Kinds Of Boundary Conditions Research Articles

Cooling of an infinite rectangular plate with an adibatically insulated side

The work is devoted to the issues of non-stationary heat transfer. The article presents a solution for the distribution of the temperature field in an infinite rectangular plate with an adiabatically isolated side. As a result, an analytical expression of the plate temperature distribution is obtained in the form of a series containing trigonometric and exponential functions. The paper also considered special cases when the internal thermal resistance of thermal conductivity is greater and when the external resistance of heat output is less. Special cases were interpreted physically. One of the special cases leads the problem to a problem with boundary conditions of the first kind, when the surface temperature is constant, which indicates the reliability of the results obtained.

Finding the temperature distribution in a rectangular plate with variable thermal conductivity in one coordinate

The work is devoted to the issues of stationary heat transfer. The article presents a solution for the distribution of the temperature field in a rectangular plate. At the same time, the law of change of thermal conductivity along one of the coordinates is known. Heat exchange at the two opposite ends of the plate occurs under boundary conditions of the third kind, there is no heat exchange at the other two. The solution is found by decomposition into a functional series. As a result, an analytical expression of the plate temperature distribution is obtained in the form of a Fourier series containing modified zero-order Bessel functions.

Heat exchange between cylindrical heating element and its shell under boundary conditions of the fourth kind

Heat exchange between cylindrical heating element and its shell under boundary conditions of the fourth kind

Heat Exchange and Aerodynamics of the Vertical Conical Funnel at the Thermal Power Plant Site

The funnel is the most important element of the thermal power station. An appropriate arrangement of the system for the removal of the gaseous products of fuel combustion inside the funnel results in no moisture condensation and proper maintenance of the operation conditions of the funnel. To measure the temperature of combustion products along the funnel height we need to define the boundary conditions of the third kind for the external surface of the funnel. The studies showed that in the case of the uniform velocity profile, the heat exchange and aerodynamics on the surface of the single vertically arranged funnel have specific features conditioned by the funnel configuration and the contact of its base with the ground surface. In particular, the rear part of the funnel is characterized by a periodic change in the static pressure, velocity and the height heat loss coefficient. The purpose of this research was to define the average heat loss along the height of the conical funnel situated at the industrial site of the thermal power station for different wind vectors and velocities. To simulate the heat exchange and aerodynamics near the funnel situated at the thermal power plant site we used the infrastructure that includes the following elements: mechanical compartment, substation, administrative building, warehouse and two cooling towers. The network model of the hydropower plant includes 1137782 units and 4741859 elements. A minimum orthogonal quality is 0.1 and a maximum biasing is 0.89. The ĝ RNG k- turbulence model, the Enhanced Wall Function and the Simplex algorithm were used for the velocity-pressure interrelation problem in steady flows. The air density is independent of the temperature at the computational volume inlet (the gravitation is neglected). The funnel surface temperature was specified as constant and equal to 100 C. The heat exchange near the funnel was studied at different wind directions in the wind velocity range of 5 to 25 m/s. Hence, the wind direction, the environmental infrastructure and the industrial thermal power plant site have an essential effect on the funnel height heat exchange distribution pattern. Evidently, it is defined to a great extent by the funnel aerodynamics. A maximum heat loss level is observed when the flow is rushed to the backside of the building at the south- to-north wind direction and a minimum heat loss level is observed for the east- to -west and west -to -east wind directions for the parallel flow. А minimum heat loss is observed at a longitudinal wind motion along the mechanical compartment building in west-to-east and east-to- west directions. These specific features should be taken into account when designing tall funnels with an optimal change in the combustion product temperature along the funnel height. We can draw a conclusion that the specific features of the heat exchange in question should be obligatory taken into account when designing tall funnels. Keywords: heat exchange, funnel, thermal power plant infrastructure, mechanical compartment, wind direction and 3D model.

Lions' representation theorem and applications

The Lions' Representation Theorem (LRT) is a version of the Lax–Milgram Theorem where completeness of one of the spaces is not needed. In this paper, LRT is deduced from an operator-theoretical result on normed spaces, which is of independent interest. As an example, we give a new characterization of dissipativity. The main part of the paper is a theory of derivations, based on LRT, which we develop. Its aim is to establish well-posedness results, not only for evolution in time but also for more general settings in terms of this new notion of derivation. One application concerns non-autonomous evolution equations with a new kind of boundary condition where values at the initial and final time are mixed.

Numerical calculation of heat transfer in a multilayer composite structure with honeycomb filler during autoclave molding at the heating stage

The production of any multilayer structure begins with the development of technological documentation, which describes, among other things, the temperature regimes during their manufacture. Conventionally, the temperature process of polymerization can be divided into three stages: preheating, temperature stabilization and cooling. The paper presents the results of a numerical calculation of temperature fields in a multilayer composite structure with a honeycomb filler during its manufacture by autoclave molding at the preheating stage. This method of manufacturing composite structures makes it possible to mold parts of varying complexity and dimensions, the demand for which is growing in such industries as mechanical engineering, aircraft building, and shipbuilding. The requirements for the quality of such products are increasing, which is greatly influenced by compliance with the temperature regime during molding. Conducting direct experiments requires large energy costs, therefore, to solve the problem of controlling heat exchange processes inside the structure, mathematical models were developed that describe these processes. A non-stationary heat conduction problem is formulated for a multilayer unbounded plate with a constant initial distribution, boundary conditions of the third kind on the outer boundaries, and boundary conditions of the fourth kind on the contact surfaces of the layers. Using the finite element method, the problem is reduced to three-point difference equations, the solution of which is found by the sweep method. The determination of the sweep coefficients is shown taking into account the thermal characteristics of the layers. The results of a numerical calculation of the temperature distribution for a nine-layer composite structure with a honeycomb core are presented. The numerical calculation was carried out using the developed program in the BorlandDelphi 7.0 object-oriented programming environment. The results obtained are presented in the form of graphic dependences of the temperature over the thickness of the sample at different points in time, as well as the dependence of temperature on time at various nodes of the sample in comparison with the theoretical curve. An analysis of these dependences was carried out, which showed that the heating of the sample occurs unevenly over its thickness. The deviation from the theoretical temperature values is observed in the layers located closer to the honeycomb layer. This can adversely affect the course of the polymerization step, which is characterized by the conversion of the binder into a polymer, and occurs at certain temperatures. Therefore, achieving the desired temperature values at the stage of heating the structure is important for the manufacture of reliable and durable structures that can withstand extreme operating conditions. The obtained temperature distributions make it possible to correct the technological process of manufacturing various multilayer structures at the stage of its development, which will reduce the economic costs of production.

Finite-difference method for solving a multidimensional pseudoparabolic equation with boundary conditions of the third kind

We study an initial-boundary value problem for a multidimensional pseudoparabolic equation with variable coefficients and boundary conditions of the third kind. The multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme by A. A. Samarsky is constructed. An a priori estimate is obtained by the method of energy inequalities, from which the uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow. For a two-dimensional problem, an algorithm for the numerical solution of the initial-boundary value problem for a pseudoparabolic equation with conditions of the third kind is developed.

Null controllability for a structurally damped stochastic plate equation

In this paper, we obtain a null controllability result for a structurally damped refined stochastic plate equation with two kinds of boundary conditions. The key point is to establish an observability inequality for the backward structurally damped stochastic plate equation. For this purpose, we derive a new global Carleman estimate for the damped stochastic plate operator with homogeneous Dirichlet boundary conditions on the solution and the laplacian of the solution and with homogeneous Neumann boundary conditions of first and third order, respectively. Such an estimate is obtained by using the Carleman estimate for stochastic parabolic operators with complex coefficients in an iteration manner.

A mathematical model of an inductively coupled radio-frequency discharge at low pressure as nonlinear partial eigenproblem

A new approach for modeling steady state inductively coupled radio frequency discharges at low pressure is described. A simple one-dimensional model is considered, which includes Maxwell’s equations and the electron balance equation with boundary conditions of the third kind. It is shown that the system of boundary value problems is a two-parameter partial eigenvalue problem. The smallest eigenvalue of the problem is the boundary value of the magnetic field strength. The second parameter of the problem is the concentration of electrons at the center of the plasma bunch. The developed approach makes it possible to calculate the inductor current required to maintain a steady state of the discharge. The results of calculations of the dependence of the inductor current, electron density, electric and magnetic fields on pressure are presented.

Modeling the convective component of the heat flow from a spill fire at railway accidence

A significant number of emergencies that occur in the chemical, processing and transport industries begin with an accidental spill and ignition of a flammable liquid. In this case, the spread of fire to neighboring objects is of particular danger. When developing fire protection measures in areas where flammable liquids are stored, as a rule, heat transfer from a fire only by radiation is taken into account. But in some cases, the convection component of the heat flow can make a significant contribution to the overall heat transfer. Ignoring it can lead to an erroneous assessment of the safety of an industrial facility. In the paper, a model of the distribution of velocity and temperature in the upward flow, rising above the spill of a burning liquid, is constructed. The model is based on the system of Navier-Stokes equations, which, by means of simplifications, is reduced to a non-linear second-order differential equation of the parabolic type. The properties of the combustion site determine the boundary conditions of the first kind. In this case, the spill of a flammable liquid can have any shape. The presence of wind is taken into account by introducing a stable horizontal component of the flow velocity.
 For the numerical solution of the equation, the method of completed differences is used. The dependence of the kinematic viscosity on the flow temperature is taken into account. An empirical formula is used as the relationship between temperature and speed. It is shown that the presence of wind leads to an inclination of the ascending flow. The angle of inclination is not constant and increases with distance from the combustion source due to a decrease in speed and cooling of the flow. An estimate of the coefficient of convection heat transfer convection of the tank wall with ascending flows over the combustion source is constructed. It is shown that the coefficient of convection heat transfer increases with increasing wind speed

Experimental assessment of the temperature state of tractor diesel pistons

INTRODUCTION: The paper presents the operating conditions of tractor diesel engines that cause the appearance of thermal fatigue cracks on the edges of the piston combustion chamber. The presence of sharp edges of the combustion chamber in the pistons, which are stress concentrators, leads to an increase in the probability of their destruction and thereby limits the engine life of the diesel engine. The main reasons for the formation of cracks in the zone of the edge of the combustion chamber are indicated.
 AIMS: The aim of this study is the assessment of the temperature state of pistons of the D-240 and the D-245 tractor diesel engines, produced by Minsk Motor Plant (MMP).
 METHODS: Temperature gauging was carried out according to the method in order to identify the nature of changing of piston heads temperatures under stationary and nonstationary operation modes of diesel engines. Transfering of thermal electromotive force from thermocouples to measuring devices was carried out by means of an intermittent current collector. Imitaion of nonstationary operation modes was carried out by means of changing the cyclic feed of a high pressure fuel pump, using a reversible electric motor.
 RESULTS: The data of the temperature state of pistons under various stationary and nonstationary operation modes of engines is provided. It is noted that the temperature state of the D-245 diesel pistons has a higher level of heat stress compared to the D-240 diesel pistons. The maximum amplitude of low-frequency temperature fluctuations at the edge of the combustion chamber and their radial differences along the piston bottom are determined, depending on the parameters of thermal loading cycles. It is noted that the most dangerous modes of diesel operation, in terms of the destruction of the edge of the combustion chamber, are sharply changing modes (eg.: theloading unloading mode).
 CONCLUSIONS: It is proposed to increase the fuel injection advance angle in the thermal loading cycle in order to conduct accelerated comparative tests of piston variants for thermal resistance. The developed thermal loading cycle, in which the total duration of the load increase is 180 s and the total duration of the load decrease is 90 s, can be recommended for accelerated motor tests of pistons for thermal cycling resistance. The obtained temperature measurement data is recommended to clarify boundary conditions of the first kind when calculating the piston using the FEM method.

A note on Leveque's solution to the Cartesian Graetz problem with free convection

In this paper, an asymptotic analysis of the Cartesian Graetz problem with third kind boundary condition is carried out. The analysis is performed using Leveque's asymptotic method, which is valid for regions near the duct entrance. The solutions for the temperature field are improved by using second order corrections for the solution; this allows to find simple algebraic expressions for the temperature field and Nusselt number, which can be easily applied to thermal engineering problems. The results obtained considering higher order corrections show a notable improvement with respect to the precision obtained considering only first order corrections, with a percentage error with respect to the exact numerical solution that is less than 0.1% for values of the axial coordinate ξ < 0.1.

COOLING OF AN INFINITE RECTANGULAR PLATE UNDER BOUNDARY CONDITIONS OF THE SECOND AND THIRD KIND

In this article, the question of finding the law of the change of the temperature field from time in a rectangular plate under different boundary conditions set on the walls of the plate is considered. At the same time, the task itself is non-stationary. In this paper, based on the Fourier method, a solution for the distribution of the temperature field in a plate of infinite length is obtained. The solution itself is obtained in the form of a series containing trigonometric and exponential functions. Special cases were also considered. The reliability of the obtained result is confirmed by the fact that one of the special cases leads the problem to a problem with boundary conditions of the first kind, when the temperature of the surface is constant.

Heat transfer analysis of compressible laminar flow in a parallel-plates channel via integral transforms

The present work analyzes a coupled nonlinear mathematical model for heat transfer in compressible laminar flow within a parallel plates channel. The governing partial differential equations are obtained considering the conservation of mass, momentum and energy, based on a two-dimensional steady laminar flow of an ideal gas with constant physical properties and taking into account viscous dissipation. Two kinds of boundary conditions at the wall were considered, namely, prescribed arbitrary temperature or heat flux longitudinal distributions. The mathematical model was solved using the hybrid numerical-analytical method known as the Generalized Integral Transform Technique (GITT). A convergence analysis of the proposed eigenfunction expansions for velocity and temperature fields was undertaken, indicating that fairly low truncation orders are required for the accurate representation of the potentials. The proposed hybrid solution was also critically compared with the Mathematica NDSolve finite difference-based routine and additional theoretical and experimental results in the literature, with good agreement in all cases considered.

Configurational entropy of colloidal particles in a confined space.

We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration. In particular, we consider systems with three kinds of boundary conditions: hard, periodic, and spherical. For small to moderate packing fraction ϕ values, we find the entropies per particle for systems with the periodic and the spherical boundary conditions tend to reach the same value with the increase of the particle number N, while that for the system with the hard boundary conditions still has obvious differences compared to them within the studied N range. Surprisingly, despite the small system sizes, the estimated entropies per particle at infinite system size from extrapolations in the periodic and spherical systems are in reasonable agreement with that calculated using thermodynamic integration. Besides, as N increases we find the pair correlation function begins to exhibit similar features as that of a thermally equilibrated hard-disk fluid at the same packing fraction. Our findings may contribute to a better understanding of how the configurational entropy changes with the system size and the influence of boundary conditions, and provide insights relevant to engineering particles in confined spaces.

Polynomial and exponential decay rates of a laminated beam system with thermodiffusion effects

In this paper we consider a laminated beam system with thermodiffusion effects with two kinds of boundary conditions, in which the mass diffusion introduces a new critical thickness in addition to the conventional critical thickness of thermoelastic damping. By using the method of semigroup, we prove the system is global well posed. The polynomial stability is established by using frequency domain method if the wave speeds are nonequal. We also establish exponential decay of energy under the assumption of equal wave speeds. At last, we present some numerical experiments to illustrate our theoretical results. The result is new and is the first time when the stabilization of the laminated beam system with thermodiffusion effects is obtained.

A simple force balance model of subduction initiation

SUMMARY The initiation and development of subduction zones are associated with substantial stress changes both within plates and at plate boundaries. We formulate a simple analytical model based on the force balance equation of a subduction zone, and validate it with numerical calculations of highly non-linear, coupled thermomechanical system. With two kinds of boundary conditions with either fixed velocity or fixed force in the far-field, we quantitatively analyse the role of each component in the force balance equation, including slab pull, interplate friction, plate bending and basal traction, on the kinematics and stress state of a subducting plate. Based on the numerical and analytical models, we discuss the evolution of plate curvature, the role of plastic yielding and elasticity, and how different factors affect the timing of subduction initiation. We demonstrate with the presence of plastic yielding for a plate of thickness, H, that the bending force is proportional to H2, instead of H3 as previously thought. Although elasticity increases the force required to start nucleating subduction it does not substantially change the total work required to initiate a subduction zone when the yielding stress is small. The analytical model provides an excellent fit to the total work and time to initiate subduction and the force and velocity as a function of convergence and time. Plate convergence and weakening rate during nucleation are the dominant factors influencing the force balance of the plate, and 200 km of plate convergence is typically required to bring a nascent subduction zone into a self-sustaining state. The closed-form solution now provides a framework to better interpret even more complex, time-dependent systems in three dimensions.

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral‐type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self‐adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral‐type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.

A DVA-Beam Element for Dynamic Simulation of DVA-Beam System: Modeling, Validation and Application

The dynamic vibration absorber (DVA) has a broad application background in slender structure vibration reduction, such as in machine gun systems, micro actuators, and so on. Rapid modeling and simulation of the DVA-beam system are of great importance for dynamic performance, evaluation, and finally, the structural design. The primary motivation for this paper is to present a reliable and convenient technique for the modeling and computation of the DVA-beam system. A novel DVA-beam element, which consists of a beam segment and a DVA, is presented. In this element, the DVA position can be arbitrarily allocated, whether on the beam node or within the beam domain. In this way, the beam can be modeled with a few elements to save on computing costs and maintain ideal modeling accuracy. An element deactivating method, which endows the DVA-beam element with the ability to simulate a bare beam and a DVA-beam structure simultaneously, is proposed. Some numerical examples were carried out to validate the reliability of the DVA-beam element in addressing different kinds of boundary conditions by comparing the beam tip responses with those simulated by ADAMS software, and good agreements were observed. Finally, two DVA optimization examples were conducted to investigate the effectiveness and applicability of the DVA-beam element in engineering optimization. The performance was impressive.

Computing static behavior of flexible rectangular von Kármán plates in fast and reliable way

In this paper, several modifications of the Bubnov–Galerkin method for studying flexible rectangular plates have been constructed and validated. This is to overcome the challenges of applying the existing approaches to a specific class of problems, bordering on accuracy and machine time efficiency. Von Kármán’s (elliptic) equations consisting of a system of nonlinear PDEs of order eight, with respect to two functions, i.e., the deflection and the stress functions, are considered. The problem is reduced by employing several modifications of the Bubnov–Galerkin method combined with the Fourier approach (separation of variables) to a system of nonlinear ODEs. The paper deals with many of such techniques and a procedure that refines their solutions based on known discrepancies. A theorem on the convergence of the proposed iterative processes is formulated and proved. The developed iterative methods are used to solve rectangular plates problem, subjected to distributed and local loads for several kinds of boundary conditions. A comparison of the obtained solutions by various methods is carried out. The optimal calculation method in terms of accuracy and minimum machine time computation is revealed. In addition, we show that the proposed novel and efficient approaches can be used to estimate the static behavior of plates of arbitrary geometry.