treatment of the neutron drip point. We compared the nuclear reactions triggered by compression in the original crust and in the accreted matter part of the hybrid crust. We discuss another crust compression astrophysical phenomenon related to spinning neutron stars. Results. The compression of the originally catalyzed outer crust triggers exothermic reactions (electron captures and pycnonuclear fusions) that deposit heat in the crust. The heat sources are cataloged as a function of the compression until the fully accreted crust approximation is reached. The pressure at which neutron drip occurs is a nonmonotonic function of the depth, leading to a temporary neutron drip anomaly. The additional potential source of energy for partially accreted crusts is the occurrence of a density inversion phenomenon between some compressed layers. Conclusions. The original crust of a neutron star cannot be neglected for the initial period of accretion, when the original crust is not fully replaced by the accreted matter. The amount of heat associated with the compression of the original crust is on the same order of magnitude as that from the sources acting in the accreted part of the hybrid crust.

Ionization cross section of noble gas atoms by electron impact

The paper presents an analysis of data on the cross sections for ionization of noble gas atoms by electron impact. For the selected sets of experimental and theoretical data an analytical formula is proposed, based on separate accounting for the knockout of electrons from the outer and inner shells, and the corresponding approximation coefficients are selected. The obtained semi-empirical formula reproduces the values of the ionization cross sections in a wide range of energies with an accuracy of several percent. The analysis of the approximation coefficients makes it possible to reduce the influence of errors in the initial experimental data and significantly increase the accuracy of estimating the ionization cross sections.

Ionization by electron impacts and ionization potential depression

We calculate the cross-section of ionization by free-electron impacts in high or moderate density plasmas. We show that the so-called ionization potential depression (IPD) strongly affects the magnitude of the cross-section in the high-density domain. We use the well-known IPD formulas of Stewart–Pyatt and Ecker–Kröll. A more recent approach based on classical molecular dynamics simulation is also investigated. The latter provides an alternative way to calculate IPD values. At near-solid densities the effects of the free-electron degeneracy should be investigated. The rates are then calculated within the Fermi–Dirac statistics. We first use the semi-empirical formula of Lotz for ionization cross-section. The results may differ significantly from measured cross-sections or calculations with reliable atomic codes. Then, in a second step, we propose a new formula that combines the Lotz formula and a polynomial expansion in terms of the ratio of the energy of the incident electron and the ionization energy. The coefficients of the polynomial expansion are adjusted to fit the cross-section provided by robust atomic codes. A great advantage of the new formula is that it allows a fully analytical calculation of the ionization rate. Our results are compared to experiments measuring IPDs, cross-sections and rate coefficients on aluminum at high and moderate densities and on Be-like CNO ions.

A temperature-independent prediction model predicts the vapor-liquid equilibrium of CO2-based binary mixtures

CO2-based binary mixtures are considered good alternative working fluids in the refrigeration and power cycle due to their excellent performance and environmental friendliness. The vapor-liquid phase equilibrium of CO2-based binary mixtures is the basis for calculating mixing enthalpy and heat capacity, which is essential for thermodynamic analysis. In this work, based on the vdW mixing rules, we proposed a kij model (kij=θjki-θikj). In the proposed model, ki and kj are defined as pure interaction factors of the pure components, θj is the influence factor of component j to component i, and θi is the influence factor of component i to component j. After continuous fitting and trial calculation, we developed a semi-empirical formula, which is called a temperature-independent prediction model. The model does not have any adjustable parameters, which are only related to the physical properties (critical pressure (kPa), critical temperature (K), and acentric factor) and molecular structure (carbon-carbon double bonds, carbon-fluorine, chlorine, iodine, and oxygen atoms) of the components. The temperature-independent prediction model can predict the bubble-point pressures, vapor phase molar fractions, and relative volatilities very well, which is better than the prediction model based on the vdW mixing rules by others and the group contribution model based on the excess free energy (GE) mixing rules (PR+MHV1+UNIFAC). The total average absolute relative deviation of pressures, the total average absolute deviation of vapor phase molar fractions, and the total average absolute relative deviation of relative volatilities by the temperature-independent prediction model are 2.23%, 0.0095, and 5.21%, respectively.

Knowledge-Enhanced Deep Learning for Simulation of Extratropical Cyclone Wind Risk

Boundary-layer wind associated with extratropical cyclones (ETCs) is an essential element for posing serious threats to the urban centers of eastern North America. Using a similar methodology for tropical cyclone (TC) wind risk (i.e., hurricane tracking approach), the ETC wind risk can be accordingly simulated. However, accurate and efficient assessment of the wind field inside the ETC is currently not available. To this end, a knowledge-enhanced deep learning (KEDL) is developed in this study to estimate the ETC boundary-layer winds over eastern North America. Both physics-based equations and semi-empirical formulas are integrated as part of the system loss function to regularize the neural network. More specifically, the scale-analysis-based reduced-order Navier–Stokes equations that govern the ETC wind field and the European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA) ERA-interim data-based two-dimensional (2D) parametric formula (with respect to radial and azimuthal coordinates) that prescribes an asymmetric ETC pressure field are respectively employed as rationalism-based and empiricism-based knowledge to enhance the deep neural network. The developed KEDL, using the standard storm parameters (i.e., spatial coordinates, central pressure difference, translational speed, approach angle, latitude of ETC center, and surface roughness) as the network inputs, can provide the three-dimensional (3D) boundary-layer wind field of an arbitrary ETC with high computational efficiency and accuracy. Finally, the KEDL-based wind model is coupled with a large ETC synthetic track database (SynthETC), where 6-hourly ETC center location and pressure deficit are included to effectively assess the wind risk along the US northeast coast in terms of annual exceedance probability.

Analysis of Electromagnetic Environment in 1200 kV Single Circuit UHVAC Transmission Line by using FACE Software and Semi-empirical Formulae

In order to ensure efficient, bulk-power transmission over a long distance, need of increasing transmission-level voltage is being thought for the past many years. As a result, Ultra High Voltage AC (UHVAC) Transmission has appeared as a feasible solution. Many countries like Italy, Japan, USA, USSR, Brazil, China, India are developing technology for UHVAC transmission systems. Operating at 1000 kV (and above voltage level) demands careful design of transmission structure, switchgear, and equipment design. This paper presents a review of several studies carried out to analyze the electro-magnetic environment in UHVAC lines. The electromagnetic environment (Field, Audible Noise, Corona Loss, Radio Interference) of 1200 kV, Wardha-Aurangabad single-circuit transmission line, India is also evaluated using FACE (Field and Corona Effect) software. The results of the study of static AC field and magnetic field of the line are found to be within prescribed limits suggested by International Commission on Non-Ionizing Radiation Protection (ICNIRP) guidelines.

Magnetic Field-Assisted Fission of a Ferrofluid Droplet for Large-Scale Droplet Generation.

With the presence of an external magnetic field, a ferrofluid droplet exhibits a rich variety of interesting phenomena notably different from nonmagnetic droplets. Here, a ferrofluid droplet impacting on a liquid-repellent surface is systematically investigated using high-speed imaging. The pre- and post-impact, including the droplet stretching, maximum spreading diameter, and final impact modes, are shown to depend on the impact velocity and the magnitude of the external magnetic field. A scaling relation involving the Weber and magnetic Bond numbers is fitted to predict the maximum spreading diameter based on the magnetic field-induced effective surface tension. The impact outcome is also investigated and classified into three patterns depending on the occurrence of the rim interface instability and the fission phenomenon. Two types of fission (i.e., evenly and unevenly distributed sizes of the daughter droplets) are first identified, and the corresponding mechanism is revealed. Last, according to Rayleigh-Taylor instability, a semiempirical formula is proposed to estimate the number of the daughter droplets in the regime of evenly distributed size, which agrees well with the experimental data. The present study can provide more insight into large-scale droplet generation with monodispersive sizes.

Stability and uniformity of gas–liquid two-phase flow in shear-thinning fluids in parallelized microchannels

The stability of gas–liquid two-phase flow and uniformity of bubble size in shear-thinning fluids within symmetrical parallelized microchannels are investigated experimentally. Air and sodium carboxymethyl cellulose (CMC) aqueous solutions with 0.3% SDS are used as the dispersed and continuous phases, respectively. The process of gas pressure buildup and rapid release is observed under conditions of exceeding the critical gas and liquid flow rates. Various slug bubble shapes are identified in different concentrations of CMC solutions, gas and liquid flow rates. The results show that the increase of liquid flow rates and the decrease of CMC concentration are beneficial for the stability of bubble formation and uniformity of bubble size in the parallelized microchannels. Finally, semi-empirical formulas are proposed, based on the rheological property of CMC aqueous solutions, to estimate the bubble length.

Self-focusing propagation characteristics of a radially-polarized beam in nonlinear media.

In this study, an analytical formula for the self-focusing length of a radially polarized beam (RPB) is first derived, which has a similar behavior to the semi-empirical Marburger formula of a Gaussian beam, and is beneficial to quantitatively and qualitatively analyze practical experimental scenarios. However, the relation of the self-focusing length with the states of polarization (SoPs) was evaluated, and it was found that RPB with spatially inhomogeneous SoP at the field cross-section can retain a further self-focusing length compared to a beam with a spatially homogeneous one. The influence of the topological charge on the self-focusing length is explored, which shows that RPB with a low topological charge can achieve a high-power density at a relatively further receiver plane. Therefore, it is demonstrated that the RPB as a laser source not only extends the self-focusing length, but also improves the power density of the target. With the help of RPB, it is possible to realize a controllable self-focusing length and a high target optical power density, which may have potential applications in fine optical manipulation, optical communication, high-power long-range laser atmospheric propagation, and related areas.

Diffusion separation of submicron particles by using magnetized spheres

The diffusion capture of submicron-sized particles in the gradient field formed by the magnetized ferromagnetic spheres in the external homogeneous magnetic field has been theoretically investigated. Magnetic field measurements were made with a specially designed prototype magnetometer. The simple semi-empirical formula of the magnetic force acting on the submicron particle was determined. The analytical solution of the diffusion equation for steady states was obtained. The concentration distribution of the particles in capturing regions was determined. A formula was obtained to evaluate the critical size of the submicron particles captured in this area. The effects of magnetic, hydrodynamic and geometric parameters of the investigated system on the critical size of the particles in the diffusion event were evaluated. The critical size of the particles decreased with the increase of the external magnetic field and was found to be independent of the dimensions of the spheres. As a result; the diffusion separation phenomenon in the gradient field formed by the magnetized ferromagnetic spheres is more effective than in the gradient magnetic field formed by the wires. The results obtained contribute to the development of the theory of magnetic separators using ferromagnetic matrix elements.

Sputter yields of monoatomic solids by Ar and Ne ions near the threshold: A Bayesian analysis of the Yamamura Model

Semi-empirical formulae like the Yamamura model provide a quick reference of sputter yields for applications such as sputter depth profiling and secondary ion mass spectrometry. Fit parameters in such models are prone to errors which can propagate into the prediction of sputter yields. We compare experimental sputter yields to predictions of the Yamamura model using a Bayesian Markov Chain Monte Carlo (MCMC) algorithm. The model parameters Q (linear scaling) and s (power-law scaling) are explored. The results from MCMC are then compared to propositions of Seah (Surface and Interface Analysis, 2005) and extended to a collection of target materials by fitting simulated yields for argon and neon using TRIDYN. Q was found to be proportional to the threshold energy and a simple relation is proposed. The simplicity notwithstanding, Q is speculated to be dependent on a multitude of parameters such as density, energy transfer and orbital filling.

Predictions of the alpha-decay half-lives of even–even superheavy nuclei using modified Gamow-like model

α-decay half-lives of superheavy nuclei have been studied using modified Gamow-like model and improved empirical formulas. The empirical formulas used in the study are Viola-Seaborg semi-empirical formula, universal decay law (UDL), Denisov-Khudenko formula (DK), Royer formula, Wang formula, AKRE formula, and new Ren B (NRB) formula. New values of the adjustable parameters in the modified Gamow-like model have been obtained using a least square fit scheme and input data from the NUBASE2020 database. We found that only two adjustable parameters are sufficient to describe the α-decay half-lives of heavy and superheavy nuclei. We also obtained new coefficients for the seven empirical formulas using similar procedure. The modified Gamow-like model (MGLM) and improved empirical formulas are then used to calculate the α-decay half-lives of measured superheavy nuclei with Z = 100–118. The calculated root mean square errors indicate that the Royer, DK, and AKRE formulas are the best three empirical formulas for evaluating the α-decay half-lives of the superheavy nuclei. The five best performing empirical formulas, together with the MGLM, are then used to predict alpha-decay half-lives of unmeasured even–even superheavy nuclei with Z = 120–126. The Q α values used for the predictions have been taken from the tabulated Weizs cker-Skyrme-4 + RBF (WS4 + RBF) Q α values. The theoretical calculations suggest that N = 184 is a possible magic number, which is in agreement with previous theoretical calculations.

Towards the uncertainty quantification of semi-empirical formulas applied to the added resistance of ships in waves of arbitrary heading

The present paper examines a semi-empirical framework for the estimation of added resistance in arbitrary wave heading under consideration of uncertainty quantification. In this respect, the calibration of the formula’s parameter vector is conducted based on particle swarm optimization as well as a database of model test results comprising 25 different ships and around 1100 samples. In the first iteration, the minimization of reducible systematic uncertainty is of interest and the effect of four objective functions on prediction accuracy is evaluated. Moreover, two different parameter combinations were obtained for blunt (CB≥0.70) and slender-type ships. Conversely, the irreducible statistical uncertainty, i.e. the inherent noise of the experimental data, is taken into account by a quantile regression procedure. Applying this approach, a 90% prediction interval for the formula’s estimates is implemented using the skewed version of the superior loss function in the previous iteration. The practical relevance of an uncertainty estimate for the prediction of the added resistance is emphasized in the final part, in which the proposed approach is validated in regular waves against model test data and other well-established prediction methods. In general, the validation studies suggest satisfactory performance and reliability of the adapted semi-empirical formulation.

Lateral Photocurrent Method for Directly Measuring the Sheet Resistance of a Junction Partner for a Chalcogenide Light Absorber in a Thin-Film Solar Cell

The purpose of this study is to develop a method for directly measuring the sheet resistance of a junction partner for a chalcogenide light absorber in a thin-film solar cell. The developed method is applied to $\mathrm{Cd}\mathrm{S}$ in a $\mathrm{Cu}(\mathrm{In},\mathrm{Ga}){\mathrm{Se}}_{2}$ (CIGS) thin-film solar cell as a proof of concept. The suggested method utilizes a combination of the lateral photocurrent flowing along $\mathrm{Cd}\mathrm{S}$ in a custom-designed cell and the current density--voltage curve of a lab-scale CIGS solar cell. By combining an equivalent circuit analysis and a derived approximate equation, a semiempirical formula is proposed for directly measuring the sheet resistance of $\mathrm{Cd}\mathrm{S}$ in the CIGS thin-film solar cell. The semiempirical formula is a function of the basic characteristics (fill factor, short-circuit current density, open-circuit voltage) of the CIGS solar cell, and the linear lateral photocurrent density measured in the custom-designed cell. The proposed semiempirical formula is validated by measuring the sheet resistance of a $\mathrm{Zn}\mathrm{O}:\mathrm{Al}$ thin film in a custom-modified cell. The sheet resistance value agrees well with the sheet resistance value measured by a four-point probe. This method can be easily extended to ${\mathrm{Cu}}_{2}{\mathrm{Zn}\mathrm{Sn}\mathrm{S}}_{4}$, $\mathrm{Sn}\mathrm{S}$, and ${\mathrm{Sb}}_{2}{\mathrm{Se}}_{3}$ thin-film solar cells, which have very similar device structures to a CIGS thin-film solar cell.

Successive Rebounds of Impinging Water Droplets on Superhydrophobic Surfaces.

When a water droplet strikes a superhydrophobic surface, there may be several to a few tens of rebounds before it comes to rest. Although this intriguing multiphase flow phenomenon has received a great deal of attention from interfacial scientists and engineers, the underlying dynamics have not yet been completely resolved. In this paper, we report on an experimental investigation into the bouncing behavior of water droplets impinging on macroscopically flat superhydrophobic surfaces. We show that the restitution coefficient, which quantifies the energy consumed during impact and rebound, exhibits a nonmonotonic dependence on the Weber number. It is the droplet-surface friction that restricts the rebound height of the impinging droplet, so its restitution coefficient increases with the Weber number when the impact velocity is below a critical value. Above this value, the viscous friction within a thin liquid layer close to the superhydrophobic surface becomes dominant, and thus, the restitution coefficient decreases sharply. On the basis of energy analyses, semiempirical formulas are proposed to describe the restitution coefficient, and these can be employed to predict the number of successive rebounds of impinging droplets on superhydrophobic surfaces.