Nonstationary Heat Conduction Equation Research Articles

RESEARCH OF THE DESTRUCTION CONDITIONS OF GLAZING OF ENCLOSING STRUCTURES UNDER THE CONDITIONS OF THERMAL INFLUENCE OF FIRE

The article presents the main results of mathematical modelling of the destruction of glazing of enclosing building structures with translucent elements under fire conditions. The authors analyse literature data on calculating the fire resistance of enclosing building structures with glazing. The results show that improving methods for predicting the fire resistance of glazing is relevant. Mathematical models for the numerical study of the behaviour of glazing under the thermal influence of the standard fire temperature regime are substantiated. Mathematical models of the heat transfer process involve using a non-stationary differential equation of heat conduction with boundary conditions of the third kind, accounting for the convective and radiant heat exchange between the air in the room with the fire and the glazing. As a criterion for the destruction of glazing under the influence of fire, the authors establish the temperature resistance parameter, which is determined based on the mechanical characteristics of the glass. The article highlights the relevant data on the study of temperature indicators and the mechanism of glazing failure, using mathematical modelling with well-founded mathematical models. It obtains the results of the thermal calculation by applying the finite difference method for solving the non-stationary differential equation of thermal conductivity. The methodology for determining the time of glazing failure under the influence of the standard temperature regime of a fire using the calculation parameters obtained based on well-founded mathematical models is substantiated. An analysis of the adequacy of the obtained calculation data was carried out by comparing them with the results of experimental studies using statistical criteria. As a result of the analysis, the authors confirmed the validity of the calculation data. A complete factorial experiment was conducted based on well-founded mathematical models, which revealed the regularities of the dependence of the parameters of thermomechanical processes in single-layer glass on its structural characteristics. The revealed regularities have the form of linear regression dependences of the destruction time on the thickness of the glazing and the maximum length of the glass panel without bars. Using the regularities of the dependence of the parameters of thermomechanical processes in single-layer glazing on its structural characteristics, the authors constructed reference tables for evaluating the fire resistance of enclosing structures with glazing. The research obtained new scientific data on the study of the behaviour of glazing of enclosing building structures under the thermal influence of fire through mathematical modelling, which is the basis for improving the methods of calculating the fire resistance of enclosing structures with translucent elements. Keywords: enclosing building structures, glazing, fire resistance limit, calculation method, glass heat resistance.

MATHEMATICAL MODELS OF THE THERMAL PROTECTION COATING OF THE GAS GENERATOR OF THE HYDROGEN STORAGE AND SUPPLY SYSTEM

The article describes the properties of the thermal protection coating of the gas generator of the hydrogen storage and supply system by two transfer functions with Hurwitz polynomials of the third order. Obtaining such transfer functions is based on the solution of the non-stationary heat conduction equation, represented in operator form using the integral Laplace transform, and in which approximating spline functions in the form of second-order polynomials are used. The article gives the solution of the non-stationary thermal conductivity equation, which describes the thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system under the thermal effect of a fire. This solution comes in the operator form for the surface temperature of the heat-protective coating on the side of the gas generator wall. The peculiarity of this decision is the presence of hyperbolic functions of an irrational argument in its composition. The structural and dynamic scheme of the thermodynamic system ‘gas generator wall – heat-protective coating’ is presented, the feature of which is the presence of two entrances. The signal at the scheme’s first input reflects the thermal effect of the fire, and the signal at the second input reflects the thermal state of the gas generator cavity. An equivalent mathematical transition to the description of thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system was carried out. This transition happened due to the use of spline functions, which approximate the hyperbolic functions of the irrational argument and are polynomials of the second order. The article gives a verbal interpretation of the algorithm for determining the transfer functions of the heat-protective coating of the gas generator of the hydrogen storage and supply system and also an example of its implementation. Furthermore, it shows that for the given conditions of functioning of the gas generator heat-protective coating, the relative errors in approximation of hyperbolic functions by second-order polynomials do not exceed 1.7 %, and the average relative error when equivalent replacement of transfer functions does not exceed 3.8 %. Keywords: gas generator, hydrogen storage and supply system, thermal protection coating.

Influence of the air gap between two cells of the storage battery on the thermal conditions of its operation: Numerical analysis and reliability assessment

To control the operating conditions of battery energy storage systems (BESS), the cells are combined into assemblies and modules located mostly in a closed space limited by the battery case. There are air gaps between the cells of the battery assembly. Energy dissipation in cells leads to an intense heat removal in the closed region of the air gap. As a result, the temperature of the battery assembly increases with possible further uncontrolled thermal runaway and subsequent battery ignition. Despite the increasing number of fires and explosions of battery ESSs every year, there is still no theory providing the ability to predict the conditions of fire and explosion safety of energy-intensive energy storage systems based on storage batteries. Particularly, there are no reliable data on the temperature fields of the cells of battery assemblies during intensive charge/discharge processes.The aim of the work was to analyze the thermal conditions of operation of a storage battery consisting of two typical prismatic cells, taking into account heat transfer in the air gap between the cells. Electrothermal modeling was performed for a fairly typical battery by solving a non-stationary heat conduction equation in a two-dimensional formulation by the finite difference method. The most realistic ranges of variation in the values of influencing factors were considered. In addition, the paper presents assessment of the thermal conditions of operation of a single cell and a comparative analysis of the obtained representative temperatures with the representative temperatures of a battery assembly of two battery cells. It has been established that the temperature of the case of cells of the battery assembly is 3–7 °C (depending on current loads) higher than the temperature of the case of a single cell at equal conditions. The obtained results show the need to take into account the heat transfer between the cells of the battery assembly when analyzing the temperatures of such assemblies.

Anticipated transients without scram in an innovative power nuclear reactor of anomalously high power

The article considers the possibility of creating a safe power reactor of anomalously high power. This may be relevant in the transition to large-scale safe and environmentally acceptable nuclear power. The safety justification is carried out based on mathematical modeling of the most dangerous emergency modes from among ATWS (anticipated transients without scram). With the use of a number of innovations (the UN-PuN-U cermets fuel used, a coolant based on lead with a high concentration of the 208Pb isotope, and tungsten fuel cladding), it was possible to ensure the safety of a fast reactor with an electric power of up to 3 GW. In contrast to previously proposed innovations, the article considers core layouts containing fuel pins with claddings made entirely of tungsten. Emergency modes are modeled approximately based on solving three groups of equations: a system of point kinetic equations, a non-stationary heat conduction equation, which, and a reactivity balance equation. The value of the void reactivity effect is estimated based on the solution of the gas-kinetic neutron transport equation. Reserves have been identified in further increasing the power and safety of the power reactor. They are associated with the optimization of the size of granules of ceramic and metal powders in the fuel composition.

Неосесимметричная нестационарная задача термоэлектроупругости для длинного пьезокерамического цилиндра

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

Axisymmetric Resonant Vibrations and Vibration Heating of an Inelastic Cylindrical Shell Compliant to Shear with Piezoelectric Actuators and Rigidly Fixed End Faces

We consider the problem of axisymmetric resonance vibrations of an inelastic cylindrical shell compliant to shear with piezoelectric actuators under the action of electromechanical monoharmonic loading. We take into account transverse shear strains, rotation inertia of a normal element, and temperature dependence of complex modules of piezopassive and piezoactive materials. The nonlinear problem is solved by using the iterative approach with respect to time within the framework of which the system of ordinary differential equations for vibrations is reduced to the integration of nonstationary heat-conduction equation by the finite-difference method. By using piezoelectric actuators, we analyze the influence of shear strains and the temperature of vibration heating on the amplitude and temperature-frequency characteristics and active damping of the resonance vibrations of the shell.

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

Heat and mass transfer in the processes of drying thin natural leathers pasted on smooth surfaces

The results of the study of drying thin natural leather pasted on a smooth surface are presented. The results of solving the differential equation of non-stationary heat conduction with constant thermophysical coefficients are used to determine the average temperature during the period of decreasing drying rate. The calculated values of temperatures for moisture-proof and moisture-free skin surfaces are given. Methods for simplifying solutions of nonlinear equations with the aim of linearizing these equations for approximating solutions with variable transport coefficients are considered. The use of the method of piecewise stepwise approximation of the transfer coefficients with constant values of these coefficients over short time intervals showed that for low-intensity drying processes, the solutions of the linear heat equation are implemented quite accurately, confirming the regularities obtained empirically. The calculation of the duration of drying according to the method of B. S. Sazhin is given. The results of the processing of experimental curves for drying leathers pasted are presented. The reliability of the obtained equations is verified and the experiment is compared with the calculated values using the formulas. The obtained approximate analytical solutions with reliable values of the transfer coefficients, confirmed by experimentally established regularities, are of practical importance. Together with experimental methods, analytical methods make it possible to establish optimal drying regimes and more accurately generalize experimental data.

Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation

<abstract><p>We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.</p></abstract>

Mathematical Simulation of Forest Fuel Pyrolysis and Crown Forest Fire Impact for Forest Fire Danger and Risk Assessment

In order to predict and assess the danger from crown forest fires, it is necessary to study the thermal degradation of different forest fuels in a high-temperature environment. In this paper, the main characteristics of pyrolysis accompanied by moisture evaporation in a foliage sample of angiosperms (birch) were investigated within conditions typical for a crown forest fire. The heat and mass transfer in the forest fuel element is described by the system of non-stationary non-linear heat conduction equations with corresponding initial and boundary conditions. The considered problem is solved within the framework of the three-dimensional statement by the finite difference method. The locally one-dimensional method was used to solve three-dimensional equations for heat conduction. The simple iteration method was applied to solve nonlinear effects caused by the forest fuel pyrolysis and moisture evaporation. The fourth kind of boundary conditions are applicable at the interface between the sub-areas. Software implementation of the mathematical model is performed in the high-level programming language Delphi in the RAD Studio software. The characteristic changes in the sample temperature field and the phase composition under high-temperature exposure from a forest fire are presented. The induction period of the thermal decomposition of dry organic matter in the sample was determined. Recommendations are made about key features of accounting for the pyrolysis and evaporation processes when predicting forest fire danger. The research results can be used in the development and improvement of systems for predicting forest fire danger and environmental consequences of the forest fires.

Coupled heat transfer in the cavity of constant volume with pulse-periodical energy supply

The possibilities of organizing a repetitively pulsed process with given frequency characteristics in a cavity of constant volume with an external energy supply to the working gas mixture are considered. Modelling of gas-dynamic and thermal processes is carried out using the numerical solution of the conjugate heat transfer problem. The gas medium is described on the basis of a viscous compressible gas model. To find the temperature field in the walls of the structure, the equation of non-stationary heat conduction is solved. The conjugation of temperature fields in a gas and a solid is carried out using an iterative procedure. In the calculations, the geometrical parameters of the cavity, the density of the energy supply, the initial pressure, and the composition of the working mixture are varied. The results of calculations obtained in the framework of the one-dimensional and two-dimensional formulation of the problem under the action of both a single pulse and a series of pulses are compared. The results obtained demonstrate the possibility of implementing the required frequency characteristics of the process for given geometric and energy parameters. Keywords: pulse-periodical energy process, gas laser, coupled heat transfer, numerical simulation, energy supply, pulse.

Сопряженный теплообмен в области постоянного объема при импульсно-периодическом подводе энергии

The possibilities of organizing a repetitively pulsed process with given frequency characteristics in a cavity of constant volume with an external energy supply to the working gas mixture are considered. Modelling of gas-dynamic and thermal processes is carried out using the numerical solution of the conjugate heat transfer problem. The gas medium is described on the basis of a viscous compressible gas model. To find the temperature field in the walls of the structure, the equation of non-stationary heat conduction is solved. The conjugation of temperature fields in a gas and a solid is carried out using an iterative procedure. In the calculations, the geometrical parameters of the cavity, the density of the energy supply, the initial pressure, and the composition of the working mixture are varied. The results of calculations obtained in the framework of the one-dimensional and two-dimensional formulation of the problem under the action of both a single pulse and a series of pulses are compared. The results obtained demonstrate the possibility of implementing the required frequency characteristics of the process for given geometric and energy parameters.

Вероятностно-детерминированный подход к расчету пожарного риска, вызванного воздействием лучистого теплового потока от фронта верхового устойчивого повального лесного пожара на жилой дом

Purpose. The purpose of the article is to create a mathematical model for calculating the fire risk caused by the impact of a radiant heat flux from the front of an independent crown wild fire on the combustible structures of a residential building. A combined calculation method has been developed that combines deterministic and probabilistic approaches. The combination of the above mentioned approaches is due to the fact that the probabilistic approach does not take into account the specific mechanism of the impact of a wild fire on the structure of a building, while the deterministic approach does not take into consideration the probability of a fire starting ABSTRACT and developing, as well as the timeliness of fire units response for fire extinguishing. Methods. The study uses theoretical methods of heat conduction and radiant heat transfer, wild fire statistics analysis, and fire risk theory. Findings. In the deterministic part of the method to calculate the temperature of the irradiated surface of the building structure, a one-dimensional non-stationary heat conduction equation is used, the boundary condition for which is determined by solving the equation of the radiant heat transfer between the wild fire front and the building. The calculated wood (pine) ignition time is compared with experimental data. The agreement between the calculation and experiment is satisfactory. The formula for calculating the fire risk for a residential building resulted from the impact of the radiant heat flux from the front of a wild fire is proposed. The above-mentioned formula takes into account the frequency of crown wild fires occurrence, the probability of ignition of combustible structures of the building when no firefighting measures are taken, as well as the probability of the active and passive fire prevention measures effectiveness. An example of fire risk calculation is presented when constructing a mineralized line, whose increase in width will reduce the fire risk by 1 000 times. Research application field. The results of the study can be applied in the development of fire prevention measures aimed at protecting communities against wild fires impact. Conclusions. The developed probabilistic-deterministic mathematical model for calculating the impact of a radiant heat flux falling from the front of an independent crown wild fire on the combustible structures of a residential building makes it possible to assess the fire risk for a building, considering specific meteorological conditions, pyrological and geometric (tree height, etc.) characteristics of the woodland, the distance from the building to the edge of the forest, as well as fire prevention measures implementation.

An Extended Analytical Solution of the Non-Stationary Heat Conduction Problem in Multi-Track Thick-Walled Products during the Additive Manufacturing Process.

An analytical model has been developed for calculating three-dimensional transient temperature fields arising in the direct deposition process to study the thermal behavior of multi-track walls with various configurations. The model allows the calculation of all characteristics of the temperature fields (thermal cycles, cooling rates, temperature gradients) in the wall during the direct deposition process at any time. The solution of the non-stationary heat conduction equation for a moving heat source is used to determine the temperature field in the deposited wall, taking into account heat transfer to the environment. The method considers the size of the wall and the substrate, the change in power from layer to layer, the change in the cladding speed, the interpass dwell time (pause time), and the heat source trajectory. Experiments on the deposition of multi-track block samples are carried out, as a result of which the values of the temperatures are obtained at fixed points. The proposed model makes it possible to reproduce temperature fields at various values of the technological process parameters. It is confirmed by comparisons with experimental thermocouple data. The relative difference in the interlayer temperature does not exceed 15%.

A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

Mathematical Simulation of Forest Fuel Pyrolysis in One-Dimensional Statement Taking into Account Soot Formation

Pyrolysis (thermal decomposition) is considered as the most important stage of a forest fire before direct forest fuel ignition. This process is accompanied by soot particle formation. Such particles have a negative impact on public health in the vicinity of forest fires. The purpose of this article was to investigate the heat and mass transfer process occurring in a typical forest fuel element (birch leaf). The pyrolysis and soot formation processes were taken into account, and various forest fire scenarios were considered. Computational experiments were carried out using the high-level programming language Delphi. Heat and mass transfer processes were described by nonlinear non-stationary differential heat conduction equations with corresponding initial and boundary conditions. The differential equations were solved by the finite difference method. Nonlinearity was resolved using a simple iteration. The main results of the research were (1) physical and mathematical models proposed for modeling forest fuel pyrolysis, taking into account soot formation and conditions corresponding to various forest fires; (2) a computer program coded in the high-level programming language Delphi; (3) the obtained temperature distributions over leaf thickness; (4) volume fractions obtained for various components dependent on time and space coordinates. The qualitative analysis of the dependencies showed that the temperature distributions in the birch leaf structure are similar for all forest fire types and differ only in absolute value. The intensity of the soot formation process directly depends on the forest fire type. The presented results should be useful in predicting and assessing forest fire danger, including near the facilities of the Russian Railways.

Mathematical Simulation of Heat Transfer in the Structures of a Passenger Carriage Under the Influence of Forest Fires

Forest fires have a negative impact on the functioning of the railway. The aim of the work is a mathematical modeling of heat transfer in the structures of a passenger carriage under the influence of radiation from a forest fire front. The originality of the study is explained by modeling heat transfer in structural materials of a passenger carriage under the influence of radiation from a forest fire front. Low and high intensity surface forest fires, crown forest fires are considered. Most of the previously published works are devoted to the analysis of the causes of forest fires, including the ones near the railway. The software implementation of the mathematical model is performed in the RAD Studio software in the high-level programming language Delphi. Mathematically, heat transfer in the materials of a passenger carriage is described by non-stationary differential heat conduction equations with corresponding initial and boundary conditions. In order to solve the partial differential equations, the finite-difference method and the locally one-dimensional method are used. Difference analogs of differential equations are solved by the marching method. The main findings of the study are: 1) physical and mathematical models of heat transfer in the structures of a passenger carriage under conditions of exposure to radiation from a forest fire; 2) the obtained temperature distributions in the inhomogeneous structure of the passenger carriage wall; 3) simulation has shown that a low-intensity surface forest fire causes a relatively safe forest fire impact scenario, while a high-intensity surface forest fire impact is a potentially dangerous scenario; 4) the scenario of the impact of a crown forest fire is unambiguously dangerous at any parameters; 5) the results obtained in this paper and the expected in future investigations will create a physical basis for the development of software for fire safety systems for rolling stock on the railway. The presented results are proposed to be used for forecasting, monitoring, and assessing forest fire dangers during the operation of the rolling stock of JSC Russian Railways.

Determining Temperature Fields in a Spatially Inhomogeneous Nonlinear Medium

We propose a method of determining temperature fields in a spatially inhomogeneous medium with temperature-dependent thermophysical properties of the material. For this purpose, point and nonlocal transformations of the nonstationary heat conduction equation are used. Examples of applying the theory for various boundary conditions in the spherical symmetric case are given.

Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process.

The work is devoted to the development of a model for calculating transient quasiperiodic temperature fields arising in the direct deposition process of thin walls with various configurations. The model allows calculating the temperature field, thermal cycles, temperature gradients, and the cooling rate in the wall during the direct deposition process at any time. The temperature field in the deposited wall is determined based on the analytical solution of the non-stationary heat conduction equation for a moving heat source, taking into account heat transfer to the environment. Heat accumulation and temperature change are calculated based on the superposition principle of transient temperature fields resulting from the heat source action at each pass. The proposed method for calculating temperature fields describes the heat-transfer process and heat accumulation in the wall with satisfactory accuracy. This was confirmed by comparisons with experimental thermocouple data. It takes into account the size of the wall and the substrate, the change in power from layer to layer, the pause time between passes, and the heat-source trajectory. In addition, this calculation method is easy to adapt to various additive manufacturing processes that use both laser and arc heat sources.

Martensitic microstructure evolution in austenitic steel: A thermomechanical polycrystalline phase field study

We have performed coupled thermomechanical phase field simulations of martensitic transformation in single-crystalline and polycrystalline 301-stainless steel by coupling Ginzburg–Landau equation to the mechanical equilibrium and non-stationary heat conduction equations. Simulations accounted for elastic anisotropy, under different thermal and mechanical BCs, i.e., clamped, embedded, and near-open, as well as adiabatic and isothermal with and without latent heat effects. To the best of authors’ knowledge, no such simulations are reported in literature on polycrystalline materials which provides the effects of various thermomechanical BCs incorporating latent heat. We have utilized a multivariant single embryo as an initial condition in both systems to mitigate the lack of nucleation criteria at the mesoscale; without any initial constraint, the model can select the martensitic transformation path and final microstructure. Simulations showed excellent capabilities in predicting influence of such thermomechanical BCs on martensitic microstructural formation including autocatalytic transformation process in polycrystalline steel and its evolution. Martensitic microstructure in a single-crystalline, and b polycrystalline steel, in deformed configuration (deformation scale factor is $$1\times$$ ) for near-open, embedded, and clamped boundary conditions (from left to right). Dataset edges in blue show initial undeformed configuration