Mathematical Model Of Heat Conduction Research Articles

Mathematical models for determining and analyzing thermal regimes in mining industry mechanism structures

Purpose. To develop linear and nonlinear mathematical models of heat conduction for isotropic heterogeneous media with internal heating. This will allow for an increased accuracy in determining temperature fields, which will subsequently impact the effectiveness of designing mechanisms, devices, and individual components of structures that have a layered structure and are subjected to heat stress. Methodology. For the development of linear and nonlinear mathematical models of the temperature field and the analysis of temperature regimes in layered media with internal thermal heating, the coefficient of thermal conductivity is described as a whole using asymmetric unit functions. This makes it possible to solve a differential equation with singular coefficients in both linear and nonlinear boundary value problems of heat conduction with appropriate boundary conditions. Findings. Quadratic equations are obtained to determine the analytical solutions of linear and nonlinear boundary problems of heat conduction for a layered plate with internal heat load. Originality. The scientific novelty lies in the given method of linearization of the nonlinear mathematical model of heat conduction and obtaining analytical solutions, in a closed form, of the corresponding linear and nonlinear boundary value problems for isotropic layered media subjected to internal heating. Practical value. The developed linear and nonlinear mathematical models for determining the temperature distribution in layered structures with internal heating make it possible to analyze heat exchange processes and ensure the thermal stability of such structures. This also makes it possible to increase the heat resistance of structures and protect them from overheating, which can lead to damage to individual components and elements of mechanisms, as well as to the entire structure as a whole. The resulting analytical solutions can be used to predict temperature fields in mine shafts, underground environments and mechanisms of mining equipment, in particular, in drilling and underground compressor stations, ventilation systems and other equipment, which improves work efficiency and reduces useful energy consumption.

INVESTIGATION OF INFLUENCE OF FACTORS ON FIRE RESISTANCE OF PARTITIONS MADE OF SANDWICH PANELS

The authors describe the results of studies of thermal processes in partitions made of sandwich panels under one-sided influence of fire. Analysis of the thermal state of multilayer partitions with mineral wool plates and development of recommendations for the design of buildings from such structures is possible using the simulation of thermal processes that occur during fire exposure to the studied samples. There is a certain number of models of the thermal state of multilayer partitions with mineral wool plates of different levels of complexity and purpose. However, in these models there are parameters that are individual for each partition and are known with insufficient accuracy. The authors provided an improved mathematical model of heat conduction in partitions, the difference of which is the application of heat transfer coefficients from the unheated surface of the partition in boundary conditions. A comparison of the fire resistance limits for a partition made of a sandwich panel indicates a strong influence of the accuracy of setting the heat transfer coefficient on the dependence of the thickness of the partition on the fire resistance limit. The accuracy of the task of the heat transfer coefficient on the dependence of the thickness of the partition on the fire resistance limit was scientifically substantiated, which showed a strong influence of the boundary condition describing the processes of heat transfer on the surface of the partition, which is not heated, on the accuracy of determining the limits of fire resistance of the structure under study. The influence of the accuracy of the assignment of the coefficient of heat transfer from the object to the environment on the fire resistance of the investigated building structures, which during the test are subject to both natural (mobility) of air and forced blowing, as, for example, during tests of air ducts with fire protection, must be taken into account convectively radiation heat exchange, which will allow to obtain the most accurate result regarding the determination of their fire resistance characteristics. Keywords: fire-resistance, methodological support, parameters, modeling.

Mathematical Model of Heat Conduction for a Semi-Infinite Body, Taking into Account Memory Effects and Spatial Correlations

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed.

Nonlocal Thermodynamics: Mathematical Model of Two-Dimensional Thermal Conductivity

Nonlocal models of thermodynamics are becoming more and more popular in the modern world. Such models make it possible to describe materials with a complex structure and unique strength and temperature properties. Models of nonlocal thermodynamics of a continuous medium establish a relationship between micro and macro characteristics of materials. A mathematical model of thermal conductivity in nonlocal media is considered. The model is based on the theory of nonlocal continuum by A.K. Eringen. The interaction of material particles is described using local and nonlocal terms in the law of heat conduction. The nonlocal term describes the effect of decreasing the influence of the surrounding elements of the material structure with increasing distance. The effect of nonlocal influence is described using the standard non-locality function based on the Gaussian distribution. The nonlocality function depends on the distance between the elements of the material structure. The mathematical model of heat conduction in a nonlocal medium consists of an integro-differential heat conduction equation with initial and boundary conditions. A numerical solution to the problem of heat conduction in a nonlocal plate is obtained. The numerical solution of a two-dimensional problem based on the finite element method is described. The influence of nonlocal effects and material parameters on the thermal conductivity in a plate under highintensity surface heating is analyzed. The importance of nonlocal characteristics in modelling the thermodynamic behaviour of materials with a complex structure is demonstrated.

Space Physical Sensor Protection and Control System Based on Neural Network Prediction: Application in Princess Elizabeth Area of Antarctica.

In the inland areas of Antarctica, the establishment of an unmanned automatic observation support system is an urgent problem and challenge. This article introduces the development and application of an unmanned control system suitable for inland Antarctica. The system is called RIOD (Remote Control, Image Acquisition, Operation Maintenance, and Document Management System) for short. At the beginning of this research project, a mathematical model of heat conduction in the surface observation chamber was established, and the control strategy was determined through mathematical relationships and field experiments. Based on the analysis of local meteorological data, various neural network models are compared, and the training model with the smallest error is used to predict the future ambient temperature. Moreover, the future temperature is substituted into the mathematical model of thermal conductivity to obtain the input value of the next input power, to formulate the operation strategy for the system. This method maintains the regular operation of the sensor while reducing energy consumption. The RIOD system has been deployed in the Tai-Shan camp in China’s Antarctic inland inspection route. The application results 4.5 months after deployment show that the RIOD system can maintain stable operation at lower temperatures. This technology solves the demand for unmanned high-altitude physical observation or astronomical observation stations in inland areas.

INVESTIGATION OF TEMPERATURE MODES IN THERMOSENSITIVE NON-UNIFORM ELEMENTS OF RADIOELECTRONIC DEVICES

Context. The non-linear boundary value problem of heat conduction for a thermosensitive non-homogeneous strip-shaped element of a radio-electronic system with a through inclusion has been solved whose analytical-numerical solution enables us to analyze temperature regimes in the element. Objective. Is to develop such a method of linearization of mathematical model of heat conduction which enables us to obtain analytical numerical solution of the corresponding non-linear boundary value problem for determination of temperature field in elements of radio electronic devices, which are geometrically represented by a thermosensitive plate with a through inclusion. Method. A linearizing function which enables us to partially linearize the initial non-linear mathematical model of heat conduction for a thermosensitive non-homogeneous element of a radio electronic system in the form of “plate-inclusion” structure has been suggested. The introduced piece-wise linear approximation of temperature on plate-inclusion interfaces has enabled us to completely linearize the corresponding partially linearized boundary value problem relative to the linearizing function. After this, it became possible to apply Fourier’s integral transformation to the obtained linear problem with respect to one of the spatial coordinates, as well as to determine the linearizing function. The linear dependence of the coefficient of heat conductivity on temperature for structure materials with the use of the linearizing function has been considered. By solving the boundary value problem, the formulae for determination of temperature field in the “plate-inclusion” thermosensetive structure have been obtained. Results. The obtained formulae for determination of temperature field in a thermosensitive non-homogeneous element of radio electronic system were used to create the software which enables us to obtain distribution of value of temperature and to analyze temperature regimes. Conclusions. A mathematical model for the calculation for the temperature field in a “plate-inclusion” thermosensitive structure is adequate to the actual physical process, because no jump of temperature at “plate-inclusion” interfaces is observed. The numerical results for the chosen materials under linear dependence of the coefficient of thermoconductivity on temperature differ by 7% from the results which are obtained for constant coefficient of heat conductivity. Prospect of further investigation will consider more complicated geometric representation of elements of radio electronic systems.

Thermal influences of stabilization on warm and ice-rich permafrost with cement: Field observation and numerical simulation

In this paper, the dry cement was mixed with warm and ice-rich permafrost to improve its mechanical and physical properties. The changes of temperature of improved soil with different cement dosage were measured by in-situ test. Then, the temperature field of the improved soil in-situ test was calculated by mathematical model of heat conduction and hydration equation with consideration of temperature effect. Based on the built calculation model, we have simulated the radius of thermal disturbance and refreezing time in the process of improvement on warm and ice-rich permafrost with deep mixing method. It can be concluded from the results that, (1) heat transfer model with temperature effect can precisely calculate the changes of temperature field during the curing of cemented frozen soil; (2) in the field test, when the cement dosage is less than 15% by weight of frozen soil, the increase of temperature in cemented soil is not obvious, the peak of temperature increase appears only when the dosage is more than 20%; (3) when the dry deep mixing method with is adopted to improve the warm and ice-rich permafrost, the maximum radius of thermal disturbance increases exponentially with the decrease of mean annual ground temperature, the refreezing time decreases hyperbolically with the decrease of mean annual ground temperature.

Parameterising continuum models of heat transfer in heterogeneous living skin using experimental data

In this work we consider a recent experimental data set describing heat conduction in living porcine tissues. Understanding this novel data set is important because porcine skin is similar to human skin. Improving our understanding of heat conduction in living skin is relevant to understanding burn injuries, which are common, painful and can require prolonged and expensive treatment. A key feature of skin is that it is layered, with different thermal properties in different layers. Since the experimental data set involves heat conduction in thin living tissues of anesthetised animals, an important experimental constraint is that the temperature within the living tissue is measured at one spatial location within the layered structure. Our aim is to determine whether this data is sufficient to reliably infer the heat conduction parameters in layered skin, and we use a simplified two-layer mathematical model of heat conduction to mimic the generation of experimental data. Using synthetic data generated at one location in the two-layer mathematical model, we explore whether it is possible to infer values of the thermal diffusivity in both layers. After this initial exploration, we then examine how our ability to infer the thermal diffusivities changes when we vary the location at which the experimental data is recorded, as well as considering the situation where we are able to monitor the temperature at two locations within the layered structure. Overall, we find that our ability to parameterise a model of heterogeneous heat conduction with limited experimental data is very sensitive to the location where data is collected. Our modelling results provide guidance about optimal experimental design that could be used to guide future experimental studies.

Heat Conduction in Porous Media Characterized by Fractal Geometry

Fractal geometry (fractional Brownian motion—FBM) is introduced to characterize the pore distribution of porous material. Based on this fractal characterization, a mathematical model of heat conduction is presented to study heat conduction behaviors in porous material with a focus on effective thermal conductivity. The role of pore structure on temperature distribution and heat flux is examined and investigated for fractal porous material. In addition, the effects of fractal dimension, porosity, and the ratio of solid-matrix-to-fluid-phase thermal conductivity (ks/kf) on effective thermal conductivity are evaluated. The results indicate that pore structure has an important effect on heat conduction inside porous material. Increasing porosity lowers thermal conductivity. Even when porosity remains constant, effective thermal conductivity is affected by the fractal dimensions of the porous material. For porous material, the heat conduction capability weakens with increased fractal dimension. Additionally, fluid-phase thermal conduction across pores is effective in porous material only when ks/kf < 50. Otherwise, effective thermal conductivity for porous material with a given pore structure depends primarily on the thermal conductivity of the solid matrix.

Mathematical modeling of static layer crystallization for propellant grade hydrogen peroxide

Hydrogen peroxide (H2O2) is an important raw material widely used in many fields. In this work a mathematical model of heat conduction with a moving boundary was proposed to study the melt crystallization process of hydrogen peroxide which was carried out outside a cylindrical crystallizer. Considering the effects of the temperature of the cooling fluid on the thermal conductivity of crude crystal, the model is an improvement of Guardani's research and can be solved by analytic iteration method. An experiment was designed to measure the thickness of crystal layer with time under different conditions. A series of analysis, including the effects of different refrigerant temperature on crystal growth rate, the effects of different cooling rates on crystal layer growth rate, the effects of crystallization temperature on heat transfer and the model's application scope were conducted based on the comparison between experimental results and simulation results of the model.

Fractional Fourier Law with Three-Phase Lag of Thermoelasticity

In this work, a new mathematical model of heat conduction for an isotropic generalized thermoelasticity with a three-phase lag is derived using the methodology of fractional calculus. Some theorems of generalized thermoelasticity follow as limited cases. This model is applied to thermoelastic interaction due to periodically varying heat source. Some comparisons are shown in the figures to estimate the effects of the fractional order parameter on all the studied fields.

ANALYTICAL SOLUTIONS OF BIO-HEAT CONDUCTION ON SKIN IN FOURIER AND NON-FOURIER MODELS

In general, the transport of thermal energy in living tissue is a complex process. The analysis of the heat conduction of skin tissue is helpful for understanding of the bio-thermo-mechanical behavior of skin tissue. So far, three kinds of conduction law — (1) the Fourier model, (2) the C-V model and (3) dual-phase-lag (DPL) model — are often investigated in bio-thermal transfer process. In this study, the mathematical model of heat conduction of the skin tissue subjected to a general transient heating at the skin surface was established. The analytical solutions of these three conduction models are presented. In addition, the measure of thermal injury of the skin tissue subjected to a harmonic heating was investigated. It was found that the phenomenon of Fourier model is greatly different to those of the C-V and DPL models. Moreover, the effects of the phase lags, the heating frequency, and the heat quantity on the temperature variation and the index of thermal injury were significant. In sum, the analytical method can be used to solve the conduction problem of any one-layer tissue.

Application of Semi-analytical Method in Transient Thermal Field of Pavement Structures

The daily temperature change of a pavement structure has an important impact on its crack production and propagation. First of all, a one-dimensional transient mathematical model of heat conduction and a heat convection boundary condition that can take solar radiation and ground radiation into account were established according to the characters of the pavement structure and heat conduction. Secondly all the computational formulae were deduced for the above model utilizing SAM. Finally, the hourly temperature field of a typical pavement structure was calculated using ANSYS and SAM respectively. Results show that SAM solutions at different depths are in good line with ANSYS solutions. Therefore, SAM is an easy-programmed, fast and high precise method, and is deserve to be popular. In addition, the daily temperature changes at different depths were discussed, and some suggestions were provided for the thermal crack control and prevention of the structure

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

Modeling of Heat Conduction in Thermoplastic Honeycomb Core/Face Sheet Fusion Bonding

Honeycomb sandwich materials have been widely used in aerospace industry as secondary structural materials or interior panels. Recently the research of full thermoplastic honeycomb sandwich materials has been of interest due to their recyclability. In their production, they usually demand an adhesive-free process, namely fusion bonding, to connect thermoplastic honeycomb core and face sheets. It is a heat induced process where a parameter of temperature should be well controlled to guarantee the product quality. This article presents a mathematical model of heat conduction to analyze the transient temperature distribution from heating tools towards inner part of the core under fusion bonding conditions. In order to simplify the complexity of 3D honeycomb geometry, a homogenization method is used to obtain average thermal properties of the honeycomb along the major heat flux direction. The model is validated by comparing with the results of in-situ temperature measurement during fusion bonding. The presented model can also be applied to analyzing general out-of-plane heat conduction through honeycomb sandwich structures made from other materials.

A new kind of phase change material (PCM) for energy-storing wallboard

A new kind of phase change material (PCM) for energy-storing wallboard is introduced in this paper. By establishing the one-dimensional non-linear mathematical model for heat conduction of the PCM energy-storing wallboard and according to the “effective heat capacity method”, simulation and calculation were made using the software MATLAB to analyze and solve the heat transfer problem of the PCM room. Meanwhile, the property can be found that the heat storing/releasing ability of the new PCM is significantly higher than that of ordinary materials by the experiment-based method. The result indicates that applying proper PCM to the inner surface of the north wall in the ordinary room can not only enhance the indoor thermal-comfort dramatically, but also increase the utilization rate of the solar radiation. So the heating energy consuming is decreased and the goal of saving energy has been achieved. If the parameters of the PCM is given as follows: the phase change temperature is set at 23 °C, the thickness is set at 30 mm, the phase change enthalpy is set at 60 kJ/kg, and the heating temperature is set at 20 °C, the energy-saving rate of heating season η can get to 17% or higher. So the energy is effectively used and saved obviously.

Extended Irreversible Thermodynamics and Generalization of the Dual-Phase-Lag Model in Heat Transfer

The process of heat conduction, with allowance made for the lag effect, is considered. A postulate of extended irreversible thermodynamics is suggested, according to which the entropy is a function of the internal energy and its time derivatives (up to the third derivative, inclusive). On the basis of the developed thermodynamic formalism, a third-order heat conduction equation is derived, generalizing the known dual-phase-lag model equation. A mathematical model of heat conduction in a package of heat-conducting plates is built, leading to the considered third-order equation. By numerical modeling it is shown that this equation describes heat wave propagation.

OPTIMIZATION OF PASTEURIZATION PROCESSES FOR A SOUS VIDE PRODUCT IN RECTANGULAR THIN PROFILE FORMS

ABSTRACTA Hawaiian mackerel and rice product was prepared for sous vide processing. Heat penetration data were gathered and used to verify the values predicted by a computer model. Heat penetration data were collected during pasteurization regimes at five degree intervals between 65C and 85C. The measured time‐temperature data were used to evaluate heat penetration parameters for the sous vide product. These data were then used to calculate the process time required for a specific pasteurization value (Pv), that is equivalent to 13 times the decimal reduction (13D) of Streptococcus faecium at 85C. The optimum time‐temperature (33 min at 75C) combination was obtained using the maximum value of a multi‐factor objective function (OF). The optimal pasteurization temperature was chosen to minimize the volume average cook value (Cav) and to maximize the OF. Results show that optimization of certain quality factors can be achieved using a mathematical model for heat conduction in rectangular plastic containers.

Heat conduction on graphs

A mathematical model for heat conduction in a graph-like object provides a setting for interpreting an algebraic eigenvalue problem associated with a graph. Applications include bounds for eigenvalues and a way to find eigenvalues and eigenvectors for a subdivision graph.

A mathematical model of heat conduction in a prolate spheroidal coordinate system with applications to the theory of welding

The steady-state heat conduction equation is solved in a prolate spheroidal coordinate system. Some simple solutions are derived in the limiting cases of both low and high Peclet numbers. The analysis was carried out in such a way as to avoid the physical details of conditions inside the weld pool, so that the solutions are restricted to the solid region outside the weld pool. This procedure was specifically adopted because these conditions are difficult to gain access to experimentally, as is the precise detailed shape of the pool; the solutions obtained can be verified experimentally. The high-Peclet-number approximation is likely to be particularly useful in the case of laser welding where large translation speeds of the weld piece are of interest. The solution of the problem is given in the form of a series as well as in an asymptotic form. The asymptotic method of solution presented can be adapted to any smooth shape of weld pool with only minor alterations, since the method involves integration in the tangent plane to the weld pool and the results of such an integration are independent of the global form of the weld pool. The asymptotic result is compared with the exact solution in a number of special geometric configurations. These are prolate spheroidal weld pool geometries with various aspect ratios and cylindrical weld pool geometries with elliptical or circular cross sections. The results of these comparisons were found to be satisfactory.