Three-dimensional Mathematical Model Research Articles

Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L2 as the time step τ tends to zero at the rate O(τ).

Helmholtz and the geometry of color space: gestation and development of Helmholtz’s line element

Modern color science finds its birth in the middle of the nineteenth century. Among the chief architects of the new color theory, the name of the polymath Hermann von Helmholtz stands out. A keen experimenter and profound expert of the latest developments of the fields of physiological optics, psychophysics, and geometry, he exploited his transdisciplinary knowledge to define the first non-Euclidean line element in color space, i.e., a three-dimensional mathematical model used to describe color differences in terms of color distances. Considered as the first step toward a metrically significant model of color space, his work inaugurated researches on higher color metrics, which describes how distance in the color space translates into perceptual difference. This paper focuses on the development of Helmholtz’s mathematical derivation of the line element. Starting from the first experimental evidence which opened the door to his reflections about the geometry of color space, it will be highlighted the pivotal role played by the studies conducted by his assistants in Berlin, which provided precious material for the elaboration of the final model proposed by Helmholtz in three papers published between 1891 and 1892. Although fallen into oblivion for about three decades, Helmholtz’s masterful work was rediscovered by Schrödinger and, since the 1920s, it has provided the basis for all subsequent studies on the geometry of color spaces up to the present time.

Study on transport phenomena and performance of proton exchange membrane fuel cell with radial flow fields

The radial flow field structure, which has the advantages of low pressure drop, good water removal and good mass transfer, is an emerging structure for proton exchange membrane fuel cell (PEMFC) flow fields. However, optimization work on the design the structure is scarce for a complex structure that is difficult to manufacture. A comprehensive three-dimensional, non-isothermal and single-phase mathematical model is developed to describe the flow and heat, mass and charge transfer processes in a PEMFC. The transport phenomena and cell performance of radial flow fields with different gradient channels and parallel flow fields are studied and compared using the commercial computational fluid dynamics (CFD) software Fluent®. The distribution of oxygen concentration, pressure drop and temperature with different radial lengths is obtained. The effects of gradient channels, gas supply modes and radial lengths on cell performance are investigated. The results show that radial flow fields could offer more uniform oxygen distributions and lower pressure drops compared with parallel flow fields. Larger gradient channel sizes contribute to larger transfer volumes and higher gas molar concentrations in the catalyst layers and lower pressures in channels. The counter-flow supply mode is superior to the co-flow supply mode because it can enable a higher oxygen velocity and hence a higher current density.

Numerical Evaluation of Wind Speed Influence on Accident Toxic Spill Consequences Scales

Abstract This study aims to evaluate numerically the influence of wind speed on scales of environmental harmful consequences caused by accidentally spilled toxic liquid evaporated from the surface of a free-form outlined spill spot. A coupled problem of the gas-dynamic movement of a toxic air-mixture cloud in the atmosphere’s surface layer under the influence of wind and a negative toxic inhalation impact on a human in an accident zone is solved by means of mathematical modelling and computer experiment. A three-dimensional non-stationary mathematical model of the turbulent movement of a gas-air mixture is used for obtaining distribution of relative mass concentration of toxic gas impurities in time and space. A probabilistic impact model based on using a modernized probit analysis method is used to obtain fields of conditional probability of a fatal human injury resulting from toxic gas inhalation. This technique allows environmental safety experts assessing the scale of considered type technogenic accident consequences numerically depending on wind speed conditions and elaborating the means to mitigate them to acceptable levels.

Quantification of Ride Comfort Using Musculoskeletal Mathematical Model Considering Vehicle Behavior

This research aims to quantify driver ride comfort due to changes in damper characteristics between comfort mode and sport mode, considering the vehicle’s inertial behavior. The comfort of riding in an automobile has been evaluated in recent years on the basis of a subjective sensory evaluation given by the driver. However, reflecting driving sensations in design work to improve ride comfort is abstract in nature and difficult to express theoretically. Therefore, we evaluated the human body’s effects while driving scientifically by quantifying the driver’s behavior while operating the steering wheel and the behavior of the automobile while in motion using physical quantities. To this end, we collected driver and vehicle data using a motion capture system and vehicle CAN and IMU sensors. We also constructed a three-dimensional musculoskeletal mathematical model to simulate driver movements and calculate the power and amount of energy per unit of time used for driving the joints and muscles of the human body. Here, we used comfort mode and sport mode to compare damper characteristics in terms of hardness. In comfort mode, damper characteristics are soft and steering stability is mild, but vibration from the road is not easily transmitted to the driver making for a lighter load on the driver. In sport mode, on the other hand, damper characteristics are hard and steering stability is comparatively better. Still, vibration from the road is easily transmitted to the driver, which makes it easy for a load to be placed on the driver. As a result of this comparison, it was found that a load was most likely to be applied to the driver’s neck. This result in relation to the neck joint can therefore be treated as an objective measure for quantifying ride comfort.

Mathematical modeling and simulation analysis of different structured cored wire feeding spheroidization by finite volume method

<abstract><p>A three-dimensional dynamic heat transfer mathematical model of the process when cored wire feed in molten iron is established based on finite volume method (FVM). The calculation area is meshed with triangles and quadrilaterals to determine nodes and control volumes, and implicit time integration method is used to ensure the stability of calculating process. For exposing the dynamic heat transfer behavior, the variation of temperature field and explosion characteristics of cored wires are studied. In addition, the melt loss rate of the top end of cored wire and the correlation among melt explosion depth, molten iron temperature and feeding speed of cored wires are theoretically calculated. More importantly, the influence of different structures of cored wires on the absorption rate of magnesium are considered. The calculation results are in good agreement with the experimental data, which indicate that the existing theoretical model has good validity and can provide theoretical guidance for spheroidization process in molten iron.</p></abstract>

Периодические режимы горения термически сопряженных СВС-систем с термокапиллярным течением расплава

This paper presents a three-dimensional mathematical model of self-propagating high-temperature synthesis (SHS) of a three-layer “sandwich” sample. The layers are formed from gasless mixtures with the addition of an inert fusible component. The mathematical model is studied numerically using the finite-difference method. The unsteady periodic regimes of gasless combustion of the three-layer sample with square cross-section are revealed with account for melting and thermocapillary flow of the melted inert component of the mixture. The unsteady periodic combustion regimes are specified depending on the relative calorific value of the mixture in the inner layer. High-temperature points move along the side faces of the sample. The velocity of the points’ motion along the combustion surface is much higher than the average burning velocity of the sample. An increase in the melt flow velocity leads to the equalization of the temperature field and stabilization of the combustion regime. The quasi-stationary regimes of control and fusion are studied during the combustion of the sample with an active inner layer, when the intrinsic burning velocities of the donor and acceptor mixtures are close to each other.

Numerical and experimental investigation on thermal performances of quartz tube gravity-driven solid particle solar receiver based on linear-focused solar furnace

This paper reported a novel quartz tube gravity-driven SPSR. A test platform of the receiver with a drop length of 4 m based on a linear-focused solar furnace was built. The optical performances of the linear-focused solar furnace were measured by a direct method. Solid particles were heated to a temperature of 156∼308°C after a single pass with an averaged irradiance of 14.9∼21.7kW/m2. A three-dimensional mathematical model with the wavelength dependent radiative properties integrated is built and discretized in OpenFOAM, which is later verified by experimental results. Lastly, the influences of several pertinent parameters on thermal performances of the receiver are studied using the proposed model. It was observed that the green-house effect of the quartz tube could reduce the radiative heat losses by a percentage from 2.1% to 9.4%. Wind speed surrounding the quartz tube has a positive effect on the green-house effect since high wind speed is helpful to decrease the outer surface temperature of the tube. An optimal thermal efficiency of 57.2% was obtained when heating particles from 27 °C to 800 °C by circulating particles for four times. The proposed model could be used for the design and optimal operation of the quartz tube gravity-driven SPSR.

MATHEMATICAL MODELING AND ECOLOGICAL DESIGN OF THE MARINE SYSTEMS TAKING INTO ACCOUNT MULTI-SCALE TURBULENCE USING REMOTE SENSING DATA

The paper considers a mathematical model of biological kinetics and geochemical cycles based on a system of convection-diffusion equations with nonlinear coefficients, supplemented by a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir, with a refined coefficient of turbulent vertical exchange. The task of monitoring the water surface in order to detect phytoplankton spots involves the creation and verification of effective methods for clustering these objects on the surface of reservoirs, in particular, restoring their boundaries based on remote sensing data. The article uses multispectral satellite images as sounding data. Based on the obtained images of plankton populations, the initial conditions for mathematical models of biogeochemical cycles can be determined, on the basis of which prognostic calculations are performed.

Development of control algorithms for magnetoelectric generator with axial magnetic flux and double stator based on mathematical modeling

The object of this research is electromechanical processes in a generator with an axial magnetic flux and a double stator and an additional non-contact excitation winding operating as part of autonomous electric power systems. The power of the additional excitation system is about 2 % of the generator power. The presence of an additional non-contact winding, which is powered by direct current, makes it possible to control the generator voltage by changing the excitation current. This resolves the task to stabilize the output voltage of the generator with permanent magnets when the load and shaft speed change. This paper reports the construction of a three-dimensional field axisymmetric mathematical model of the generator under study, which has made it possible to calculate and investigate its characteristics and parameters, in particular the magnitude of magnetic induction in all structural elements. The model built takes into account the influence of finite effects, magnetic scattering fields, and the radial-axial nature of the closure of the main magnetic flux and the magnetic flux of the additional excitation winding. The use of a structure with a double stator makes it possible to more efficiently utilize the usable volume of the generator and to increase its power. A mathematical model of the generator in the d-q coordinate system was built, which has made it possible to synthesize algorithms for controlling the automatic voltage stabilization system of the generator voltage under conditions of change in load and shaft speed. Control algorithms were developed on the basis of the concept of inverse dynamics problems in combination with minimizing local functionalities of instantaneous energy values, which ensures that the system is robust when changing generator parameters and that regulators are implemented in a simple way, due to the lack of differentiation operations. Based on the models built and algorithms developed, the quality of control of the generator's output voltage when the load and frequency of the generator change was investigated by modeling in the MATLAB/Simulink environment. When setting a jump in the rated load and changing the rotational speed within ±15 % of the rated value, the automatic stabilization system provides astatic voltage control at a given level of 48 V. The results can be practically used in the design of autonomous electric power systems with high energy conversion efficiency, in particular wind turbines and hydraulic units

Modeling Study on Melt Flow, Heat Transfer, and Inclusion Motion in the Funnel-shaped Molds for Two Thin-Slab Casters

For the purpose of studying compact strip production (CSP) funnel-shaped mold and flexible thin-slab rolling (FTSR) funnel-shaped mold, a three-dimensional (3D) multi-field coupling mathematical model was established to describe the electromagnetic braking (EMBr) continuous casting process. To investigate the metallurgical effect of EMBr in the CSP and FTSR funnel-shaped thin-slab molds, a Reynolds-averaged Navier–Stokes (RANS) turbulence model, together with an enthalpy–porosity approach, was established to numerically simulate the effect of ruler EMBr on the behaviors of melt flow, heat transfer, solidification, and inclusion movement in high-speed casting. The simulation results indicate that the application of ruler EMBr in the CSP and FTSR molds shows great potential to improve the surface temperature of molten steel and reduce the penetration depth of downward backflow. This contributes to the melting of the slag rim near the meniscus region and facilitates the floating removal of the inclusions in the molten pool. In addition, in comparison with the case of no EMBr, the parametric study shows that the braking effect of ruler EMBr with an electromagnetic parameter of 0.5 T can enhance the upward backflow in the two high-speed thin-slab molds. The enhanced upward backflow can successfully entrain the inclusions to the top of the mold and improve the activity of surface fluctuations to avoid the formation of the slag rim. For instance, for the ruler EMBr applied to the FTSR mold, the maximum amplitude of surface fluctuation and the floatation removal quantity of inclusions with a diameter of 100 μm are increased by 4.6 percent and 51 percent, respectively.

Numerical Simulation of a Three-dimensional Subsea Cable with Buoyancy Module

Extreme formation of slacks and entanglement with risers induced from a small diameter and high structural flexibility of subsea cable is often being a huge challenge in subsea cable installation. In many cases, the cable is prone to breaking, bending, or twisting due to the rough and dynamic underwater condition. One of the alternatives is the use of buoyancy module to facilitate the cable installation and to reduce cable tension and false slacks. This paper proposes reliable three-dimensional mathematical model and simulation to specifically investigate the behaviour of subsea cable with buoyancy module during laying operation. A finite difference method (FDM) and optimization algorithm is used to discretize the formulation and simulate this type of subsea cable problem concerning on the tensional analysis. The methodology is specifically designed to allow cable configuration to be changed over time as a result from the buoyancy module. The viability subsea cable configuration with existence of buoyancy module has been showed successfully reduce the cable tension especially at the hang-off section.

Об абсолютной сходимости рядов в топологических пространствах

Calculation and evaluation of wind currents hydrophysical characteristics of large reservoirs is the purpose of this work. Adequate dynamic mathematical models are necessary for conducting these studies and constructing appropriate algorithms implemented on supercomputers. The shallow water reservoirs hydrodynamic model is developed on the basis of three-dimensional mathematical model that includes the Navier — Stokes equations of motion in three coordinate directions, the equations of continuity of an incompressible fluid, the equations of heat transport and the equations of state or the density dependence of the aqueous medium on temperature. The discretization of the hydrodynamic equations is carried out by the pressure correction method. The developed numerical algorithms and the software package that implements them are used to study the pressure field and the aquatic medium velocity vector field the for a given section of the reservoir water area. The practical significance lies in the possibility of their application for studying the hydrophysical processes of these territories, assessing the hydrodynamic impact on the large reservoirs’ coastal zone.

Dynamics of a simple mathematical model for unemployment with time delay

Abstract A three-dimensional mathematical model is analyzed in order to study the labour market, with an emphasis on unemployment, employment and the number of open positions. The rate of change of new positions created by the public and private sectors depends on the historical levels of unemployment, taking into account the distributed time delay. The non-dimensional mathematical model is done and the existence of the equilibrium points is examined. The findings related to the positivity and boundedness of solutions are presented. In addition the global asymptotic stability of the positive equilibrium is provided. The theoretical results are exemplified by the numerical simulations.

Численное моделирование воздействия ветровых течений на прибрежную зону крупных водохранилищ

Calculation and evaluation of wind currents hydrophysical characteristics of large reservoirs is the purpose of this work. Adequate dynamic mathematical models are necessary for conducting these studies and constructing appropriate algorithms implemented on supercomputers. The shallow water reservoirs hydrodynamic model is developed on the basis of three-dimensional mathematical model that includes the Navier — Stokes equations of motion in three coordinate directions, the equations of continuity of an incompressible fluid, the equations of heat transport and the equations of state or the density dependence of the aqueous medium on temperature. The discretization of the hydrodynamic equations is carried out by the pressure correction method. The developed numerical algorithms and the software package that implements them are used to study the pressure field and the aquatic medium velocity vector field the for a given section of the reservoir water area. The practical significance lies in the possibility of their application for studying the hydrophysical processes of these territories, assessing the hydrodynamic impact on the large reservoirs’ coastal zone.

Implementation of a three-stage scheme for the co-combustion of pulverized coal and coal-water slurry in an industrial boiler to reduce NOx emissions

In this paper, the co-combustion process of coal-water slurry (CWS) and pulverized coal fuel (PCF) in E500 pilot boiler when using staged afterburning scheme, was investigated based on numerical simulation. A complex three-dimensional mathematical model and numerical methodology was used to describe the CWS and PCF co-combustion, as well as subprocesses in the combustion chamber. The proposed mathematical model was tested based on data obtained from a full-scale experiment. A good compliance was shown between the computational results and the experimental data. A detailed comparative analysis of the effect of changes in the design of a pilot boiler when using a three-stage combustion scheme and the flare-drip combustion technology for CWS on the physical-chemical processes in the combustion chamber and environmental indicators was carried out for the first time. The change of circulation zones depending on the method of coal fuel supply during three-stage combustion was investigated. Calculations have shown that the implementation of the proposed changes reduced the underburning of solid carbon from 1.9% to 0.51%. It was revealed that using CWS as a reducing agent diminishes the amount of harmful NOx emissions by more than 40% compared to the basic version.

Designing a voltage control system of the magnetoelectric generator with magnetic flux shunting for electric power systems

The object of research in this work is a three-phase magnetoelectric generator with magnetic flux shunting based on industrial induction electric motors. The presence of a magnetic shunt makes it possible to control the voltage of the generator by changing the excitation current in the non-contact electrical winding of the magnetic shunt, which is powered by direct current. Thus, the problem of stabilization of the output voltage of the generator with permanent magnets is solved when the speed of rotation and load change. This paper reports the construction of a three-dimensional field mathematical model of the generator, which allows for electromagnetic calculations of the generator with specified parameters, taking into consideration the influence of final effects, magnetic scattering fields, as well as their radial-axial nature. The results of the calculation of the electromagnetic field are the initial parameters for building a simulation model in the MATLAB-Simulink environment. A simulation model of a magnetoelectric generator with magnetic flux shunting under conditions of changing rotational speed and load has been constructed in the MATLAB-Simulink environment. On the basis of the built models, the performance characteristics of a magnetoelectric generator with magnetic flux shunting were established, which show the limits of control of the output voltage. Adjusting characteristics were determined at zero and rated shunt current for different types of load. The adjusting characteristics of the generator are presented at the rated voltage of the generator for different types of load and with an increase to 150 % of the rated value. The study's results show the high efficiency of the voltage control system of a magnetoelectric generator with a magnetic shunt at different speeds of rotation and load

DEVELOPING A «LAKE – CATCHMENT» INFORMATION ANALYSIS SYSTEM AS A METHOD FOR BASIC RESEARCH AND A TOOL FOR DECISION-MAKING (LAKE ONEGO CASE STUDY)

An information analysis system (IAS) is being developed to assess the state and predict the dynamics of a lake-catchment system. Forecasting of the state of the lake ecosystem in the context of global climate change and active exploitation of its resources (water, energy, recreational and biological resources, transport) is done using a three-dimensional mathematical model of the lake ecosystem, coupled with a model of runoff from the catchment. Lake and catchment models are combined with a database and a knowledge base to form a single IAS. The prognostic assessments of the seasonal and long-term dynamics of the lake ecosystem performed using the IAS serve as the basis for identifying the possible economic, social and cultural consequences of regional and global changes. Operational IAS is necessary to create a management decision support system.

Air dehumidification performance study of a desiccant wheel by a three-dimensional mathematical model

Hitherto air-conditioners have severe shortcomings because of the inherent need to maintain the cooling coil temperature significantly below the air's dew point. Their energy efficiency can be improved by employing desiccant wheel dehumidifiers that offer independent humidity control. Although numerous experimental and theoretical studies have evaluated desiccant wheels’ performance, no existing investigation yields a detailed account of its localized temperature and humidity distributions. Therefore, a new three-dimensional mathematical model was developed to study the time-dependent and steady-state temperature, humidity, and reaction distributions. The model predicted the performance with the maximum discrepancy of ±10% in both temperature and humidity ratio. It also revealed that the countercurrent airflow configuration has a more uniform reaction distribution than the concurrent configuration. The parametric analysis using the model revealed that higher regeneration temperatures require smaller regeneration angles. With the regeneration temperatures ranging between 110 and 50 °C, the optimal regeneration angles span 120–180°. Additionally, the dehumidification performance could be easily improved by increasing the wheel's rotation speed at low desiccant mass since coating more desiccant on the wheel is costly and inconvenient.

A Model-Based Investigation of the Performance Robustness of the Deflector Jet Servo Valve

Although it possesses the capabilities of both anti-contamination and rapid response, the deflector jet servo valve is still confronted with the issue of weak performance robustness, usually manifesting as considerable uncertainty of the product pressure indices. To elaborate on the complex internal mechanism of deflector jet valves and figure out the origins of performance fluctuations, a three-dimensional mathematical model of the deflector jet pilot valve is constructed, in which a series of assumptions are presented to apply specific theorems to different regions of the flow field. Numerical simulations and experiments show that this theoretical system can provide a distinct and logical explanation for both the internal flow distribution and the external performance of the servo valve. On this basis, the causes of performance fluctuations are discussed, concerning the installation error of the deflector and the machining error of the shunt wedge. Calculations show that the latter can bring about remarkable performance variation. Quantificationally, a 10 micron width error of the shunt wedge will induce 7.4% and 3.6% drifts of the receiver pressure and the pressure gain, respectively. However, further analyses confirm that a decrease in the deflector jet distance will lead to dramatic deterioration of the valve’s susceptibility to errors. Hence, it is concluded that to enhance the performance robustness of servo valve products, the machining accuracy of the shunt wedge and non-negative errors of the deflector jet distance should both be guaranteed.