Unsteady Heat Conduction Problems Research Articles

Modeling methodology of locally non-equilibrium heat conductivity processes

With the development of laser technologies and the ability to carry out processing steps under extreme conditions (ul-trahigh temperatures, pressures and their gradients), the interest in studying the processes that occur under locally non-equilibrium conditions has grown significantly. The key directions for the description of locally non-equilibrium pro-cesses include thermodynamic, kinetic and phenomenological ones. The locally non-equilibrium transfer equations can also be derived from the Boltzmann equation by using the theory of random walks and molecular-kinetic methods. It should be noted that some options of locally non-equilibrium processes lead to conflicting results. This study aims to develop a method for mathematical modeling of locally nonequilibrium heat conduction processes in solids, which allows determining their temperature with high accuracy during fast and high-intensity heat transfer processes. As applied to heat transfer processes in solids, a generalized heat equation that takes into account the relaxation properties of materials is formulated. The exact analytical solution is obtained using the Fourier method of separation of variables. The methodology for mathematical modeling of locally non-equilibrium transfer processes based on modified conservation laws has been developed. The generalized differential heat equation which allows performing N-fold relaxation of the heat flow and temperature in the modified heat balance equation has been formulated. For the first time, an exact analytical solution to the unsteady heat conduction problem for an infinite plate was obtained taking into account many-fold relaxation. The analysis of the solution to the boundary value problem of locally nonequilibrium heat conduction enabled to conclude that it is impossible to instantly has establish a boundary condition of the first kind. It has been demonstrated that each of the following terms in the relaxed heat equation has an ever smaller effect on the heat transfer process. The obtained results can be used by the scientific and technical personnel of organizations and higher educational institutions in the study of fuel ignition processes, the development of laser processing of materials, the design of highly efficient heat transfer equipment and the description of fast-flowing heat transfer processes.

Heat Conduction at a Variable Heat-Transfer Coefficient

Some practically important problems of unsteady heat conduction with a time-variable relative heat-transfer coefficient are considered. Various approaches to finding a solution to the analytical problem are systematized: decomposition of the generalized integral Fourier transform, the serial expansion of a sought temperature function, and the reduction of the problem to an integral, second-order Volterra equation. It is demonstrated that the solution is reduced to an infinite series of successive approximations of different functional form in all cases, and the main objective of each of these approaches is to find the most advantageous first approximation. Particular cases of the time dependence of the relative heat-transfer coefficients are considered: linear, exponential, power, and root cases. Analytical solutions and numerical experiments are described, and some specific features of the temperature curves of a number of mentioned dependences are revealed. It is established that the picture of the change in the temperature curve for the linear time law of the heat-transfer coefficient becomes appreciably different from the classic case of a constant coefficient, whereas the exponential dependence does not introduce any essential changes.

Temperature Fluctuations in a Half-Space under Spatially Inhomogeneous Heat-Transfer Conditions

An unsteady two-dimensional heat-conduction problem without initial conditions is considered for a half-space under boundary conditions of the I, II, and III types. It is assumed that the boundary-value functions (including heat-transfer coefficient) are periodic functions of time and a spatial variable. Cyclic solutions to the problem are found in the form of double trigonometric Fourier series. The application of the obtained dependences is demonstrated with several specific examples. The conditions under which the quasi-one-dimensional solutions can be used are studied.

NUMERICAL MODELING AND SOFTWARE FOR DETERMINING THE STATIC AND LINKAGE PARAMETERS OF GROWING BODIES IN THE PROCESS OF NONSTATIONARY ADDITIVE HEAT AND MASS TRANSFER

Currently, methods of three-dimensional printing of products from polymer, metal and ceramic materials are widely used. Layer-by-layer synthesis technologies make it possible to obtain high-quality products with sufficiently high mechanical characteristics close to those realized using traditional technological processes. A related problem is solved, including calculating the temperature in the product and the operation area surrounding it in a flat setting based on two different approaches. For a layer-by-layer extensible body, the unsteady heat conduction problem is stated. At each step, the height of the computational region increases due to the filling of a new layer of powder. One time step of the calculation is determined by the interval between two consecutive passes of the laser. The temperature distribution found at each step is used as the initial conditions for the calculations at the next step. The temperature field obtained as a result of solving the problem that is implemented in the product in question with a step of 1000 (after applying and melting 1000 layers) is further compared with the solution of the quasi-stationary problem for the finite dimensions product. Using the example of the simple geometry taken, it is shown that a quasi-stationary solution can provide a satisfactory estimate of the macroscopic thermal state of the growing product.

Meshless local B-spline collocation method for heterogeneous heat conduction problems

Several numerical issues still pertain in the modeling of heterogeneous heat conduction, particularly from the viewpoints of material discontinuity and its handling and the presence of heat source. In this paper, a meshless local B-spline collocation method is presented for unsteady heat conduction problems of heterogeneous media. Unknown field variables are approximated by using B-spline basis functions within overlapped compact domains covering the geometry of materials. The present method is a truly meshless approach. The proposed approach is mainly coming with the following advantage that it is straightforward in dealing with discontinuity across the interface of heterogeneous materials. Treatment of discontinuity by using non-crossing interface compact domains allows material discontinuity to be handled geometrically without enforcing additional term/function at the interface. Several heat conduction problems in 2D and 3D heterogeneous media with arbitrary discontinuity shapes are considered. Attention is given for heterogeneous heat conduction problem accompanied by the presence of crack as well. The analysis is then completed by simulating effect of heat generation, in particular which produces high temperature rise inside a heterogeneous structure/component. Simulation results show that the proposed method is a simple and accurate numerical technique for solving unsteady heat conduction problems of heterogeneous media.

Нестаціонарна змішана задача теплопровідності для півбезмежного порожнистого циліндра

Analysis of temperature fields is important for many engineering applications. The account of actual operating conditions of these structures frequently leads to mixed heating condition. The authors of this paper developed a new effective method of solutions derivation for mixed boundary-value unsteady heat conduction problems. This paper considers the cylinder with at the part of surface of which the temperature distribution is known. Outside this area the heat transfer by Newton's law is performed. To the heat conductivity problem it is applied the Laguerre integral transformation in time variables and integral Fourier transformation in spatial variable. As a result the triangular sequence of ordinary differential equations is obtained. The general solution of these sequences is obtained in the form of algebraic convolution. Taking into account the mixed boundary conditions leads to dual integral equations. For solution of this problem it is proposed the method of Neumann's series. By this method the problem is reduced to the infinite system of algebraic equations, for which the convergence of reduction procedure is proved. Finally, the unknown temperature is submitted as a series of Laguerre polynomials. The coefficient of these series is Fourier integrals.

INTEGRAL TRANSFORMATION IN A PARTIALLY BOUNDED REGION WITH A RADIAL THERMAL FLOW

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

Solving of Two-Dimensional Unsteady Inverse Heat Conduction Problems Based on Boundary Element Method and Sequential Function Specification Method

The boundary element method (BEM) and sequential function specification method (SFSM) are used to research the inverse problem of boundary heat flux identification in the two-dimensional heat conduction system. The future time step in the SFSM is optimized by introducing the residual error principles to get the more accurate inversion results. For the forward problems, the BEM is used to calculate the required temperature value of discrete point; for the inverse problems, the impacts of different future time steps, measuring point position, and measuring error on the inversion results are discussed. Furthermore, the comparison is made for the optimal future time step obtained by introducing the residual error principle and the inherent future time step. The example analysis shows that the method proposed still has higher accuracy when the measuring error exists or the measuring point position is far away from the boundary heat flux.

Heat transfer during cooling of high temperature spheres in subcooled water at different pressures

In our previous publications Yagov et al. (2016) we have confirmed an existence of a particular mode of film boiling heat transfer in subcooled water, featured with extremely high intensity in comparison to saturated film boiling. The experiments with other liquids (ethanol, isopropanol, perfluorohexane) did not reveal this film boiling regime even at huge subcoolings (up to 160K). Basing on the analyzes of the revealed regularities it is possible to assume that the regime of intensive heat transfer in film boiling of subcooled water occurs only in cooling processes, when heat supply to the surface is controlled with a metal thermal effusivity. This regime appears easier during cooling of spheres with lower value of this property. Probably, some points of the surface (ledges of roughness) can protrude a vapor film and contact with liquid, when the average surface temperature is much higher than that of homogeneous nucleation. At present, there is no clear model of mechanisms of incipience of the regime discussed, and cumulation of the experimental facts continues to be necessary. The present paper sets forth the experimental results on heat transfer during cooling the spherical patterns from nickel, stainless steel, and copper in subcooled water at different pressures. The experiments at atmospheric pressure revealed marked influence of the metal thermal effusivity on the cooling process in highly subcooled water. The new experimental data evidence that the regimes of high intensity heat transfer are found under the conditions of impossibility of direct liquid/solid contacts at cooling water temperatures 100–150°C, if the liquid subcooling to saturation exceeds 20K. The cooling time decreases at the same water temperature with subcooling increase due to the pressure growth. The cooling process in saturated and slightly subcooled water occurs in the commonly known regime of film boiling independently on an absolute value of the saturation temperature. The approximate method based on averaging the surface temperatures and solving 1D inverse unsteady heat conduction problem was applied for calculations of heat transfer coefficient and heat flux density during cooling. The calculated HTCs in the regimes of intensive film boiling of subcooled water are an order of magnitude higher than those in the saturated and slightly subcooled liquid.

Streamlining the teaching of unsteady heat conduction in regular solid bodies with heat convection to neighboring fluids

In the analysis of unidirectional, unsteady heat conduction for simple solid bodies (large slab, long cylinder and sphere), the modern tendency adopted by authors of heat transfer textbooks is to calculate the temperatures and total heat transfer with “one-term” series accounting for the proper eigenquantities, which are expressed in terms of the Biot number. The supporting information is available in tables for a large slab, a long cylinder and a sphere. To avoid linear and quadratic interpolation for the Biot numbers listed in the tables, the goal of the present study is to use regression analysis in order to develop compact correlation equations for the first eigenvalues, the first eigencontants and the first constants (for the total heat transfer) varying with the Biot number for large slabs, long cylinders and spheres, all in the ample range 0 < Bi ≤ 100. This direct approach will speed up the step-by-step calculations of a multitude of unsteady heat conduction problems for engineering students.

New Approach to Simulation of Heat State of Compartments from Lattice Composite Shells for Space Engineering Products

A new approach to the simulation of the heat state of the compartment of lattice polymer composite materials (PCM), not providing for the use of known commercial software packages, has been proposed. The simulation has been performed using the PCM interstage of the Proton rocket as an example with due account of aerodynamic heating, solar radiation and acting of jets of auxiliary propulsion units. At the first stage of numerical analysis, a problem of unsteady heat conduction in the system “skin-air gap-heat insulation” has been solved. An effect of changing a pressure inside a compartment on thermal conductivity of heat insulation was taken into account. The effective thermal conductivity in gaps was used. An effect of a temperature of equipment on a value of radiant heat flux was also taken into account. At the second stage, the heat state of the system “skin-rib” was analyzed. A mathematical model in the form of a system of nonlinear equations for heat balance of control elements on which a rib and a skin section were partitioned, including an information about a temperature of heat insulation received at the first stage of the simulation, was used.

Solving of spatial problem of non-stationary heat conduction based on semi-analytical finite element method

Based on the semi-analytical finite element method it is developed solution for solving of spatial problem of non-stationary heat conduction for prismatic bodies of complex shape cross-section. The basis of the initial equation is correlations of spatial unsteady heat conduction problem in curvilinear coordinates in differential and variational formulations. The formula for determining the matrix elements of thermal conductivity and heat capacity based on semi-analytical finite element method are obtained based on the presentation of temperature distribution along the coordinate by x 3 Mikhlin polynomials allowing to use effective algorithm of the iterations block of upper relaxation systems for linear algebraic equations and implement efficient algorithm solution for solution system of differential equations in time. The value of semi-analytical finite element method and algorithms implemented in the form of problem-oriented subsystems for computer modeling of unsteady thermal processes are obtained. The reliability of results is substantiated by the interpretation of test cases with analytical and numerical results.

THE THERMOELECTRIC SYSTEM MODEL FOR THERMAL COSMETIC PROCEDURES

The model of thermoelectric systems for thermal cosmetic procedures that are based on solving two-dimensional unsteady heat conduction problem for a system of complex configuration. The dependences of temperature change at different points of the object of influence – the area of the skin exposed to the heat of cosmetic procedures, depending on the cooling and heating capacity of thermoelectric modules, external conditions.

Radial basis function (RBF)-based interpolation and spreading for the immersed boundary method

Immersed boundary methods are efficient tools of growing interest as they allow to use generic CFD codes to deal with complex, moving and deformable geometries, for a reasonable computational cost compared to classical body-conformal or unstructured mesh approaches. In this work, we propose a new immersed boundary method based on a radial basis functions framework for the spreading–interpolation procedure. The radial basis function approach allows for dealing with a cloud of scattered nodes around the immersed boundary, thus enabling the application of the devised algorithm to any underlying mesh system. The proposed method can also keep into account both Dirichlet and Neumann type conditions. To demonstrate the capabilities of our novel approach, the imposition of Dirichlet boundary conditions on a 2D cylinder geometry in a Navier–Stokes CFD solver, and the imposition of Neumann boundary conditions on an adiabatic wall in an unsteady heat conduction problem are considered. One of the most significant advantage of the proposed method lies in its simplicity given by the algorithmic possibility of carrying out the interpolation and spreading steps all together, in a single step.

Development of approximate analytical method for modeling transient thermal processes using S-functions

A new methodology for mathematical modeling of transient thermal processes in structural components is proposed. It is based on the joint application of the structural method, the Bubnov-Galerkin method and S-functions to solving heat conduction problems with unsteady boundary conditions of the third kind. The analytic structures for solving these problems, accurately satisfying unsteady boundary conditions at any given time dependence of the heat transfer coefficient and ambient temperature are constructed. These qualitative features of the analytic structures for solving heat conduction problems have allowed first proposed solution methods to obtain approximate analytical solutions of these problems.Using the Bubnov-Galerkin method has allowed to reduce solving heat conduction problems with unsteady boundary conditions to solving the system of ordinary differential equations with respect to the unknown time-dependent coefficients of problem solution structures. Herewith, in the given system of ordinary differential equations, known time-dependent coefficients contain the heat transfer coefficients and the ambient temperature in analytical representation at their any given time dependence. This first allows one-dimensional unsteady heat conduction problems (infinite plates, cylinder, hollow sphere) to obtain an approximate analytical solution of unsteady heat conduction problems for those options of time dependence of the heat transfer coefficient and the ambient temperature, for which the operating method is not applicable.

Investigation of V&V process for thermal fatigue issue in a sodium cooled fast reactor – Application of uncertainty quantification scheme in verification and validation with fluid-structure thermal interaction problem in T-junction piping system

Thermal fatigue caused by thermal mixing phenomena is one of the most important issues in design and safety assessment of fast breeder reactors. A numerical simulation code MUGTHES consisting of two calculation modules for unsteady thermal-hydraulics analysis and unsteady heat conduction analysis in structure has been developed to predict thermal mixing phenomena and to estimate thermal response of structure under the thermal interaction between fluid and structure fields. Although verification and validation (V&V) of MUGTHES has been required, actual procedure for uncertainty quantification is not fixed yet. In order to specify an actual procedure of V&V, uncertainty quantifications with the grid convergence index (GCI) estimation according to the existing guidelines were conducted in fundamental laminar flow problems for the thermal-hydraulics analysis module, and also uncertainty for the structure heat conduction analysis module and conjugate heat transfer model was quantified in comparison with the theoretical solutions of unsteady heat conduction problems. After the verification, MUGTHES was validated for a practical fluid-structure thermal interaction problem in T-junction piping system compared with measured results of velocity and temperatures of fluid and structure. Through the numerical simulations in the verification and validation, uncertainty of the code was successfully estimated and applicability of the code to the thermal fatigue issue was confirmed.

Study on Heat Conduction of Microwave Oven

For the heat distribution problem of the baking dish food in microwave oven, we established a partial differential equation model for the unsteady heat conduction problems, and solved this model by using the finite element method. Then, we drew the figures of temperature field distribution for the circular pan and the rectangular pan at a certain time respectively. Based on the conjecture that the part of their internal circular area for the graphics between the circle and rectangle are uniformly distributed, we verified the relationship between the pan evenly heated area and the shape of the pan, then we got the maximum heat distribution area for the circular pan and the rectangular pan, rectangular pan and regular hexagon pan.

Transient three-dimensional thermal model for batteries with thin electrodes

A three-dimensional analytical model is proposed to investigate the thermal response of batteries, with a plurality of thin electrodes, to heat generation during their operation. The model is based on integral-transform technique that gives a closed-form solution for the fundamental problem of unsteady heat conduction in batteries with orthotropic thermal conductivities, where the heat generation is a function of both temperature and depth-of-discharge. The full-field solutions take the form of a rapidly converging triple infinite sum whose leading terms provide a very simple yet accurate approximation of the battery thermal behavior with modest numerical effort. The accuracy of the proposed model is tested through comparison with numerical simulations. The method is used to describe spatial and temporal temperature evolution in a sample pouch type lithium-ion polymer battery during galvanostatic discharge processes while subjected to convective–radiative cooling at its surfaces (the most practical case is considered, when surrounding medium is at a constant ambient temperature). In the simulations, emphasis is placed on the maintenance of the battery operational temperature below a critical temperature. Through definition of a surface-averaged Biot number, certain conditions are highlighted, under which a two-dimensional thermal analysis is applicable.

Numerical solution of transient heat conduction problems using improved meshless local Petrov–Galerkin method

An improved meshless local Petrov–Galerkin (MLPG) method is presented and applied to calculate the two-dimensional unsteady state heat conduction problems. In this method, the moving Kriging interpolation is employed instead of the traditional MLS approximation to construct the MLPG shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions and then, the Heaviside step function is used as the test function over a local sub-domain. Since no mesh is needed either for integration of the local weak form, or for construction of the shape functions, the presently developed MLPG method is a truly meshless method. Several examples are performed to illustrate the accuracy and efficiency of the present method. A good agreement can be found among the proposed, analytical and finite element methods.

J053015 散逸粒子動力学法による非定常温度応答の解析(〔J053-01〕マイクロ・ナノスケールの熱流体現象(1))

Dissipative particle dynamics (DPD) has been regarded as one of an effective simulation method for mesoscopic system because the treatment of multi-body hydrodynamic interactions among particles is easy. Traditionally, DPD simulations have been used only for simulation of uniform temperature system because the temperatures of each particle are not defined. In this paper, energy conserved DPD simulation for one dimensional unsteady heat conduction problem has been conducted in order to verify the applicability for energy transport process. Energy conserved DPD simulation is accomplished by introducing an internal energy of each particles and the temperature of each particles are connected with internal energy. DPD simulation result of unsteady temperature response is compared with analytical solution of heat conduction equation and it is found that both results agree well. Furthermore, thermal conductivities are calculated from simulation results of steady heat conduction. It is clarified that the thermal conductivity of simulation system in DPD simulation is easily controllable by adjusting the energy transport coefficient k_o in formula of energy exchange between particles.