Differential Equation Of Heat Conduction Research Articles

Semi-analytical solution for interfacial debonding of high-speed railway ballastless track under thermal loading using a quasi-dynamic method

Interfacial debonding of multi-layered ballastless track systems has proved to be one of the most common diseases in the high-speed railway during long-term operations. This paper focuses on the theoretical modeling of track temperature field and consequent bond-slip behavior between track layers under thermal loading. By establishing heat conduction differential equations, whose boundary conditions are combined with complicated geographical and meteorological factors, the temperature field of a multi-layered track structure is obtained via the integral transform technique. The reliability of the heat conduction model is well-validated by an on-site experiment in a high-speed railway line. Then, we originally extend an explicit integration algorithm for dynamics analysis to solve nonlinear statics problems with small deformation by proposing a quasi-dynamic method (QDM). Equilibrium equations of a track slab subjected to quasi-dynamic equivalent thermal loads and mixed-mode interfacial cohesive forces are constructed based on the Mindlin plate in-plane and flexural vibration equations. The application of the QDM to solve interface responses is verified by comparing results obtained from finite element (FE) software. Finally, numerical discussions are conducted to reveal the nonlinear distribution characteristics of temperature field, evolution process of the interface damage and failure modes under time-varying thermal loads, which provide a better understanding of the interfacial debonding phenomenon encountered in ballastless track systems.

Fast response characteristics of fiber Bragg grating temperature sensors and explosion temperature measurement tests

Transient temperature measurement technology with fast response, harsh measurement environment, and high technical challenges has attracted research interest in the field of temperature measurement. Based on the differential equation of heat conduction at the interface of an optical fiber, the dynamic-response-time formula of the optical fiber was derived, the dynamic response capability of the optical fiber with different diameters was analyzed, and the transient thermal response simulation of the optical fiber was conducted. The theoretical dynamic test response times of optical fibers with diameters of 80, 125, and 145 µm were 30.21, 60.78, and 85.97 ms, respectively. Fast-response fiber Bragg grating (FR-FBG) temperature sensors were designed and manufactured by femtosecond-fabricated FBGs with diameters of 80 and 125 µm. The average response times of the 80 and 125 µm-FR-FBG temperature sensors were 33.2 and 65.3 ms, and the errors between the data and theoretical values were 9.89 % and 7.44 %, respectively. From explosion experiments, it is concluded that the femtosecond grating can be applied to the monitoring of rapid temperature response during explosion, and the response time is related to the diameter of the grating.

Coupled Thermoelasticity of Hemitropic Media. Pseudotensor Formulation

The present paper deals with the problem of deriving the constitutive equations for the micropolar thermoelastic continuum GN-I in terms of the standard pseudotensor formalism. In most cases, the pseudotensor approach is justified in modeling hemitropic micropolar solids, the thermomechanical properties of which are sensitive to mirror reflections of three-dimensional space. The requisite equations and notions from the theory of pseudotensors are revisited. General thermodynamic approaches are used, entropy and energy balance equations are discussed. The weights of the main thermomechanical pseudotensors are established. In a linear approximation, the constitutive equations of the hemitropic micropolar thermoelastic continuum (GN-I) of the first type are derived. A coupled system of differential equations of heat conduction and dynamic equations of the micropolar thermoelastic continuum GN-I is obtained.

Research of Tribological and Thermodynamic Parameters of the WC-Cu Braking System of the Experimental Modular Electric Vehicle

Abstract The authors of this manuscript present the development of a braking system with friction material base WC-Cu coating for the electric vehicle. This manuscript follows on from the original development of an AGV multi-disc braking system and an experimental investigation of the friction factor of WC-Cu coatings. In addition to developing the mechanical elements and construction of the electric vehicle, the tribological parameters of three samples of the steel substrate, the C45 with WC-Cu coating, were investigated in the tribological laboratory. A metallic coating of the WC-Cu base was applied on the C45 steel substrate using electro-spark deposition coating technology. The experiment used three samples with different percentage ratios of chemical elements in the coating structure. The tribometer working on a “Ball on Plate” principle was an investigation of the friction factor of all samples during the experiment. Subsequently, the surface of the samples was modified structure WC-Cu with laser technology. The microhardness of modified and unmodified coatings according to the Vickers methodology was investigated in the next stage. At the end of the experimental investigation, a braking simulation was created in the programming environment of the Matlab® software, considering all driving resistances. The researchers also focused on the simulation of heat conduction during braking for some considered driving modes with braking on a level and with a 20% slope roadway. The simulation of heat flow was carried out in the Matlab® programming environment using the Fourier partial differential equation for non-stationary heat conduction.

Analytical Solution of Transient Heat Conduction through a Hollow Spherical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, spherical coordinate, non-linear partial differential equation of transient heat conduction through a hollow spherical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

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.

Combined Thermal and Wear Analysis of Linear Rolling Guide Subjected to Rigid–Flexible Coupling Motion Stage

The local heat and wear generated due to friction in a precision motion stage in electronic packaging equipment have a significant impact on its positioning precision. In this paper, we studied the temperature changes over time and the wear of a linear rolling guide (LRG) in a rigid–flexible coupling motion stage (RFCMS), both analytically and experimentally, and we proposed an evaluation method for LRG wear in an RFCMS. According to Fourier’s law and the law of conservation of energy, the differential equation of heat conduction for the LRG and the thermal boundary conditions were established. Steady-state and transient thermal simulations were carried out using ANSYS Workbench to predict the temperature increase in the LRG due to friction. Finally, a test apparatus was built to demonstrate that an RFCMS reduced the operating temperature of the LRG, which also reduced the wear on the contact surface. Through response surface methodology, the levels of the influence of different flexure hinge thicknesses, strokes, velocities, and accelerations on the temperature change rate (TCR) of the LRG were obtained, as well as the approximate regression equation of four variables of the TCR. This provided a new research method for precision maintenance, life design, and operational parameter selection for high-precision motion stages.

Analytical modelling of transient thermal characteristics of precision machine tools and real-time active thermal control method

Thermal error is one of the primary factors affecting the machining accuracy of precision machining tools. Therefore, it is important to study the transient thermal characteristics of machine tools and the thermal-error control strategies. Thus far, a transient analytical modelling method for characterising the thermal characteristics of machine tools was proposed and an active error control strategy was provided. First, temperature-field modelling was conducted using an analytical method based on the Fourier series method and partial differential equations of heat conduction. Second, using the derived temperature field, the thermal deformation field was calculated based on finite element theory. Subsequently, the continuous real-time effect of the thermal power per unit heat source on the temperature and deformation fields of precision machine tools was studied. The proposed analytical modelling method not only predicts the machine tool heat deformation based on the working conditions of the heat source, but also matches the thermal control source power with the demand of the machine tool heat deformation. The optimal real-time power of the thermal control source is dynamically iterated and matched, such that the thermal deformation caused by the heat and thermal control sources can be balanced in real time at the displacement control point. Finally, the volumetric thermal error was actively controlled by adjusting the temperature field of the machine tool.The simulated and experimental results indicate that the transient analytical model can accurately predict the real-time thermal characteristics of the machine tool and that the real-time active thermal control method can effectively reduce volumetric thermal errors. Using active thermal control, the squareness error in the YZ-plane was reduced by approximately 45%, the spindle thermal elongation was reduced from 23 μm to 7 μm, and the volumetric thermal error in the X, Y, and Z directions were reduced by approximately 16, 14, and 17 μm, respectively.

Experiment on local convective heat transfer for successive droplets impacting on heated cylindrical surface

An experiment was conducted to investigate the underlying heat transfer mechanism when successive droplets impacting on the heated cylindrical surface. The hydrodynamics of droplets were captured by visual method and the local convective heat transfer characteristics were obtained by combining direct measurement and numerical solution of three-dimensional differential equation of heat conduction. The influences of impact frequency, impact velocity and input heat flux were experimentally discussed. It is found that, based on the evolution of local wall temperature, there are three zones along the circumference, i.e., impingement zone, heat diffusion zone and tail detachment zone. Increasing the impact frequency and input heat flux enhances the convective heat transfer performance. However, when the impact frequency is high enough, continuously increasing the impact frequency, the degree of convective heat transfer improvement gradually decreases. The increase of impact velocity mainly enhances the convective heat transfer on the impingement zone and heat diffusion zone, but has little effect on the tail detachment zone. The continuous increase of impact velocity results in the splashing phenomenon and the decrease of convective heat transfer performance. Furthermore, some dimensionless correlations were proposed for predicting the droplet splashing threshold and the local convective heat transfer performance along circumference, respectively.

Deterministic–Probabilistic Prediction of Forest Fires from Lightning Activity Taking into Account Aerosol Emissions

Forest fires arise from anthropogenic load and lightning activity. The formation of a thunderstorm front is due to the influence of a number of factors, including the emission of aerosol particles from forest fires. The purpose of this study is mathematical modeling of heat and mass transfer in vegetation firebrand carried out from a forest fire front, taking into account the formation of soot particles to predict forest fire danger from thunderstorm activity. Research objectives: (1) development of a deterministic mathematical model of heat and mass transfer in a pyrolyzed firebrand of vegetation, taking into account soot formation; (2) development of a probabilistic criterion for assessing forest fire danger from thunderstorms, taking into account aerosol emissions; (3) scenario modeling of heat and mass transfer and the formation of a thunderstorm front; (4) and the formulation of conclusions and proposals for the practical application of the developed deterministic–probabilistic approach to the prediction of forest fires from thunderstorms, taking into account aerosol emissions. The novelty of this study lies in the development of a new model of heat and mass transfer in a pyrolyzed vegetation firebrand and a new probabilistic criterion for forest fire danger due to thunderstorm activity, taking into account aerosol emission. The distributions of temperature and volume fractions of phases in a firebrand are obtained for various scenarios. Scenarios of surface fires, crown forest fires, and a fire storm are considered for typical types of coniferous vegetation. Cubic firebrands are considered in the approximation of a two-dimensional mathematical model. To describe the heat and mass transfer in the firebrand structure, a differential heat conduction equation is used with the corresponding initial and boundary conditions, taking into account the kinetic scheme of pyrolysis and soot formation. Variants of using the developed mathematical model and probabilistic criterion in the practice of protecting forests from fires are proposed. Key findings: (1) linear deterministic–probabilistic mathematical model to assess forest fire occurrence probability taking into account aerosol emission and lightning activity; (2) results of mathematical modeling of heat and mass transfer in firebrand taking into account soot formation; (3) and results of scenario modeling of forest fire occurrence probability for different conditions of lightning activity and aerosol emission.

Groundwater Temperature Modelling at the Water Table with a Simple Heat Conduction Model

This study aimed at the analysis and modelling of the groundwater temperature at the water table in different regions of Slovakia. In the first part, the analysis of the long-term trends of air and soil/ground temperature to a depth of 10 m is presented. The average annual soil/groundwater temperatures at different depths were the same but lower than the annual average air temperature by about 0.8 °C. The long-term trend analysis of the air temperature and soil temperature at a depth of up to 10 m in Slovakia showed that the air and soil/ground water temperature have risen by 0.6 and 0.5 °C, respectively, per decade over the past 30 years. The second part of the study aimed at modelling the daily groundwater temperatures at depths of 0.6–15 m below the surface. The simple groundwater temperature model was constructed based on a one-dimensional differential Fourier heat conduction equation. The given model can be used to estimate future groundwater temperature trends using regional air temperature projections calculated for different greenhouse gas emission scenarios.

Performance of Concrete Structure after Fire Disaster Based on Thermal Stress Coupling Field Analysis

Concrete is a kind of non-combustible material, but its mechanical properties are quite sensitive to temperature, once the temperature exceeds the limit temperature of steel and concrete, the structure of the building will be broken, therefore it’s an urgent task to study the fire resistance, bearing capacity and stability of the concrete structure after fire disaster, and to respond to these questions, this paper studied the performance of concrete structure after fire disaster based on thermal stress coupling field analysis. At first, this paper gave the method for collecting displacement changes of each surface of concrete structure under high temperature during the experiment, and analyzed the thermal expansion strain of the concrete under thermal stress coupling effect. Then, to investigate the changes in the stability of concrete structure under high temperature, this paper simulated the thermal stress coupling of concrete structure after fire. After that, parameters of the concrete structure under high temperature were selected to further simulate the fire disaster temperature field of the concrete structure under high temperature and construct the corresponding heat conduction differential equations of the structure. At last, this paper analyzed the experimental results, proposed conclusions, and verified the effectiveness of the proposed parameter calculation method.

Efficient separation of cathode materials and Al foils from spent lithium batteries with glycerol heating: A green and unconventional way

The recycle and reuse of spent lithium-ion batteries are of great environmental and economic significance. Here we succeeded in separating cathode materials from Al foils under an innovative method by the combination of glycerol heating and mechanical stirring. The peel-off rate of cathode material from aluminum foil reaches more than 95% after the treatment under the optimized conditions: heating temperature 200 °C, stirring speed 350 r/min, stirring time 15min. The heat transfer mechanism of cathode materials was determined to adapt theoretical analysis method based on heat conduction differential equation, and the relationship between heating temperature and residence time were accurately calculated. The results infer that the positive electrode sheets can quickly gain the same heat and reach targeted temperature through heated glycerol. The forces on the cathode materials stirred in glycerol were counted into the experiment as well. The stirring speed leading to the isolation of cathode materials from aluminum foils was calculated theoretically based on boundary layer theory and Blasius solution, and the theoretical stirring speed of 239 r/min was concluded by fluid mechanics.

Information-modeling forecasting system for thermal mode of top converter lance

On basis of mathematical modeling and object-oriented programming, a computer information-modeling forecasting system (IMFS) for thermal mode of top lance barrel (TLB) of oxygen converter was developed in order to fulfill the urgent and economically feasible task of determining the compliance of input technological parameters with certain safety criteria for conducting converter melting.The program was created in the form of a Windows-oriented application by refining the previously developed mathematical model of the temperature mode of the top converter lance barrel using the object-oriented programming language C# in Microsoft Visual Studio 2019 IDE. The mathematical model provides the solution of differential heat conduction equation in cylindrical coordinates (two-dimensional formulation) with assignment of the initial (temperature distribution in the computational domain) and boundary conditions of the II and III kind (respectively, on the outer and inner surfaces of the TLB). The finite-difference approximation of the heat conduction equation and boundary conditions was obtained by the integro-interpolation method (balance method). A numerical sweep method (modified Gauss method) and an unconditionally stable implicit scheme were used to calculate the temperature field. Thermophysical values were obtained by approximating the corresponding tabular values. The application does not put forward special requirements for the computer infrastructure, operates locally (without the need for access to Internet), does not require special skills to work with it, having an intuitive user interface: the working area of the program consists of three windows (sections), in which the results of calculating the thermal mode of the TLB are displayed. The developed IMFS allows evaluating the design and technological parameters of the top blowing device as a criterion for its safe operation. Its application in the “advisor” mode ensures the optimal design of the top oxygen lances with a rational water cooling system in order to ensure the proper thermal mode of the TLB throughout the entire operation period, as well as trouble-free operation of the blowing device, which is especially important for the conditions of converter shops in Ukraine equipped with outdated designs of top lances with low service life.

Biot’s Variational Method to determine the thermal strain in layered glazings

A critical issue in the structural design of glazed surfaces is the evaluation of the strain consequent to temperature variations due to environmental actions such as solar radiation, which represents one of the main causes of breakage. In the practice, approximate solutions are used, where the temperature profile across the glass thickness is constant or linear, but the consequent thermal stress cannot be adequately estimated from these. On the other hand, sophisticated thermal software is available only for important tasks.Here, we propose a semi-analytical approach, easily implementable in a simple FEM code, to evaluate the time-dependent temperature profile through the thickness of layered glazing, which is based on the variational method proposed by Biot in the Fifties. A prompt evaluation not only of the temperature field, but also of the heat flux, can be obtained. Compared to other numerical approaches, this method rigorously accounts for energy conservation and, since it does not involve temperature gradients in the formulation, it is particularly efficient for problems with steep temperature variations. Temperature profiles that are not necessarily linear can be approximated by Hermite–splines, for a precise evaluation of the thermally-induced stress. Comparisons with a direct numerical solution of the heat-conduction differential equations confirm the accuracy and the effectiveness of the proposed approach.

MATHEMATICAL MODELING OF THE FOREST FIRE IMPACT ON THE BRANCH OF A CONIFEROUS TREE

It is necessary to develop quantitative methods to assess the formation of thermal burns in the morphological parts of coniferous trees. The purpose of the study is mathematical modeling of heat transfer in the layered structure of a coniferous tree branch under the influence of a forest fire front. The heat propagation in the «branch — needles — flame zone» system is described by a system of non-stationary differential equations of heat conduction with the corresponding initial and boundary conditions. As an object of research, a digital model of a branch of a coniferous tree for various species, namely, pine, larch and fir, was used. Temperature distributions are obtained for different variants of the branch structure and conditions of the impact of the forest fire front. Conclusions are made about the need for further modernization of the mathematical model. The developed model is the basis for creating software tools for specialized geographic information systems.

Three‐dimensional thermal modeling and dimensioning design in the nuclear waste repository

AbstractA mathematical model is established by using the cylindrical heat source to simulate the temperature field in the nuclear waste repository. By means of the integral transformation method, the partial differential equations of heat conduction were transformed into ordinary differential equations to obtain the semi‐analytical solutions to temperature change. The radial temperature distribution around the nuclear waste canister was analyzed and compared with other analytical models. The built model was applied to determine the canister spacing with variable tunnel spacing for different type canister panel. Then, the quantitative effects of decay heat power at initial installation and of other parameters on the canister spacing were specified. The results show that the built model can predict the near‐field and far‐field temperature around the waste canister at the same time. The decay heat power at the initial installation and thermal properties of buffer layer and surrounding rock have strong effects on the canister spacing.

Thermo-vibrational analyses of skin tissue subjected to laser heating source in thermal therapy

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation in the tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of vibration and heat transfer in the tissue with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

Research on Thermal Characteristics of Ball Screw Feed System Considering Nut Movement

The heat generated by the ball screw feed system will produce thermal errors, which will cause the positioning accuracy to decrease. The thermal simulation modeling of the ball screw feed system is the basis for compensating thermal errors. The current thermal characteristic modeling method simplifies the reciprocating movement of the nut pair on the screw shaft to varying degrees, which leads to a decrease in simulation accuracy. In this paper, the nut is regarded as a moving heat source, and a novel method is adopted to make the moving process of the heat source closer to the actual nut movement process. The finite difference method is used to simulate the temperature field and thermal error of the ball screw feed system under different working conditions. Firstly, based on the heat transfer theory, the heat conduction differential equation of the feed system is established and discretized. The thermal error model of the ball screw feed system is established. Then, the relationship between nut heat source position and operating time is established to simulate nut reciprocating motion. Finally, the temperature and thermal error experiments of the ball screw feed system were carried out, and the temperature experiment results were compared with the simulation results of the finite difference method. The results show that the maximum simulation error of the average temperature in the operating interval is 11.4%, and the maximum simulation error of thermal error is 16.4%, which verifies the validity and correctness of the method. The thermal characteristic modeling method of the ball screw feed system proposed in this paper has a substantial application value for accurately obtaining the temperature field of the feed system.

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.