Unsteady Heat Conduction Problems Research Articles

Effects of flow direction and thermal short-circuiting on the performance of coaxial ground heat exchangers

The effects of flow direction and thermal short-circuiting on the performance of coaxial ground heat exchangers are studied by finite-element simulations, performed through COMSOL Multiphysics 3.4 (©Comsol, Inc.). The real 2-D axisymmetric unsteady heat conduction and convection problem is considered. The distribution of the fluid bulk temperature in the inner circular tube is determined by means of the “weak boundary form” boundary condition available in COMSOL Multiphysics; the laminar convective heat transfer in the outer annular passage is simulated directly. Two Coaxial Ground Heat Exchangers (CGHEs) with the same length but different cross sections are examined; moreover, two values of the ground thermal conductivity, as well as two materials for the inner tube wall are considered. The results point out that the annulus-in flow direction (fluid inlet in the outer annular passage) is more efficient than the center-in flow direction (fluid inlet in the inner circular tube) and that the effect of short circuiting is not very important, if the annulus-in flow direction is employed. Indeed, with this inlet configuration the energy loss for thermal short-circuiting is less than 1% for time intervals longer than 1 hour.

A variable future-time-steps method for solving nonlinear unsteady inverse heat conduction problems

In some non-linear unsteady inverse problems, the inverse solution will oscillate violently in the whole time domain due to the sharp change of the sensitivity coefficients. To deal with this problem, a new sequential function specification method with variable future time steps is proposed in this paper. The future time steps are adjusted by the error amplification coefficients which are defined as the reciprocal of the square sum of the sensitivity coefficients. When the error amplification coefficients are small, a small number of future time steps is used to reduce the deterministic error. While in the period with large error amplification coefficient, a large number of future time steps is used to reduce stochastic error. Finally, the total error of estimated heat flux is reduced. Avoid the sharp fluctuation of estimated heat flux in time domain due to the sharp change of sensitivity coefficients. The variable future-time-steps method is applied to the estimation of 1-D non-linear unsteady heat flux without and with ablation through numerical experiments. Numerical experiments show that the proposed method can not only estimate various forms of heat flux, but also its inversion results are significantly better than those of the fixed future time steps method based on the discrepancy principle, and also better than those of the fixed future time step method based on the minimum relative error of heat flux.

Transient heat conduction modeling in continuous and discontinuous anisotropic materials with the numerical manifold method

Transient heat conduction is a common problem in engineering. However, related studies in anisotropic materials remain relatively scarce for their complexity, compared with isotropic ones. Benefiting from the use of the two cover systems, i.e., the mathematical cover and the physical cover, the numerical manifold method (NMM) provides a unified framework for both continuous and discontinuous analyses. Herein, the NMM is applied to solve transient heat conduction problems with anisotropic materials for continuous and discontinuous cases. Firstly, the fundamental formulas for anisotropic heat conduction are displayed. Then, the NMM approximations for heat conduction in continuous and discontinuous situations are given, and the discrete formulations are derived based on the Galerkin-form weighted residual method and solved by the backward difference scheme. Finally, typical numerical examples are conducted with non-conforming mathematical covers, and the results show that the proposed method can tackle both continuous and discontinuous unsteady anisotropic heat conduction problems with high accuracy and convenience.

Research on Unsteady Inverse Heat Conduction Based on Dynamic Matrix Control

For the unsteady multi-boundary inverse heat conduction problem, a real-time solution method for boundary heat flux based on dynamic matrix control is proposed in the paper. The method solves the heat flux at the boundary in real-time by measuring the temperature information at the measurement points of the heat transfer system. A two-dimensional direct heat conduction model of the heat transfer system is established in the paper, and is solved by the finite difference method to obtain the temperature information of the measurement points under any heat flux boundary. Then, the correspondence between the heat flux of boundary and the temperature information is presented by means of a step-response model. The regularization parameters are introduced into the method to improve the stability of the inversion process, and the effect of real-time inversion on the heat flux of the boundary is achieved through rolling optimization. The experimental results show that the proposed method can achieve real-time inversion of the heat fluxes of the two-dimensional boundary with good accuracy.

On the proofs of orthogonality of eigenfunctions for heat conduction, wave propagation, and advection-diffusion problems

The orthogonality of eigenfunctions in problems of unsteady heat conduction in an infinite slab with symmetric and nonsymmetric convective boundary conditions are demonstrated by performing actual integration of the products of the derived forms of eigenfunctions and by implementing the corresponding eigenvalue conditions. The analysis also applies to longitudinal vibrations of an elastic rod attached at its ends to linear elastic springs, and to advection-diffusion problems under appropriate boundary conditions. The same form of eigenfunctions and the same type of eigenvalue condition apply in all three considered cases. The proofs are compared with the well-known general proof of orthogonality of eigenfunctions, which circumvents the actual integration. The presented analysis is pedagogically appealing for use in engineering and applied physics education.

Real-Time Solution of Unsteady Inverse Heat Conduction Problem Based on Parameter-Adaptive PID with Improved Whale Optimization Algorithm

To solve the problem of the common unsteady inverse heat conduction problem in the industrial field, a real-time solution method of improving the whale optimization algorithm (IWOA) and parameter-adaptive proportional-integral-differential (PID) is proposed in the paper. A feedback control system with IWOA-PID, which can inversely solve the boundary heat flux, is established. The deviation between the calculated temperature and the measured temperature of the measured point obtained by solving the direct heat conduction problem (DHCP) is used as the system input. The heat flux which is iteration-solved by IWOA-PID is used as system output. The method improves the initial solution distribution, global search capability and population diversity generalization of the traditional whale optimization algorithm (WOA), which effectively improves the parameter-adaptive capability of PID. The experimental results show that the solution method of inverse heat transfer proposed in the paper can accurately retrieve the variation of the boundary heat flux in real time and has good resistance and self-adaptability.

Analytical solution of unsteady heat conduction in multilayer internal combustion engine walls

An analytical solution of the unsteady heat conduction problem in multilayer walls was developed and applied to study thermal barrier coatings for reciprocating internal combustion engines. The mathematical solution was derived using the matrix method and complex analysis/residue-calculus Laplace transform inversion techniques. The one-dimensional domain includes a time-varying heat flux on the combustion chamber surface, and a time-varying temperature on the backside coolant surface. These boundary conditions are simulated as the superposition of two adjacent triangular, unit-magnitude pulses, which gives a piece-wise linear representation of the applied flux or temperature history. The temperature at any location of interest is found as the convolution of the transfer function, which results from the inversion, with the discrete-time heat flux or backside temperature history. The transfer functions describe the exact response and only depend on multilayer architecture, therefore, they can be computed a priori. The triangular pulse method provides a more than two order of magnitude reduction in computation time compared to a finite difference solution at the same level of accuracy. A 104-fold speed increase relative to a finite difference solution was realized when using the Overlap-add convolution technique for a full drive cycle simulation, i.e., a long time record. The surface response or transfer function acts like a finite impulse response and can be viewed in the frequency domain to reveal complementary information about coating performance.

Robust RRE technique for increasing the order of accuracy of SPH numerical solutions

This study presents the use of a post-processing technique called repeated Richardson extrapolation (RRE) to improve the accuracy of numerical solutions of local and global variables obtained using the smoothed particle hydrodynamics (SPH) method. The investigation focuses on both the steady and unsteady one-dimensional heat conduction problems with Dirichlet boundary conditions, but this technique is applicable to multidimensional and other mathematical models. By using all the variables of the real type and quadruple precision (extended precision or Real*16) we were able to, for example, reduce the discretization error from 1.67E−08 to 3.46E−33 with four extrapolations, limited only by the round-off error and, consequently, determining benchmark solutions for the variable of interest ψ(1/2) using the SPH method. The increase in CPU time and memory usage owing to post-processing was almost null. RRE has proven to be robust in determining up to a sixteenth order of accuracy in meshless discretization for the spatial domain.

Unsteady Heat Conduction Problem for a Plane with a Crack at the Interface between Two Inhomogeneous Materials

Unsteady Heat Conduction Problem for a Plane with a Crack at the Interface between Two Inhomogeneous Materials

Use of the Method of Integral Relations for the Analytic Solutions of Hyperbolic Models of Thermal Conductivity

The study of the processes of unsteady heat conduction, the calculation of the parameters of media under conditions of unsteady heat conduction of the latter is an important direction, which is used in applied problems of heat and mass transfer. When solving a mathematical model under various boundary conditions, there is a problem of the reliability of numerical calculations, therefore there is a need to solve the mathematical model by an analytical method. For example, a mathematical model of heat and mass transfer processes in a heat accumulator during its charging and discharging is solved analytically by the Green's function method, similarly, a mathematical model of heat carrier heating processes in solar collectors is solved. The specific definition of the Green's function corresponds to a specific problem in mathematical physics. Green's function contains complete information about the studied equation, and with its help one can construct a solution for any inhomogeneity. The development of the method of Green's functions for solving boundary value problems of unsteady heat conduction of generalized type on the basis of the Maxwell-Cattaneo-Lykov law is proposed. On the basis of the introduced Green's function of the differential equation, the Green's function of the boundary value problem is determined. Green's function of a boundary value problem is considered as an element of the set of Green's functions of an equation or a system of equations. Boundary conditions are formulated in accordance with the specified law. When considering specific problems, in a number of cases, it is expedient to transfer the integral form of writing boundary conditions of the second or third kind into a differential form equivalent to the integral one. The proposed integral relations for analytical solutions of boundary value problems of unsteady heat conduction for equations of hyperbolic type. Necessary and sufficient conditions for the existence and uniqueness of the Green's function of the boundary value problem are given and its analytical representation is given in terms of the fundamental system of solutions and boundary conditions. Boundary conditions are formulated for hyperbolic models of heat conduction in integral and differential forms. Boundary value problems for a semi-infinite region are considered, analytical solutions are obtained, their analysis is carried out, and temperature jumps at the heat wave front are calculated. Illustrative problems for a semi-infinite region are considered, and the heat wake region and the unperturbed region are described.

Matrix method to solve the inverse problem of heat transfer in heat exchangers

Along with verification calculations of known designs of heat exchangers, in design engineering and when we develop new technologies, design calculations are necessary to solve the inverse problems of choosing the optimal designs and operating modes of equipment. Previously, the formulation and solution of inverse problems of classification and unsteady heat conduction have been considered, while the inverse problems of heat transfer in the design of heat exchange equipment are poorly presented in the literature. The development of methods to solve inverse problems in the design of heat exchange equipment is an urgent task of power industry. Matrix models of heat transfer based on mass and energy balance equations are used to formulate and solve inverse problems of heat exchange systems. Methods of mathematical programming are applied to solve inverse and optimization problems. For design calculations, a matrix method to solve inverse problems for choosing the design of devices and parameters of heat carriers that ensure the effective operation of the system is proposed. The inverse problem is formulated for the case of the sliding boundary of the beginning of the phase transition with the countercurrent type of movement of heat carriers. The obtained results can be used in power energy, chemical and food industries to improve the efficiency of designing resource-and energy-saving technologies. The solutions obtained can be implemented when developing measures to improve resource and energy saving technologies.

Unsteady Heat Conduction Study During Underground Coal Gasiflcation For Constructing an In-situ Pyrolysis Project.(Dept.M)

Underground coal gasification (UCG) Is a potential coal utilization technology, receiving renewed interest around the world. In the last .five years there has been Increasing Interest In tile world energy and mining Industry for tile development of a new operating underground coal gasification systems to generate synthetic gas from underground coal seams. In this paper, the problem of unsteady heat conduction In the underground coal and rock layers during underground gasification Is considered. The heating of coal results In releasing considerable amount of gases mainly hydrogen and methane which would be collected and pumped through pipelines for use in different Industrial applications. In the present analysis, a spherical type heat source having a surface temperature of (1000OC, 1500OC, 1750OC and 2000OC) is used for heating the coal layer for achieving tile pyrolysis process. The differential equations describing the unsteady heat conduction in both coal and rock layers are solved numerically using the discretization method (control volume formulation). A computer program Is constructed for calculating the temperature fields and the gasified volumes of mineral coal as well as the heating velocity in the coal seams. In the experimental part, a model of an underground coal layer Is constructed. To validate the numerical model, an electrical spherical heat source Is Inserted directly into the coal layer for heating. The temperature distribution In coal layer is measured In different locations using the Thermocouples. The results show that the temperature profiles and the maximum attainable temperature in the coal seams depend on the time and the surface temperature of the heating reactor. Furthermore, the gasified volume of coal increase with the increase of both the surface temperature of the heating source and the running time. It is found also that the heating velocity in the underground coal. Lies between the mean value (2 to 10-3 K/min.) and increase with the time to reach a maximum value, after this it decrease to approach zero at the steady-state conditions. This study indicates that geometry, position, surface temperature and distribution of heating sources are very important parameters which should be carefully chosen for realizing an efficient pyrolysis of underground coal and avoiding thereby the combustion of a part of the coal seams. The analysis indicated that the best temperature is between (700-800oC) at which the high efficiency of pyrolysis process, the analysis might be useful in constructing an in-situ pyrolysis project for utilizing the underground coal seams.

Bayesian Method for Parameter Estimation in Transient Heat Transfer Problem

Like many other scientific fields, an inverse technique is used to find unknown material properties (such as thermal conductivity, specific heat) or boundary conditions (e.g. heat flux, heat transfer coefficient) in the heat transfer problem. The role of the inverse technique is even more important if the properties or boundary conditions change with location and/or time. In this study, a Bayesian inference based inverse technique is used to find local heat transfer coefficient as well as local steam temperature at the inner surface of a steam header where the header was first heated by supplying steam followed by the cooling of the header due to the release of steam from the chamber. A finite volume based numerical model is developed to solve the forward unsteady heat conduction problem in the steam header. For inverse problem, a Markov chain Monte Carlo method is used to evaluate the posterior probability density functions (PPDFs) of unknown parameters in Bayesian framework, while a Metropolis-Hastings algorithm is used for random sample generation. Here PPDFs are used to extract the point estimates of the parameters and their credible intervals from their possible distributions. With the estimated parameters, the calculated outer surface temperatures agree well with experimental measurements in both heating and cooling process. At the end of the discharge process, the temperature distribution in the radial direction approaches flat profiles, but a steady state is only achieved at the bottom of the header. Our stochastic based Bayesian technique can be used to estimate unknown boundary conditions efficiently at an inaccessible location.

Cylindrical body temperature field simulation which made in the transitional thermal process conditions out of polymer material

The advancing requirements for strength, relaxation, thermophysical, electrical, and other structural elements characteristics actualizes the polymer composite material use for the soft part and node point manufacture, which improves performance index. This paper reported the need to take into account relaxation phenomena in predicting the body’s thermal field development that is made of polymeric materials, and the thermal relaxation time and the thermal damping time proportional to the duration of transient thermal process certain periods. In this article three-period thermal process in a cylindrical body mathematical model is presented. cylindrical body made of a low-heat-conducting material by using a heat conduction hyperbolic equation that is reflecting the heat flow relaxation and thermal damping phenomenon. A numerical solution to the problem of unsteady heat conduction in a circular disk for a two-phase delay equation is presented, which is based on the grid method implementation by using a three-layer implicit difference scheme and the finite difference method use. Calculation formulas for the run-through coefficients as well as the temperature values at the outer boundaries are concluded using the boundary conditions approximation for the intermediate and upper time layers, taking into account the multi-period of the process. The implementation of the modified run-through method when solving the non-stationary heat conduction problem in a cylindrical body, taking into account the finite heat propagation speed and thermal damping is described. The calculation results for the cylindrical body temperature field are obtained by using the polymethyl methacrylate example upon sudden heating based on a model with a two-phase delay. The results presented in this paper aid in an increase in predicting temperature field accuracy in polymer composite materials in the transient thermal processes study.

A non-overlapping optimized Schwarz method for the heat equation with non linear boundary conditions and with applications to de-icing

When simulating complex physical phenomena such as aircraft icing or de-icing, several dedicated solvers often need to be strongly coupled. In this work, a non-overlapping Schwarz method is constructed with the unsteady simulation of de-icing as the targeted application. To do so, optimized coupling coefficients are first derived for the one dimensional unsteady heat equation with linear boundary conditions and for the steady heat equation with non-linear boundary conditions. The choice of these coefficients is shown to guarantee the convergence of the method. Using a linearization of the boundary conditions, the method is then extended to the case of a general unsteady heat conduction problem. The method is tested on simple cases and the convergence properties are assessed theoretically and numerically. Finally the method is applied to the simulation of an aircraft electrothermal de-icing problem in two dimensions.

Modeling of thermophysical processes with moderate cryogenic effect on the periodontal zone using a thermoelectric cooling system

The aim of the article is to develop a mathematical model and conduct a numerical experiment of a thermoelectric system (TPP) for the treatment of in^ammatory periodontal diseases by moderate cryotherapy. Materials and methods . A design, a physical and mathematical model of TES for the treatment of in^ammatory periodontal diseases by moderate cryotherapy, developed on the basis of solving the unsteady heatconduction problem fora multilayer system, has been developed. Results . /4s a result of a numerical experiment, data were obtained that describe the temperature distribution over the thickness of each of the layers in the system, taking into accountthe heatfluxes on the junctions of the thermoelectric battery (ТЕБ), the parameters of the periodontal region, the temperature change atvarious points of the thermal power plant system - the periodontal region in time. It has been established that the necessary level of procedures associated with lowering the surface temperature of the periodontal zone to -40°C can be realized with a cooling capacity of ТЕБ of 6000 W/m2, which corresponds to the power of modern standard produced thermoelectric modules. Conclusions. It was estabished that the selection of the geometrical parameters TEB and power supply should be guided by the restrictions on the operation of the device, as well as medical norms and standards in order to avoid frostbite of adjacent tissues. A method for increasing the efficiency of the system is proposed, according to which the preliminary cooling of thermal power plants by an external source of cold is used, as well as by the use of forced operating modes of thermal fuel cells TEB.

Mathematical Modeling of Heat Transfer Processes in a Solid With Spherical Layer-type Inclusion to Absorb Penetrating Radiation

The paper deals with determining a temperature field of an isotropic solid with inclusion represented as a spherical layer that absorbing penetrating radiation. A hierarchy of simplified analogues of the basic model of the heat transfer process in the system under study was developed, including a “refined model of concentrated capacity”, a “concentrated capacity” model, and a “truncated model of concentrated capacity”. Each of the mathematical models of the hierarchy is a mixed problem for a second-order partial differential equation of the parabolic type with a specific boundary condition that actually takes into account the spherical layer available in the system under study.The use of the Laplace integral transform and the well-known theorems of operational calculus in analytically closed form enabled us to find solutions to the corresponding problems of unsteady heat conduction. The “concentrated capacitance” model was in detail analysed with the object under study subjected to the radiation flux of constant density. This model is associated with a thermally thin absorbing inclusion in the form of a spherical layer. It is shown that it allows us to submit the problem solution of unsteady heat conduction in the analytical form, which is the most convenient in terms of both its practical implementation and a theoretical assessment of the influence, the spherical layer width has on the temperature field of the object under study.Sufficient conditions are determined under which the temperature field of the analysed system can be identified with a given accuracy through the simplified analogues of the basic mathematical model. For simplified analogues of the basic model, the paper presents theoretical estimates of the maximum possible error when determining the radiated temperature field.

Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions

A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented.

Real-time estimation of thermal boundary of unsteady heat conduction system using PID algorithm

Based on the principle of the proportional-integral-derivative (PID) control in feedback control theory, an inverse method for unsteady heat conduction process is established in this paper. The inverse problems of unsteady heat conduction are transformed into a kind of PID control problem, and the PID parameters are tuned by the complex optimization method. The deviation signal between the output of the direct heat conduction model and the measured temperature is used to drive the PID controller, and the boundary heat flux of the heat conduction system is estimated in real time through the PID control law. The numerical simulation results and experiment results show that the inverse algorithm has strong anti-interference ability.

Study on heat transfer between the rod and the environment under conditions of forced convection

Obtaining analytical solutions to unsteady heat conduction problems is of great scientific and practical interest. Such solutions make it possible to do an in-depth analysis of thermal processes, such as isotherm fields analysis, study of the thermally stressed states of structures, parametric identification, etc. This article deals with a simple method for obtaining approximate analytical solutions of one-dimensional heat conduction problems. In particular, an algorithm for solving the problem for a rod (plate) with a given boundary condition of the third kind on one of the surfaces is presented. It is shown that solving the equation at isolated points of the spatial variable allows obtaining the high-precision solutions to this problem with a minimum amount of computational work. The relations for determining the temperature have a simple form and do not contain special functions and parameters. It should be noted that the exact solution to a similar problem based on the Fourier separation method is an infinite series containing eigenvalues (roots of the transcendental equation).The practical application of such solutions is very limited. The paper also contains the convergence analysis of the method, the residuals of the initial differential equation for various approximations. The method developed can be used to solve more complex problems that allow separation of variables in the initial l differential equation.