Partial Analytical Solution Research Articles

Решение краевой задачи тепломассопереноса методом источников для условий лучистого обогрева почвы

The purpose of the research is to find, predict and regulate the temperature and humidity regime of the soil for the conditions of radiant heating of cultivation facilities using dark-type ceiling infrared emitters. A system of differential equations (in dimensional and dimensionless forms) is presented, reflecting the relationship of thermal and mass transfer processes in colloidal capillary-porous bodies in the case of surface radiant heating. A partial analytical solution of this system of differential equations for a semi-bounded body is considered with the exclusion of the phenomenon of thermal and thermal conductivity and period diffusion processes. On the example of milling peat, taking into account the initial data, the solution of the boundary value problem of heat and mass transfer by the method of sources is obtained, which is one-dimensional non-stationary fields of moisture content and temperature. It is established that under the given initial and boundary conditions, as well as taking into account the thermophysical properties of milling peat, the required moisture content will be achieved in six hours, the temperature in three hours. By adjusting the thermal power of the infrared radiation source, and, therefore, the intensity of moisture evaporation from the soil surface into the environment, it is possible to control the rate of heating and drying of the layer (for example, to determine the time or frequency of watering the soil). To clarify the values of moisture content and soil temperature by coordinate and over time, it seems expedient to consider analytical solutions that took into account not only basic, but also conjugate processes of heat and mass diffusion

A coupled model of composite synthesis in combustion regime

The work proposes a model for synthesising a ‘metallic matrix–reinforcing inclusions’ composite in a simplified reaction scheme. A suggested mathematical model takes into account the interdependence of thermal, chemical and mechanical phenomena. The problem includes kinetic equations for determining the composition of the synthesis products. The reactions are initiated by a thermal impulse from a surface. It was established that after some mathematical transformations and partial analytical solution, the model includes equivalent parameters of the reactions and equivalent thermophysical properties that depend on the mechanical modules. The model allowed us to study two variants of composite synthesis: in the quasi-stationary mode for a sample whose dimensions exceed the size of the heating and reaction zones, and in the unsteady mode for a finite-sized sample. It is shown that the consideration of the coupling of thermal, chemical and mechanical processes changes all the characteristics of conversion propagation and the values of stresses and strains in the reaction zone. The stresses in the reaction zone and in the products are affected by the parameters characterising the reaction itself. It is shown that there is a range of parameters at which a stable combustion wave is not formed.

Two temperature generalized thermoelasticity involving memory-dependent derivative under fuzzy environment

A one-dimensional two-temperature model of generalized thermoelasticity theory in half-space has been studied in this present discussion with memory-dependent derivative. The fuzzy approach has been applied to construct the governing equations. The fuzzy variables such as stress, strain and so on are represented in r-cut. The coupled partial differential equations' analytical solutions are obtained in Laplace transform domain subject to a stress-free boundary with a time-dependent imprecise thermal shock. A suitable numerical inverse Laplace transformation technique is applied to get the space time-domain results for different time delay parameter values and several kernel functions. Two examples of numerical results in the fuzzy environment are performed and are compared with the crisp results graphically. The concluding remarks are made based on numerical results and graphs.

1D Finite Volume Scheme for Simulating Gas–Solid Reactions in Porous Spherical Particles with Application to Biomass Pyrolysis

Models for gas–solid reactions in porous particles typically consist of a set of mass and energy balances in the form of conservation equations. For spherical or close to spherical particles, these equations are formulated in 1D spherical coordinates. In the case where accumulation of gas inside the particle is significant, the balance equations contain convective terms. The present work presents a simple numerical scheme based on flux limited finite volume methods for discretizing conservation equations for convective and diffusive transport of mass and energy in radial direction in a porous sphere. The velocity is governed by Darcy’s law coupled to an equation of state. The proposed scheme is applied to a series of test problems that admit full or partial analytical solutions. For the cases where only partial analytical solutions are available, a Comsol model is adopted for comparison. It is found that the scheme is able to resolve step gradients without generating oscillations and that it properly handles changes in the sign of the convective velocity. Applying the scheme for solving a common model for biomass pyrolysis reveals the importance of convective gas transport in the pyrolysis of thermally thick biomass particles.

К ОГРАНИЧЕННОЙ ЗАДАЧЕ ТРЕХ ТЕЛ

The paper analytically investigates the classical restricted three-body problem in a special non-inertial central coordinate system, with the origin at center of forces. In this coordinate system, an analytical expression of the invariant of the centre of forces is given. The existence of the invariant of the centre of forces admits the correct division of the problem into two problems. The first is a triangular restricted three-body problem. The second is a collinear restricted three-body problem. In this paper the collinear restricted three-body problem is investigated. Using the properties of the invariant of centre of forces of the restricted three-body problem in the special non- inertial central coordinate system, the basic differential equations of motion for the collinear restricted three-body problem are obtained when three bodies lie on the same line during all motion. Differential equations of the collinear restricted three-body problem in the rotating non-inertial central coordinate system in pulsating variables are derived. New differential equations of motion for the collinear restricted three-body problem in three regions of possible location of the massless body with stationary solutions corresponding to the three Euler libration points have been derived. The circular collinear restricted three-body problem is investigated in detail. The corresponding Jacobi integrals are obtained. New exact non-stationary partial analytical solutions of the obtained new differential equations of motion of the collinear restricted three-body problem have been found for the considered case.

Results of studies of processes of clearing of gaseous contaminants and ventilating the atmosphere of habitable pressurized modules in a space station

The paper presents partial analytical solutions for equations describing variation in trace amounts of carbon dioxide in the atmosphere of habitable spaces within pressurized modules (PM) of a space station. The solutions may find practical application in calculations of concentrations for any contaminants, or for air flow through the purification system. It is shown that in a case where low-toxic contaminants are released into the PM atmosphere, it would be enough to cycle 3 volumes of the PM air through the purification system when the system operates without the breakthrough concentration (without the residual concentration of the contaminant at the outlet from the purification system), in order to achieve the 95% purification of the atmosphere. For highly toxic contaminants this value should be significantly increased depending on the maximum allowable concentration of the substance (47 volumes and more — up to 99.9% purification). The paper also considers variation in the concentration of the contaminant in the atmosphere during intermixing of atmospheres between PMs using intermodular ventilation. As a result, new analytical solutions were obtained for practical calculations which make it possible to determine gaseous contaminant concentration at any point in time and the time of the final equalization of the contaminant concentration in the space station atmosphere. It was determined that the time needed for complete mixing of gaseous contaminants through intermodular ventilation between two PMs does not depend on the initial concentrations of the contaminants (and only depends on the PM volumes and the intermodular ventilation flow rate). Key words: space station, pressurized module atmosphere, carbon dioxide, atmosphere purification, variation in concentration, air flow, atmosphere mixing.

RESULTS OF STUDIES OF THE PROCESSES OF CLEARING OF GASEOUS CONTAMINANTS AND VENTILATING THE ATMOSPHERE OF HABITABLE PRESSURIZED MODULES IN A SPACE STATION

The paper presents partial analytical solutions for equations describing variation in trace amounts of carbon dioxide in the atmosphere of habitable spaces within pressurized modules (PM) of a space station. The solutions may find practical application in calculations of concentrations for any contaminants, or for air flow through the purification system. It is shown that in a case where low-toxic contaminants are released into the PM atmosphere, it would be enough to cycle 3 volumes of the PM air through the purification system when the system operates without the breakthrough concentration (without the residual concentration of the contaminant at the outlet from the purification system), in order to achieve the 95% purification of the atmosphere. For highly toxic contaminants this value should be significantly increased depending on the maximum allowable concentration of the substance (47 volumes and more — up to 99.9% purification). The paper also considers variation in the concentration of the contaminant in the atmosphere during intermixing of atmospheres between PMs using intermodular ventilation. As a result, new analytical solutions were obtained for practical calculations which make it possible to determine gaseous contaminant concentration at any point in time and the time of the final equalization of the contaminant concentration in the space station atmosphere. It was determined that the time needed for complete mixing of gaseous contaminants through intermodular ventilation between two PMs does not depend on the initial concentrations of the contaminants (and only depends on the PM volumes and the intermodular ventilation flow rate). Key words: space station, pressurized module atmosphere, carbon dioxide, atmosphere purification, variation in concentration, air flow, atmosphere mixing.

Совершенствование методики расчёта нормативов упражнений с пожарно-техническим оборудованием

Introduction. The article deals with improvement of the differential model of training for decision-making when setting standards of training with fire-fighting equipment. Using a differential training model, the study of the setting training standards with new models of technology and fire-technical equipment was carried out. Goals and objectives. The aim of the study is to improve the training model for calculating the training standards and predicting the number of approaches to ensure the successful development of the training in the framework of the management of professional training of firefighters. Methods. The process of obtaining new scientific results can be divided into three stages. At the first stage, when improving the training model, methods for solving differential equations with separable variables were used. At the second stage of the study, the Student's criterion and elements of probability theory were used in the development of training standards with fire-fighting equipment. At the third stage, the Pearson determination criterion was used in the process of assessing the adequacy of the results. Results and discussion. The authors obtained a partial analytical solution of the general differential training model for the case of linear dependence of the training function on the results of the exercise. A method for calculating standards for exercises with fire-fighting equipment based on truncated data is proposed and the results of calculating standards are compared with the generally accepted method for setting standards for exercises with fire-fighting equipment. A method has been developed for determining the values of the function of training based on the results of repeated exercises, taking into account the training factor. Conclusions. Thus, the proposed modification of the training model consists in establishing a linear dependence of the results of the exercises on the number of approaches, which allows planning actions for the successful implementation of exercises with fire-fighting equipment. The directions of further research in terms of the development of the results obtained in the management of professional training of personnel in fire departments have been determined. Keywords: firefighter training management, fire-fighting equipment, training standards, differential training model, decision-making

Solving procedure for the dynamics of charged particle in variable (time-dependent) electromagnetic field

In this challenging analytic survey, we present a new approach for solving equations of the dynamics of a charged particle (describing its motion in variable, time-dependent electromagnetic field) which has been applied earlier in various fields of mechanics for solving equations of hydrodynamics, Euler–Poisson equations of rigid body rotation and even in celestial mechanics (solution of equations of small body’s motion in the CR3BP problem near librations points): a new type of the solving procedure is implemented in all of these equations as well as in case of solving momentum equation for the aforementioned dynamics of a charged particle, determined by Lorentz force in non-relativistic case. Meanwhile, in each case the system of momentum equations has been successfully solved analytically. The main result of the current research should be formulated as follows: the analytic algorithm is pointed out for solving momentum equation, which has been reduced to the analytical solution of three nonlinear ODEs with respect to the components of velocity of the particle. Moreover, absolutely new partial analytical solutions have been obtained for the special cases of magnetic field (with zero electric field’s components). In addition to this, we conclude that system of Lorentz momentum equations for dynamics of a charged particle has not the analytical presentation of solution in case of nonzero, time-dependent electric field.

Rotating AdS-Einstein universes from constrained metrics

The imposition of a constraint between the metric tensor elements in both three- and four-dimensional, rotating AdS space-times is shown to reduce the number of independent equations of motion and to result in new families of solutions to the equations of motion. For the geometries investigated analytic solutions or partial analytic solutions of the equations of motion are obtained. In all cases the number of independent field equations is less than the number of independent functions, resulting in an undetermined function which can be freely specified. For rotating, asymptotically $AdS$ space-times the reduction of the number of field equations to be solved holds for vacuum black hole solutions and for black hole solutions obtained from space-times containing matter.

Photon electrodynamics and photon structure as a bunch of one of many possible states of electromagnetic field

Fundamental laws of conservation are used to show that electromagnetic field is generally represented (even in vacuum at ρ = 0 and j = 0) using four vectors D, E, B, and H with different equations of state (material equations) that are linear for electromagnetic waves and nonlinear for photons and particles. An equation that describes different states of electromagnetic field (i.e., different but not arbitrary relationships of field vectors E, H, D, and B) is derived. It is shown that electromagnetic wave and photon are different states of electromagnetic field that exhibit different dependences of energy density on field vectors. Partial analytical solutions are obtained for a photon (spatially localized bunch of electromagnetic field energy) that propagates at a velocity of light along a single (as distinct from electromagnetic wave) direction.

Generational Spreading Speed and the Dynamics of Population Range Expansion.

Some of the most fundamental quantities in population ecology describe the growth and spread of populations. Population dynamics are often characterized by the annual rate of increase, λ, or the generational rate of increase, R0. Analyses involving R0 have deepened our understanding of disease dynamics and life-history complexities beyond that afforded by analysis of annual growth alone. While range expansion is quantified by the annual spreading speed, a spatial analog of λ, an R0-like expression for the rate of spread is missing. Using integrodifference models, we derive the appropriate generational spreading speed for populations with complex (stage-structured) life histories. The resulting measure, relevant to locations near the expanding edge of a (re)colonizing population, incorporates both local population growth and explicit spatial dispersal rather than solely growth across a population, as is the case for R0. The calculations for generational spreading speed are often simpler than those for annual spreading speed, and analytic or partial analytic solutions can yield insight into the processes that facilitate or slow a population's spatial spread. We analyze the spatial dynamics of green crabs, sea otters, and teasel as examples to demonstrate the flexibility of our methods and the intuitive insights that they afford.

Kinetics of Solid–Solid Reactions: Influence of the Number of Contact Points

Dalvi and Suresh (AIChE J. 2011, 57, 1329−1338) proposed a model for the kinetics of a reaction between two particulate solids A and B, where B is the diffusing component, in terms of the average number of contact points NAB between the two types of particles. In the present work, the applicability and validity of the model is studied, taking the formation of tricalcium aluminate from its immediate precursors as a candidate reaction. Partial analytical solutions to the Dalvi–Suresh model have been derived, and a simple parameter-estimation methodology has been developed. The data clearly show an influence of NAB, and use of the Ginstling–Brounshtein model (Appl. Chem. USSR 1950, 23, 1327−1338) for parameter estimation leads to diffusivity values that depend on NAB in a systematic way. Parameters estimated using the Dalvi–Suresh model, on the other hand, show only a random scatter and also show reasonable agreement with the values of diffusivity of Ca in oxides.

An unconventional adaptation of a classical Gaussian plume dispersion scheme for the fast assessment of external irradiation from a radioactive cloud

This article focuses on derivation of an effective algorithm for the fast estimation of cloudshine doses/dose rates induced by a large mixture of radionuclides discharged into the atmosphere. A certain special modification of the classical Gaussian plume approach is proposed for approximation of the near-field dispersion problem. Specifically, the accidental radioactivity release is subdivided into consecutive one-hour Gaussian segments, each driven by a short-term meteorological forecast for the respective hours. Determination of the physical quantity of photon fluence rate from an ambient cloud irradiation is coupled to a special decomposition of the Gaussian plume shape into the equivalent virtual elliptic disks. It facilitates solution of the formerly used time-consuming 3-D integration and provides advantages with regard to acceleration of the computational process on a local scale. An optimal choice of integration limit is adopted on the basis of the mean free path of γ-photons in the air. An efficient approach is introduced for treatment of a wide range of energetic spectrum of the emitted photons when the usual multi-nuclide approach is replaced by a new multi-group scheme. The algorithm is capable of generating the radiological responses in a large net of spatial nodes. It predetermines the proposed procedure such as a proper tool for online data assimilation analysis in the near-field areas. A specific technique for numerical integration is verified on the basis of comparison with a partial analytical solution. Convergence of the finite cloud approximation to the tabulated semi-infinite cloud values for dose conversion factors was validated.

Genesis and evolution of polarization of light in the ocean [Invited

The radiative transfer of sunlight through the deep oceans of the world is a complex and only partially solved environmental optical problem. Empirically, in situ systematic measurements of key parameters such as polarization of deep open seawater have been very sparse in recent decades. Although we have the necessary equation of transfer to solve this complex problem, until it can be solved explicitly, only approximations and partial analytic solutions are possible in addition to some successful computer modeling. Further complexity is added by the diversity of researchers' interests from academic to international policy making, as well as the ineffective communication between the different disciplines concerned, ranging from mathematics to endangered species. As a result, isolated focused pockets of good data and theory have been developed in recent decades without the needed breadth of understanding. This present review intends to bring together some visual biology and optical physics in order to understand the role of polarization in navigation, communication, and identification of marine animals as well as a possible tool for remotely sensing underwater objects.

Scale transition theory: Its aims, motivations and predictions

Scale transition theory is an approach to understanding population and community dynamics in the presence of spatial or temporal variation in environmental factors or population densities. It focuses on changes in the equations for population dynamics as the scale enlarges. These changes are explained in terms of interactions between nonlinearities and variation on lower scales, and they predict the emergence of new properties on larger scales that are not predicted by lower scale dynamics in the absence of variation on those lower scales. These phenomena can be understood in terms of statistical inequalities arising from the process of nonlinear averaging, which translates the rules for dynamics from lower to higher scales. Nonlinearities in population dynamics are expressions of the fundamental biology of the interactions between individual organisms. Variation that interacts with these nonlinearities also involves biology fundamentally in several different ways. First, there are the aspects of biology that are sensitive to variation in space or time. These determine which aspects of a nonlinear dynamical equation are affected by variation, and whether different individuals or different species are sensitive to different extents or to different aspects of variation. Second is the nature of the variation, for example, whether it is variation in the physical environment or variation in population densities. From the interplay between variation and nonlinearities in population dynamics, scale transition theory builds a theory of changes in dynamics with changes in scale. In this article, the focus is on spatial variation, and the theory is illustrated with examples relevant to the dynamics of insect communities. In these communities, one commonly occurring nonlinear relationship is a negative exponential relationship between survival of an organism and the densities of natural enemies or competitors. This negative exponential has a biological origin in terms of independent actions of many individuals. The subsequent effects of spatial variation can be represented naturally in terms of Laplace transforms and related statistical transforms to obtain both analytical solutions and an extra level of understanding. This process allows us to analyze the meaning and effects of aggregation of insects in space. Scale transition theory more generally, however, does not aim to have fully analytical solutions but partial analytical solutions applicable for circumstances too complex for full analytical solution. These partial solutions are intended to provide a framework for understanding of numerical solutions, simulations and field studies where key quantities can be estimated from empirical data.

Representation reduction and solution space contraction in quasi-exactly solvable systems

In quasi-exactly solvable problems partial analytic solutions (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters. To obtain a larger solution space one varies the energy over a discrete set (the spectrum) by simply changing the value of a given integer. A unified treatment that includes the standard as well as the new class of quasi-exactly solvable problems is presented and a few examples are given. The solution space is spanned by discrete square integrable basis functions in which the matrix representation of the Hamiltonian is tridiagonal. Consequently, the wave equation gives a three-term recursion relation for the expansion coefficients of the wavefunction. Imposing quasi-exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component. The finite is real and exactly solvable, whereas the infinite is complex and associated with zero norm states. Consequently, the whole physical space contracts to a finite-dimensional subspace with normalizable states.

Destructive aggregation: Aggregation with collision-induced breakage

Mean-field population balance equations are used to describe the evolution of particle size distributions in a wide variety of systems undergoing simultaneous aggregation and breakage. In this paper we develop a population balance that includes aggregation combined with collision-induced particle breakage for arbitrary fragment distribution functions, provided that this distribution function depends only on the total mass of the particles undergoing a collision. We then develop a specific distribution function for arbitrary two-body collisions by postulating that each collision produces a transition-state aggregate having the morphology of a linear polymer. The behavior of the resulting equation is then analyzed for the case in which the collision kernel is a constant, and partial analytical solutions are derived and compared to corresponding Monte-Carlo simulation results. The computer simulations are then used to validate a proposed scaling law for the steady-state particle size distribution. Lastly, the behavior of the aggregation with collision-induced-breakage population balance equation is compared and contrasted with the behavior of an analogous aggregation with linear-breakage population balance equation.

Investigation of the charge and potential of a dust grain in a low-pressure plasma with allowance for ionization in the intergrain space

The formation of the ion flow to a dust grain and the distribution of the electric potential in a low-pressure dusty plasma are investigated theoretically with allowance for ionization in the intergrain space. Poisson’s equation similar to the Langmuir plasma-sheath equation is solved numerically with the use of partial analytic solutions at the boundary of the Seitz-Wigner cell and in thin layers in the intergrain space. The charge and potential of a dust grain are found as functions of the grain radius and cell size. The grain potential and the total cell potential energy as functions of the cell size display weak minima, whose positions correspond to the observed intergrain distance in dusty crystals.

Multi-Monte Carlo approach for general dynamic equation considering simultaneous particle coagulation and breakage

Particle size distribution is described by general dynamic equation (GDE). A new multi-Monte Carlo (MMC) method is promoted to solve GDE for simultaneous particle coagulation and breakage. MMC method is based on “time-driven” Monte Carlo technique and conserves constant number of fictitious particles and constant volume of computational domain with the evolution of time. Firstly, MMC method is described in details, which includes the scheme of simultaneous coagulation and breakage, the introduction of “weighted fictitious particle”, the setting of time step, the judgment of the occurrence of coagulation and breakage event, the choice of fictitious coagulation partner, and dealing with the consequence of particle coagulation and breakage event. Then MMC method is used to simulate four kinds of special cases in which complete or partial analytical solutions exist; the simulation results of MMC method for GDE agree with analytical solutions well, which proves that MMC method has high and stable statistical precision.