Two-term Approximation Research Articles

Verification of direct coupling code system using FRENDY version 2 and GENESIS for light water reactor lattices

ABSTRACT This study newly established a direct coupling code system consisting of the nuclear data processing code FRENDY version 2, and the three-dimensional heterogeneous transport code GENESIS (FRENDY-V2/GENESIS) for easy implementation of the random-sampling-based uncertainty quantification considering the implicit effect due to nuclear cross-section (XS) perturbations. The multi-group macroscopic XSs prepared for GENESIS were generated by FRENDY version 2, where the Dancoff factor was calculated by the neutron current method. Then the background XSs were evaluated based on the Carlvik two-term rational approximation. The infinite multiplication factor (k-infinity) and the fission reaction rate distribution in UO2 and MOX lattice geometries were compared with MVP3 to verify the calculation accuracy of FRENDY-V2/GENESIS. The sensitivity analyses on the discretization conditions such as the ray tracing of the method of characteristics were also carried out. Through several comparisons between FRENDY-V2/GENESIS and MVP3, FRENDY-V2/GENESIS with the SHEM 361-group structure calculates the k-infinity within approximately 50 pcm and the fission reaction rate distribution within approximately 0.1% by the root mean square, respectively. Consequently, the applicability of FRENDY-V2/GENESIS was verified, and FRENDY-V2/GENESIS can be used to discuss the implicit effect due to multi-group XS perturbations.

Numerical investigation of plasma properties in Ar/SiH4 inductively coupled plasmas considering electron energy distribution functions

In thin film deposition, Ar/SiH4 mixtures are widely used to make polysilicon (poly-Si) and hydrogenated amorphous silicon (a-SiH) layers. Despite extensive research conducted on this mixture, little research has focused on the variations in plasma properties, radicals, and ions that occur during plasma discharge in inductively coupled plasma (ICP) equipment compared to capacitive coupled plasma equipment. In this paper, we investigate the properties of the plasma generated through mathematical modeling of Ar/SiH4 inductive coupled plasma discharge by using the electron energy distribution function (EEDF) obtained by solving the Boltzmann equation. We closely examine the variation in plasma properties and the correlation of plasma variables by controlling the radio frequency power and gas pressure during the process conditions. The Boltzmann equation was computed by assuming the two-term approximation, resulting in a Druyvesteyn-like EEDF due to the high-pressure conditions. To validate the simulation model, the 2D simulation results were compared with probe measurements performed in a two-turn ICP chamber. The results demonstrated encouraging agreement with the measured data. This research not only enhances our comprehension of the discharge characteristics but also establishes a framework for optimizing the discharge conditions to enhance the process and effectively regulate external variables.

Two-term Boltzmann approximation versus Monte-Carlo simulation: effect of magnetic field

In this paper, we investigate the spatial-local electron energy distribution function (eEDF) interacting with a background gas at the sub-atmospheric pressure in a wide range of applied crossed electric and magnetic fields using the Boltzmann kinetic equation. We compare solutions obtained using two numerical approaches (deterministic two-term approximation and stochastic Monte Carlo method) to identify their applicability in the context of determining drift velocity and reaction constants for electrons. For argon and helium, the upper limit of the reduced electric field applicability of the two-term approximation is discussed. It has been shown that the presence of a magnetic field can reduce this limit. Two explanations are given, one is based on the math of two-term formalism, and the other is based on velocity-space analysis. Two-term approximation fails due to it’s inability to resolve underlying cyclotron oscillation (it should result in an energy variation along the electron’s trajectory). The absence of this feature causes an incorrect estimation of momentum-transfer rate. This results an inaccuracy in the estimation of the angle between electric field and drift velocity.

The Mathematical Modeling, Diffusivity, Energy, and Enviro-Economic Analysis (MD3E) of an Automatic Solar Dryer for Drying Date Fruits

Date fruit drying is a process that consumes a significant amount of energy due to the long duration required for drying. To better understand how moisture flows through the fruit during drying and to speed up this process, drying studies must be conducted in conjunction with mathematical modeling, energy analysis, and environmental economic analysis. In this study, twelve thin-layer mathematical models were designed utilizing experimental data for three different date fruit varieties (Sakkoti, Malkabii, and Gondaila) and two solar drying systems (automated solar dryer and open-air dryer). These models were then validated using statistical analysis. The drying period for the date fruit varieties varied between 9 and 10 days for the automated solar dryer and 14 to 15 days for open-air drying. The moisture diffusivity coefficient values, determined using Fick’s second law of diffusion model, ranged from 7.14 × 10−12 m2/s to 2.17 × 10−11 m2/s. Among the twelve thin-layer mathematical models, we chose the best thin drying model based on a higher R2 and lower χ2 and RMSE. The Two-term and Modified Page III models delivered the best moisture ratio projections for date fruit dried in an open-air dryer. For date fruit dried in an automated solar dryer, the Two-term Exponential, Newton (Lewis), Approximation diffusion or Diffusion Method, and Two-term Exponential modeling provided the best moisture ratio projections. The energy and environmental study found that the particular amount of energy used varied from 17.936 to 22.746 kWh/kg, the energy payback time was 7.54 to 7.71 years, and the net CO2 mitigation throughout the lifespan ranged from 8.55 to 8.80 tons. Furthermore, economic research showed that the automated solar dryer’s payback period would be 2.476 years.

Drying kinetics and mass transfer parameters of mung beans dried using a convective dryer

Convective drying of mung bean seeds was conducted to investigate drying kinetics and mass transfer parameters at temperatures of 40, 50, 60, 70, and 80°C and a fixed air velocity of 1.2 m s-1. Drying characteristics were determined by drying rate, moisture ratio, diffusion coefficient, activation energy, Biot number, and mass transfer coefficient. Drying models such as Newton, Page, Henderson and Pabis, Logarithmic, Midilli, Two-term, and Diffusion approximation were applied and fitted to the moisture ratio of the experimental data. The drying rate was significantly affected by the drying temperature. The Midilli model was considered the most suitable for experimental drying curves. The effective moisture diffusivity ranged from 3.19×10-9 to 10.76×10-9 m2 s-1, with an activation energy of 59.78 kJ mol-1. Similar to the mass transfer coefficient, the Biot number increased with increasing drying temperature. The simultaneous transfer of heat and mass was controlled by diffusion and surface.

Secure control for remote networked stochastic systems via integral sliding mode

This paper deals with the secure control problem for a class of networked stochastic systems with false data injection attacks via an integral sliding mode control technique. The networked control system is with a hierarchical structure, and the main controller and a remote controller are considered to realize the secure control against false data injection attacks on the network between a main controller and the plant. A mode-shared event-triggering controller is designed as the main controller, by utilizing a time delay approach. An input–output model based on a two-term approximation is applied to cope with the formulated time-varying delay. Then, the scaled small gain theory is employed to analyze the stability of the resulting system. Sufficient conditions on ensuring the desired system performance are derived and then the controller parameters are synthesized. Moreover, an elaborated sliding mode control law is proposed for the desired secure control action. Finally, two simulation examples are permitted to verify the effectiveness of the theoretical derivations.

Calculation of the velocities induced by the trailing vorticity in the rotor plane of a horizontal-axis turbine or propeller

Lifting line (LL) analysis of propellers and horizontal-axis turbines requires the axial and circumferential velocities induced by the vortex system representing the blades and the trailing vorticity. If the blades are straight and radial, the induced velocities along the LLs are due only to the trailing vorticity. Accurate two-term approximations for these velocities have been developed from the exact Kawada–Hardin (KH) equations for the velocity field of a doubly infinite helical vortex of constant pitch and radius, Wood et al. (Ocean Engineering, 2021, 235). This paper describes a straightforward extension of the approximations to give the induced velocities anywhere in the equivalent of the rotor plane for a doubly infinite helix. The third term in the approximation of the KH equations is derived and compared to an alternative third term due to Okulov (Journal of Fluid Mechanics, 2004, 521, 319–342). Both three-term approximations produce a small improvement in accuracy over the two-term approximations for a range of operating conditions for turbines and propellers. Okulov’s third term is superior. To determine the induced velocities for a singly infinite trailing vortex behind a rotor, an additional equation is derived from the Biot–Savart law. Numerical examples show that the resulting equations provide accurate estimates for the induced velocities over the rotor plane. The main application of the analysis is to account for blade sweep and coning by including the angle between the vortex origin and the control point at which the velocities are required, often the center of each blade element.

Efficient modeling of electron kinetics under influence of externally applied electric field in magnetized weakly ionized plasma

We present a theory based on the conventional two-term (i.e. Lorentzian) approximation to the exact solution of the Boltzmann equation in non-magnetized weakly ionized plasma to efficiently obtain the electron rate and transport coefficients in a magnetized plasma for an arbitrary magnitude and direction of applied electric field and magnetic field . The proposed transcendental method does not require the two-term solution of the Boltzmann equation in magnetized plasma, based on which the transport parameters vary as a function of the reduced electric field , reduced electron cyclotron frequency , and angle between and vectors, where N is the density of neutrals. Comparisons between the coefficients derived from BOLSIG+’s solution (obtained via the two-term expansion when ) and coefficients of the presented method are illustrated for air, a mixture of molecular hydrogen (H2) and helium (He) representing the giant gas planets of the Solar System, and pure carbon dioxide (CO2). The new approach may be used in the modeling of magnetized plasma encountered in the context of transient luminous events, e.g. sprite streamers in the atmosphere of Earth and Jupiter, in modeling the propagation of lightning’s electromagnetic pulses in Earth’s ionosphere, and in various laboratory and industrial applications of nonthermal plasmas.

Estimating two groups of fracture weaknesses using azimuthal differences in partially incidence-angle-stacked seismic amplitudes

Geophysical and geologic data, e.g., results obtained using rock cores, outcrops, and borehole images, reveal the presence of multiple sets of fractures in rock. To analyze how fracture interactions affect seismic anisotropy and dispersion, we consider an effective model that contains two sets of orthogonal fractures. In seismic amplitude analysis, we focus on the case of subsurface target zones containing a fracture network composed of primary and secondary fracture sets. Focusing on gas-bearing rocks with small fracture densities, we formulate simplified stiffness parameters in terms of two groups of fracture weaknesses, which are related to the primary and secondary fractures, respectively. Using the simplified stiffness parameters, we derive the PP-wave reflection coefficient and the azimuthal elastic impedance (AEI) as functions of the two groups of normal and tangential fracture weaknesses. Based on the derived PP-wave reflection coefficient and AEI, we propose a method and workflow in which the azimuthal variations of the partial incidence-angle-stacked seismic data are used to estimate the AEI and differences in AEI (which we refer to as DEI) is input to a nonlinear inversion for two groups of fracture weaknesses. In the nonlinear inversion, the initial values of fracture weaknesses are obtained based on a two-term approximation of the PP-wave reflection coefficient. First- and second-order derivatives of DEI with respect to fracture weakness parameters are calculated to generate the update in the unknown parameter vector. Synthetic seismic gathers with varying signal-to-noise ratios are used to validate the robustness of the estimation. Applying the inversion method to a real data set, we obtain what we interpret to be reliable fracture weakness estimates that produce azimuthal amplitude differences matching those extracted from real seismic data. The results provide a potential tool for determining if two sets of fractures have developed in a reservoir.

Динамика плоского движения космического крана-манипулятора типа руки с учетом изгиба звеньев

The paper considers the plane motion dynamics of a three-link crane-manipulator consisting of two rods elastic in bending and a massive solid body attached to them at their end (load capture). All links are interconnected by hinge joints with the given (controlled) relative rotation angles. The first link is also connected to the mobile base. The system mathematical model was developed to calculate its non-stationary oscillations and dynamic loading of links at arbitrary in time and large relative angles of the links rotation as the solid bodies. The rods bending deformations (elastic transverse displacements and rotation angles of the axes) were considered to be insignificant (linear). Each rod ending was represented by the Ritz method using two given shape functions with the unknown coefficients, which were taken as the generalized coordinates. In this case, the “cantilever” bending forms of each rod with the given transverse displacement and angle of rotation at their ends were used. A simplified version was considered: each rod bending was represented by only one given form of its bending by a single moment applied at the end. System oscillation equations were obtained based on the principle of probable displacements in the generalized coordinates (four equations or two in a simplified form). Since rotation angles at the ends of the elastic rods were added to the given rotation angles of the non-deformable rods, total angles were left under the signs of sines and cosines without linearization for convenience of calculations and increasing accuracy. As a result, such equations of non-stationary oscillations remained non-linear. They were integrated by numerical methods using standard programs in one of the computer algebra systems. Examples are provided with the comparative results estimates obtained in the two-term and one-term approximations of the function in regard to the bending form of the system bending rod joints.

Electron-neutral collision cross sections for H2O: II. Anisotropic scattering and assessment of the validity of the two-term approximation

This work proposes a complete and consistent set of cross sections (CS) for electron collisions with water gas molecules to be published in the IST-Lisbon database on LXCat. The set is validated by the electron swarm analysis technique. The anisotropic angular distribution of electrons in rotational collisions is considered by means of the Born approximation in a two-term Boltzmann solver (LisbOn KInetics two-term Boltzmann solver (LoKI-B)) and a Monte Carlo simulations code (LoKI-MC), both freely available as open-source codes. The inclusion of electron anisotropic scattering in rotational collisions significantly improves the agreement between calculations and measurements of the electron drift velocity, reduced mobility, characteristic energy, reduced Townsend ionisation coefficient, reduced effective Townsend coefficient and reduced attachment coefficient. The MC simulations are deemed more accurate and shown to yield similar results as LoKI-B with the proposed set. The use of LoKI-MC also validates the set of CSs against parameters that cannot be obtained by LoKI-B, such as the longitudinal diffusion coefficient or the bulk transport coefficients.

The comparison of two-term Boltzmann approximation and Monte-Carlo solutions for e+Ar0 plasma

In this paper, we examine the energy distribution function of electrons in the case of a very weakly ionized argon plasma at sub-atmospheric pressure and external electric field using Boltzmann kinetic equation. We consider a spatially uniform model because our primary interest is the behavior of the collisional part of the equation. The purpose of the study is to compare two different numerical approaches: a deterministic one (using two-term local non-stationary approximation) and a stochastic approach (using the Monte Carlo method) over a wide range of reduced electric fields. We compare steady-state and time-dependent solutions, isotropic and anisotropic parts of the electron energy distribution function, and reaction constants. The research will also help to identify any limitations and challenges of these methods.

Numerical study of nitrogen bulk dissociation in DC magnetron discharge in Ar+N2 mixture

The effect of the composition of the gas feed mixture on the dissociation rate constant of molecular nitrogen is examined by numerical solving the Boltzmann equation written in a two-term approximation for conditions corresponding to nitride aluminum films deposition. By solving of macroscopic balance equations in volume occupied with plasmas of stationary low-pressure glow discharge in a nitrogen-argon mixture the atomic nitrogen density was determined. It’s shown that there is a maximum of atomic nitrogen density at certain stoichiometry of gas feed mixture.

Third-order transport coefficients for electrons in N2 and CF4: effects of non-conservative collisions, concurrence with diffusion coefficients and contribution to the spatial profile of the swarm

Using a multi-term solution of the Boltzmann equation and Monte Carlo simulation technique we study behaviour of the third-order transport coefficients for electrons in model gases, including the ionisation model of Lucas and Saelee and modified Ness–Robson model of electron attachment, and in real gases, including N2 and CF4. We observe negative values in the E/n 0-profiles of the longitudinal and transverse third-order transport coefficients for electrons in CF4 (where E is the electric field and n 0 is the gas number density). While negative values of the longitudinal third-order transport coefficients are caused by the presence of rapidly increasing cross sections for vibrational excitations of CF4, the transverse third-order transport coefficient becomes negative over the E/n 0-values after the occurrence of negative differential conductivity. The discrepancy between the two-term approximation and the full multi-term solution of the Boltzmann equation is investigated for electrons in N2 and CF4. While the accuracy of the two-term approximation is sufficient to investigate the behaviour of the third-order transport coefficients in N2, it produces large errors and is not even qualitatively correct for electrons in CF4. The influence of implicit and explicit effects of electron attachment and ionisation on the third-order transport tensor is investigated. In particular, we discuss the effects of attachment heating and attachment cooling on the third-order transport coefficients for electrons in the modified Ness–Robson model, while the effects of ionisation are studied for electrons in the ionisation model of Lucas and Saelee, N2 and CF4. The concurrence between the third-order transport coefficients and the components of the diffusion tensor, and the contribution of the longitudinal component of the third-order transport tensor to the spatial profile of the swarm are also investigated. For electrons in CF4 and CH4, we found that the contribution of the component of the third-order transport tensor to the spatial profile of the swarm between approximately 50 Td and 700 Td, is almost identical to the corresponding contribution for electrons in N2. This suggests that the recent measurements of third-order transport coefficients for electrons in N2 may be extended and generalized to other gases, such as CF4 and CH4.

Функция распределения электронов по энергиям в высоковольтном импульсном разряде с протяженным полым катодом

The dynamics of a nanosecond discharge with an extended hollow cathode in argon is studied and the electron energy distribution function (EEDF) in a plasma column is calculated. The issues of the formation of the EEDF in the discharge column and inside the cathode cavity are considered. The conditions of the validity the two-term approximation of the EEDF in the plasma column are analyzed and the appropriate evaluations are given.

Electron energy distribution function in a high-voltage pulsed discharge with an extended hollow cathode

The dynamics of a nanosecond discharge with an extended hollow cathode in argon is studied and the electron energy distribution function (EEDF) in a plasma column is calculated. The issues of the formation of the EEDF in the discharge column and inside the cathode cavity are considered. The conditions of the validity the two-term approximation of the EEDF in the plasma column are analyzed and the appropriate evaluations are given. Keywords: hollow cathode, pulse discharge, argon, electron energy distribution function.

Benchmarking the calculation of electrically insulating properties of complex gas mixtures using a multi-term Boltzmann equation model

The accurate calculation of DC breakdown voltage thresholds solely from elementary electron-neutral interactions in complex gas mixtures using a multi-term Boltzmann equation (BE) kinetic model is demonstrated. SF6:N2 mixtures in the 100 Td < E/N < 400 Td field regime are explored to benchmark the model’s effectiveness. A ten-term BE model is found to yield DC breakdown voltages which, on average, agree within 3% of experimental measurements. A two-term BE model is also applied in order to characterize the error introduced in all calculations by the two-term approximation. These discrepancies are largest in pure N2 where error is greater than 10% for diffusion coefficients, within 6% for particular vibrational rate coefficients, and within 5% for breakdown voltages. However, this error falls to within 1% for most parameters and breakdown voltages in mixtures with large SF6 content.

Zero-dimensional and pseudo-one-dimensional models of atmospheric-pressure plasma jets in binary and ternary mixtures of oxygen and nitrogen with helium background

A zero-dimensional (volume-averaged) and a pseudo-one-dimensional plug-flow (spatially resolved) model are developed to investigate atmospheric-pressure plasma jets operated with He, He/O2, He/N2 and He/N2/O2 mixtures. The models are coupled with the Boltzmann equation under the two-term approximation to self-consistently calculate the electron energy distribution function. An agreement is obtained between the zero-dimensional model calculations and the spatially averaged values of the plug-flow simulation results. The zero-dimensional model calculations are verified against spatially resolved simulation results and validated against a wide variety of measurement data from the literature. The nitric oxide (NO) concentration is thoroughly characterized for a variation of the gas mixture ratio, helium flow rate and absorbed power. An ‘effective’ and a hypothetical larger rate coefficient value for the reactive quenching are used to estimate the role of the species N2(A3Σ, B3Π; v > 0) and multiple higher N2 electronically excited states instead of only N2(A3Σ, B3Π; v = 0) in this quenching. The NO concentration measurements at low power are better and almost identically captured by the simulations using the ‘effective’ and hypothetical values, respectively. Furthermore, the density measurements under the same operation conditions are also better predicted by the simulations adopting these values. It is found that the contribution of the vibrationally excited nitrogen molecules N2(v ⩾ 13) to the net NO formation rate gains more significance at higher power. The vibrational distribution functions (VDFs) of molecular oxygen O2(v < 41) and nitrogen N2(v < 58) are investigated regarding their formation mechanisms and their responses to the variation of operation parameters. It is observed that the N2 VDF shows a stronger response than the O2 VDF. The sensitivity of the simulation results with respect to a variation of the VDF resolutions, wall reaction probabilities and synthetic air impurity levels is presented. The simulated plasma properties are sensitive to the variation, especially for a feed gas mixture containing nitrogen. The plug-flow model is validated against one-dimensional experimental data in the gas flow direction, and it is only used in case an analysis of the spatially resolved plasma properties inside the jet chamber is of interest. The increasing NO spatial concentration in the gas flow direction is saturated at a relatively high power. A stationary O2 VDF is obtained along the direction of the mass flow, while a continuously growing N2 VDF is observed until the jet nozzle.

Large-deformation reduced order homogenization of polycrystalline materials

In this manuscript, we present a finite strain formulation of a reduced order computational homogenization model for crystal plasticity. The proposed formulation leverages and generalizes the principles of the Eigenstrain-based reduced order homogenization (EHM) approach. Asymptotic analysis with multiple scales is employed to describe the microscale problem in the deformed configuration. A two-term Taylor series approximation of the constitutive behavior along with a geometry-based basis reduction is employed to arrive at the reduced order model. An efficient implementation scheme is proposed to evaluate the multiscale system without the need to recompute the reduced basis as a function of evolving deformation. The ability of the proposed modeling approach in capturing homogenized and localized behavior as well as texture evolution is demonstrated in the context of single crystal and polycrystal microstructures.

Effect of anisotropic scattering for rotational collisions on electron transport parameters in CO

The role of anisotropic scattering in rotational collisions of electrons with CO molecules is investigated numerically with Monte Carlo (MC) simulations and with calculations using the Lisbon KInetics two-term Boltzmann solver (LoKI-B). The study adopts integral cross sections taken from the IST-Lisbon database of LXCat or extracted from Biagi’s code Magboltz v11.10. Different angular scattering models for rotational collisions are implemented and compared in MC simulations, and a novel anisotropic scattering model is derived from the dipole-Born differential cross sections, to describe the strongly forward-peaked nature of rotational collisions. This model is also implemented in LoKI-B, to describe the anisotropic inelastic/superelastic scattering in dipole rotational collisions, using coherent expressions for the corresponding integral and momentum transfer cross sections. The comparison between MC and LoKI-B results shows that the calculation of swarm parameters is more influenced by the choice of the angular scattering model than the adoption of the two-term approximation, yielding deviations up to 50% in the reduced mobility for different angular distributions. The consequences in the swarm derivation of cross sections are also discussed. Finally, it is shown that inclusion of electric-quadrupole interactions, usually neglected in electron swarm studies, can improve the agreement between numerical results and measurements.