Instantaneous Point Source Research Articles

Studying the Effect of the Instantaneous and Continuous Pollutants Sources on the Advection-diffusion Equation and Its Applications

This study investigates instantaneous and continuous sources as point, line, and area sources. Gaussian concentration in the case of the puff model with an instantaneous point source inhomogeneous longitudinal diffusion is investigated. The concentration is calculated using different dispersion parameters to get the proposed normalized concentration of the puff model at ground level around the centerline, which is compared with observed data by the Copenhagen experiment and previous work [1]. Also, the continuous point source is used to get the Gaussian plume model in three dimensions using dispersion parameters to compare with the observed concentration data measured by the Egyptian Atomic Energy Authority for Iodine-135 (I135) in an unstable condition.

Simulation of thermal processes in a half-space under heating by a moving source with a uniformly distributed heat flow

Using the method of applying instantaneous point sources, a solution was obtained to the problem of heat conduction during surface heating of a body in the form of a half-space by a uniformly distributed highly concentrated heat flux moving at a constant speed along a rectilinear trajectory with a different shape of the heating spot at constant thermophysical characteristics of the material. The effect of temperature loading modes and the shape of the heating spot on thermal processes in the heat-affected zone is studied. The surfaces and lines of the temperature level are constructed for different moments of time and speed loading modes in different planes of the heating zone. Time dependences of temperatures, heating and cooling rates for body points are given. The shortcomings of the methods used for linear thermal conductivity, the lack of direct consideration in the design scheme of the surface melt zone of the material do not allow one to reliably assess the effect of heat treatment modes on changes in material properties, focusing only on the level of the maximum design temperature. In this regard, the structure formation of metal in the zone of thermal action is proposed to be associated with a thermal impulse, i.e. the total thermal energy perceived by the material at a given point of the body, as well as with the effective structurization impulse introduced into consideration, which characterizes the energy spent on the process of structural transformations of the material, and the structurization time at a point and some volume of the body. The dependencies of these values on the speed of movement and the shape of the heating spot are presented. The considered approaches can be applied to various metals and alloys. The research results can be used to develop more effective methods for determining the optimal modes of surface hardening of metal products with a high-energy source.

Source Parameters of Strong Turkish Earthquakes on February 6, 2023 (<i>M<sub>w</sub></i> = 7.8 and <i>M<sub>w</sub></i> = 7.7) from Surface Wave Data

Abstract—Based on the amplitude spectra of surface waves, the source parameters of the strong Turkish earthquakes of February 6, 2023 (Mw = 7.8 and Mw = 7.7) were calculated in two approximations: an instantaneous point source and an elliptical shear dislocation. As a result, rupture planes were identified, data were obtained on the scalar seismic moment, moment magnitude, focal mechanism, and source depth of the considered seismic events, and the integral parameters characterizing the rupture geometry and its development in time were estimated. It is shown that the sources of the earthquakes under study were formed under the influence of the regional stress field and their focal mechanisms were left lateral faults with a strike direction close to the strike of the East Anatolian fault zone for the first event and close to the strike of the Sürgü-Çardak fault system for the second. For the first earthquake, our estimates of the rupture duration and its length (t = 52.5 s, L = 180 km) probably refer not to the entire rupture, but only to its main phase, confined to the northeastern segments of the East Anatolian fault and characterized by maximum displacements and values of the released seismic moment. The values of t = 30 s and L = 180 km that we obtained for the second earthquake fully characterize the entire rupture.

Mathematical model of the heat distribution process in a steel part during surface modification by the electro-spark alloying method

Problem. Electro-spark alloying (ESA) is a promising process of surface modification, which allows obtaining a high-quality coating, is environmentally safe and energy efficient. Wide application in production is hindered by the need to develop the technology, as well as the lack of a mathematical model for predicting the properties of coatings depending on materials and processing modes. It is known that the high adhesive bond of the coatings obtained by the ESA method with the base is explained both by the intensive mixing of the electrode materials in the liquid phase and by the diffusion of the anode material into the cathode in the solid phase. The goal is to develop the mathematical model that allows describing the process of heat propagation in the coating during electro-spark alloying of the metal surface. Methology. The direction of technological thermal physics of N.M. Rykalin is widely known, it is based on the theory of thermal conductivity of the method of heat sources. Regarding the thermal conductivity of solid bodies, the most universal Green's function is the Kelvin solution - the fundamental solution of the heat conduction equation, which describes the temperature distribution in an unbounded solid body, which occurs after the release of a finite amount of heat in the form of an instantaneous point source. The mathematical model of the process of heat propagation in the product during ESA was described using the Laplace operator. It is taken into account that the temperature of the surface of the strengthened part depends on the thermophysical characteristics of the anode and cathode, the diameter of the electrode, as well as the speed of movement of the alloying electrode. Results. The original mathematical model that allows describing the process of heat propagation in the coating during electro-spark alloying of the metal surface is developed. It is shown that the larger the diameter of the electrode, the higher the temperature at the coating-base interface, and therefore the better the adhesion of the coating. The optimal speed of moving the electrode is 0.5-1.5 m/s. Practical value. To obtain the best adhesion of the coating to the base, it is necessary to use an electrode with the largest diameter of 4.5 mm in combination with a low speed of moving the electrode.

Analytical solution of the heat equation for an instantaneous point source in a hollow sphere

Analytical solution of the heat equation for an instantaneous point source in a hollow sphere

Coupling remote sensing and particle tracking to estimate trajectories in large water bodies

Propelled by the rapid development of equipment, technology and computational power, the monitoring and simulation of the hydrodynamics in lakes have steadily advanced. In contrast, water quality simulations are more difficult to implement, due to the difficulty in obtaining large-scale, spatially resolved field observations for model validation and the number of interacting processes to be parameterized. Here we demonstrate that remote sensing data can be used to inform Lagrangian particle tracking in a large lake, and vice versa. We used total suspended matter (TSM) as a parameter that can be both estimated from the backscattering in satellite images and modelled in terms of particle abundance. Specifically, we compared TSM concentrations in Lake Geneva deduced from images taken by Sentinel-2 and Sentinel-3 satellites to those estimated from Delft3D hydrodynamic and particle tracking models. TSM concentrations obtained from both methods were compared over a time span of up to 5 days in several scenario studies, including instantaneous and continuous point sources and large-scale TSM simulations. The results demonstrate that remote sensing images can serve to calibrate and validate particle tracking models with independent observations. The model was able to capture both the position of a TSM cloud arising 5 days after an instantaneous point source release, and the direction of particle transport and TSM plume size resulting from a continuous source. Even when simulating the whole lake domain, model results closely approximated the satellite-derived TSM concentrations along lake transects within 9%. In return, the particle tracking model was able to complete partially impaired satellite images, and fill in a four-day image gaps between satellite revisits. The synergy of remote sensing techniques and particle tracking modelling allows a rapid, continuous and more accurate analysis on solute transport in lakes.

Optimal emission control and identification of an unknown pollution source

The advection-diffusion-reaction equation is used for describing the dispersion of a quasi-passive contaminant from industrial point sources in a limited area. The conditions established on the open boundary ensure that the problem is correct in the sense of Hadamard, that is, its solution exists, is unique, and is stable to initial perturbations. The Lagrange identity is used to construct the adjoint operator and formulate an adjoint problem. Equivalent direct and adjoint estimates are derived to assess the concentration of the pollutant at monitoring sites of the area. Formulas obtained on the basis of adjoint estimates are useful in analysing the sensitivity of the model to both variations in the intensity of pollution sources and variations in the initial distribution of the pollutant concentration in the area. New optimal emission control strategies based on using the adjoint estimates are developed in order to prevent violations of existing sanitary standards by timely reduction of emission rates of operating sources. Optimal control here lies in minimizing these reductions. In addition, this control is primarily aimed at reducing the intensity of emissions from sources that most pollute the monitoring site. Also, new methods are proposed for identifying the main parameters of an unknown point source that arose as a result of a dangerous incident (accident, explosion, etc.). These methods allow determining the location and intensity of a constant or non-stationary point source, as well as the moment of emission of a pollutant in the case of an instantaneous point source. This helps to quickly assess the scale of the incident and its consequences. Numerical results show the effectiveness of the methods.

Migration of buoyancy-controlled active particles in a laminar open-channel flow

The migration of active particles in a laminar open-channel flow is significant for many ecological and biological processes associated with ambient water, for example, the harmful algae blooms in the long-distance water conveyance project. This paper presents an analytical analysis of the migration of the buoyancy-controlled active particles in a laminar open-channel flow. The analytical solution of concentration distribution has also been derived for the active particle cloud in the laminar open-channel flow under the combined action of ambient flow and self-migration of active particles, by combing Chatwin’s long-time asymptotic expansion and central concentration moments. The analytical solution shows that the active particle cloud accumulates near the surface rather than uniform distribution in the cross-section even for the long-time evolution. The analytical solutions are also been rigorously derived to reflect the holistic characteristics of the active-particle cloud, including the total quantity over each streamline, moving velocity of mass center, longitudinal dispersivity, skewness, and kurtosis have been rigorously derived for an instantaneous point source in a laminar open-channel flow. It is found that the distribution of total active particles over each streamline, centroid motion, and longitudinal dispersion eventually reach a stable status with a necessary time dependent on the relative strength of the vertical migration velocity and the total vertical diffusion. However, the self-migration of active particles results in the deviation of the moving velocity of mass center from the depth-averaged flow velocity. The longitudinal dispersivity of the active particle cloud gradually increases to the peak and then decreases to a constant value, in contrast to the monotonous increase for the passive particle cloud. It is also found that the strong vertical self-migration can result in changes of skewness and kurtosis of concentration distribution: negative skewness and positive kurtosis during the initial stage. However, both of them gradually tend to zero for the long-time evolution of the active particle cloud.

Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.

Subsecond Time Evolution of Type III Solar Radio Burst Sources at Fundamental and Harmonic Frequencies

Abstract Recent developments in astronomical radio telescopes opened new opportunities in imaging and spectroscopy of solar radio bursts at subsecond timescales. Imaging in narrow frequency bands has revealed temporal variations in the positions and source sizes that do not fit into the standard picture of type III solar radio bursts, and require a better understanding of radio-wave transport. In this paper, we utilize 3D Monte Carlo ray-tracing simulations that account for the anisotropic density turbulence in the inhomogeneous solar corona to quantitatively explain the image dynamics at the fundamental (near plasma frequency) and harmonic (double) plasma emissions observed at ∼32 MHz. Comparing the simulations with observations, we find that anisotropic scattering from an instantaneous emission point source can account for the observed time profiles, centroid locations, and source sizes of the fundamental component of type III radio bursts (generated where f pe ≈ 32 MHz). The best agreement with observations is achieved when the ratio of the perpendicular to the parallel component of the wavevector of anisotropic density turbulence is around 0.25. Harmonic emission sources observed at the same frequency (∼32 MHz, but generated where f pe ≈ 16 MHz) have apparent sizes comparable to those produced by the fundamental emission, but demonstrate a much slower temporal evolution. The simulations of radio-wave propagation make it possible to quantitatively explain the variations of apparent source sizes and positions at subsecond timescales both for the fundamental and harmonic emissions, and can be used as a diagnostic tool for the plasma turbulence in the upper corona.

A Theoretical Study about Ergodicity Issues in Predicting Contaminant Plume Evolution in Aquifers

A large-time Eulerian–Lagrangian stochastic approach is employed to: (1) estimate centroid position uncertainty of contaminant plumes that originate from instantaneous point sources in statistically stationary and isotropic porous formations; (2) assess the time needed for achieving ergodic conditions, which would allow for the evaluation of local concentration values based on the only ensemble mean distribution; (3) derive the concentration coefficient of variation (CV) as a function of asymptotic macro-dispersion coefficients and centroid trajectory variances. The results indicate that the decay time of plume position uncertainty is so large that there is practically no chance for effective ergodicity. The concentration coefficient of variation is zero at the centroid but rapidly increases when moving away from it. The dissipative effect of local dispersion in the presence of relatively high Péclet numbers is considerably exalted by marked flow field heterogeneity, which confirms the previously postulated synergic, non-additive effect of advection and local dispersion in passive solute dilution. A further result from this study is the derivation of the power law that relates dimensionless concentration micro-scale to dimensionless local dispersive area. The exponent of this power law is the same that appears in the relationship between dimensionless Kolmogorov turbulent micro-scale and flow Reynolds number.

ANALYTICAL SOLUTION OF A NON - HOMOGENEOUS ONE - DIMENSIONAL ADVECTION DIFFUSION EQUATION WITH TEMPORALLY VARYING COEFFICIENTS .

Advection Diffusion Equation is a partial differential equation that describes the transport of pollutants in rivers. Its coefficients (dispersion and velocity) can be constant, dependent on space or time or both space and time. This study presents an analytical solution of a one dimensional non - homogeneous advection diffusion equation with temporally dependent coefficients, describing one dimensional pollutant transport in a section of a river. Temporal dependence is accounted for by considering a temporally dependent dispersion coefficient along an unsteady flow assuming that dispersion is proportional to the velocity. Transformations are used to convert the time dependent coefficients to constant coefficients and to eliminate the advection term. Analytical solution is obtained using Fourier transform method considering an instantaneous point source. Numerical results are presented. The findings show that concentration monotonically decreases with increasing distance and increasing time.

Improvement of Instantaneous Point Source Model for Simulating Radionuclide Diffusion in Oceans under Nuclear Power Plant Accidents

Simulation methods have become an important tool to reveal radionuclide migration during accidental radionuclide releases and predict influences of accidents on the marine environment. The instantaneous point source model is a useful method to simulate the large-scale radionuclide diffusion in marine areas. However, the simulation accuracy of this method requires improvement as it didn’t take radionuclide decay into account. In this study, an improved instantaneous point source model considering radionuclide decay was proposed on the basis of the original model. Furthermore, the instantaneous point source model and the improved version were used to simulate the concentrations of 131I and 137Cs following the Fukushima Dai-ichi nuclear power plant accident. The results showed that the relative error of 131I concentrations decreased from 136.03% to 37.59% when using the improved model; and improvements in relative errors for 137Cs concentrations were not apparent as the simualtion period was much shorter than its half-life period. Therefore, the improved model can accurately simulate the diffusion process for radionuclides following an accident and provides an efficient decision support tool for risk assessment managers and for use in safety guarantees of nuclear power plants during siting and operational phases.

A numerical model with Stokes drift for pollutant transport within the surf zone on a plane beach

This study examines the effects of Stokes drift on pollutant transport within the surf zone on a plane beach both numerically and experimentally. Firstly, the numerical model is described. The wave-induced current is modeled using the concept of the radiation stress. The wave propagation model is based on the wave energy conservation equation. And the advective diffusion model including the Stokes drift is used to describe the pollutant transport in the surf zone. Model validation was achieved in this case versus an analytical solution for an instantaneous point source in a uniform horizontal flow. This study also describes a laboratory experiment on dye release in the surf zone over a plane beach. We examined the final inclination angle required by a continuously released pollutant plume to reach the shoreline under both cases, and transport velocities in the alongshore and cross-shore directions were estimated by linearly fitting the location of a dye-patch front at different time. Results show that this dye patch moved shoreward with an approximate speed of 0.05 m/s (0.017 m/s) between 10 s and 40 s and 0.001 m/s (0.011 m/s) after 40 s for Case 1 (2). This model was then used to simulate pollutant transport in the surf zone on a plane beach as reproduced in the current experiment. Comparisons between our dye transport experiment and numerical results were then also conducted; the data showed that the numerical results including Stokes drift agreed more closely with experimental results than those without it. The data showed that the pollutant was generally transported obviously shoreward in addition to its expected drift along the shore. We also suggest that Stokes drift plays an important role in pollutant movement in the surf zone, especially shoreward.

Temperature Distribution in the “Material-Coating” Interface from a Fast-Moving Heat Emission Source for Electromagnetic Surfacing

During simulation of the surfacing process, it is proposed to present the heat emission source in the form of the system of instantaneous point sources flickering for some time; and the period between flashes tends to zero. The final results of the temperature distribution are determined by summing the solitary impacts according to the proposed computational pattern. The obtained characteristics and analytical curves allow presenting, at a first approximation, the value and intensity of the temperature fields in the working zone during the electromagnetic surfacing of the large size blanks with sufficient heat removal from the restored surface. The proposed method may be applied for repair and modernization of the railway vehicles elements.

Analytical Models for 3-D Diffusion of Ions from Salt-Contaminated Aggregates

Abstract This article presents an analytical investigation of the mobility of salts from contaminated aggregates in concrete. Salt-contaminated aggregates may have varied effects on the mechanical properties and durability of concrete. These depend primarily on the mobility of salts within the concrete matrix. Existing diffusion-based models for the mobility of salts in concrete focus on their intrusion from external sources (e.g., chloride penetration from deicing salts and brines). Such problems are well described by the closed-form solution of Fick’s law for diffusion in one dimension from a continuous source. Salt-contaminated aggregates represent a case of diffusion from a finite internal source rather than intrusion from a continuous external source. When the source is internal—as in the case of salt-contaminated aggregates—diffusion occurs in three dimensions and the source is finite. The 1-D solution is ill conditioned to model this problem, so new diffusion models must be introduced. This article presents two models of varying complexity that are well conditioned to model diffusion of salts from contaminated aggregates. First, the problem is modeled using Fick’s law for the simplified case of 3-D diffusion from an instantaneous point source in an infinite medium. Contaminated aggregates are treated as infinitesimal slugs of diffusing material. The same problem is then modeled using Fick’s law for the more realistic case of 3-D diffusion from an instantaneous spherical source, where contaminated aggregates are treated as approximately spherical. The intent of this article is to present an academic discussion on how chloride ions might migrate within the concrete matrix if they are assumed to do so by diffusion alone.

Temperature field due to a moving heat source in machining orthotropic composites with arbitrary fiber orientation

Milling of fiber reinforced plastics (FRP) is challenging with respect to surface integrity and tool wear due to high process temperatures. The maximum temperatures occurring in the workpiece determine the extent of the matrix decomposition area. Improving the workpiece quality therefore requires an understanding of its internal temperature distribution during milling. However, steep temperature gradients and high mechanical stress make measurements of temperature fields in the cutting zone difficult. In addition, thermal properties change depending on the fiber orientation in the case of orthotropic FRP.In this paper, an existing model describing the temperature field for isotropic materials is extended to unidirectional, orthotropic FRP. Here, the thermal impact of an end mill upon the machined surface is represented by a strip-shaped heat source which moves with the feed velocity along a semi-infinite space. Starting with the temperature field of an instantaneous point source, multiple integration steps and a coordinate transformation lead to the temperature field of the moving strip source for orthotropic materials with arbitrary fiber orientation.Using this analytical approach, two-dimensional temperature fields within the workpiece can be calculated for various feed velocities, heat source widths and fiber orientation angles. A cross verification of the analytical solution is successfully carried out by comparing it to a numerical simulation. Furthermore, temperature measurements during end milling of carbon fiber reinforced plastics using thermocouples confirm these results for different fiber orientations.The derived model can be applied to a variety of heat flow problems relevant for orthotropic materials, e.g. other machining technologies.

Method for determination of technological mode parameters of electron-beam welding based on the application of optimality criterion with the view on the weld pool uniform heating

The paper presents the results of mathematical modelling of temperature fields when the AMG-6 alloy is heated with energy sources equivalent to an electron beam in electron-beam welding. The evaluation of the results was performed using the criterion of optimality proposed by the authors. Used as a functional for calculations was the normalized standard deviation of the temperature from heating by a complex heat source consisting of mobile instantaneous point and linear energy sources. Proposed in this work are both an algorithm for finding solutions using an optimality criterion and the results of calculations for plates with thicknesses of 0.12cm and 1.2cm from the AMG-6 material, which were compared with experiments carried out under laboratory conditions.

Modelling of the welded seam parameters at electron-beam welding

The article presents the results of technological parameters mathematical modeling when heating a titanium alloy by energy sources equivalent to an electron beam in electron-beam welding. Analysis and evaluation of simulation results was carried out using the criterion of optimality proposed by the authors. As a tool for calculations, the authors used a functional based on mathematical models of metal heating by a complex heat source consisting of mobile instantaneous point and linear energy sources. The results of calculations for a 0.12 cm thick plate are given in the paper. They were compared with the data obtained from prototypes from the VT-14 material using the design welding regimes carried out in the laboratory on an A306.13 electron beam unit.

Time evolution of the heat diffusion phenomenon from the point source near the interface

The characteristic of the mathematical solution describing linear heat diffusion from an instantaneous point source near the interface of two semi-infinite domains in perfect thermal contact is discussed. Because spherical symmetry is lacking and the solution cannot be scale-transformed into a time-independent shape, the only way to grasp the geometrical information of the solution is by ad-hoc computation. The solution which was first derived more than 65 years ago is re-arranged to permit physical interpretation. Through the term-by-term analysis, the time evolution of the solution is divided into three stages and this enables systematic understanding of its nature.