Solution Of Transient Heat Conduction Research Articles

Parameterized Reduced-Order Models for Probabilistic Analysis of Thermal Protection System Based on Proper Orthogonal Decomposition

The thermal protection system (TPS) represents one of the most critical subsystems for vehicle re-entry. However, due to uncertainties in thermal loads, material properties, and manufacturing deviations, the thermal response of the TPS exhibits significant randomness, posing considerable challenges in engineering design and reliability assessment. Given that uncertain aerodynamic heating loads manifest as a stochastic field over time, conventional surrogate models, typically accepting scalar random variables as inputs, face limitations in modeling them. Consequently, this paper introduces an effective characterization approach utilizing proper orthogonal decomposition (POD) to represent the uncertainties of aerodynamic heating. The augmented snapshots matrix is used to reduce the dimension of the random field by the decoupling method of independently spatial and temporal bases. The random variables describing material properties and geometric thickness are also employed as inputs for probabilistic analyses. An uncoupled POD Gaussian process regression (UPOD-GPR) model is then established to achieve highly accurate solutions for transient heat conduction. The model takes random heat flux fields as inputs and thermal response fields as outputs. Using a typical multi-layer TPS and thermal structure as two examples, probabilistic analyses are conducted. The mean square relative error of a typical multi-layer TPS is less than 4%. For the thermal structure, the averaged absolute error of the radiation and insulation layer is less than 25 °C and 6 °C when the maximum reaches 1200 °C and 150 °C, respectively. This approach can provide accurate and rapid predictions of thermal responses for TPS and thermal structures throughout their entire operating time when furnished with input heat flux fields and structural parameters.

Finite integral transform with homogenized boundary conditions solution of transient heat conduction applied to thermal plug and abandonment of oil wells

After the end of productive life of a geological oil reservoir, a set of operations for well Plug and Abandonment (P&A) is performed to recover the soil layers to its natural state. As traditional P&A techniques are expensive activities, alternative technologies are gaining renewed interest for capital expenditure reduction and leak risks mitigation. Hence, the Thermal Plug and Abandonment (TP&A) procedure is here investigated by focusing on the fulfillment of two intermediate milestones, combining new technology with conventional P&A practices. The first milestone consists in applying a thermite exothermic reaction to generate enough heat for melting the entire thickness of the production tube. The second milestone aims to determine if the resulting molten section has sufficient dimensions to allow for the passage of the cement pumped downhole, thus ensuring the complete sealing of the oil well's cross-section. The thermal investigation was conducted by applying a homogenized form of the Finite Integral Transform (FIT) analytical framework to solve the transient heat conduction equation in 2-D polar coordinates. The well assembly was geometrically discretized as a multilayered circular domain. The thermite reaction was modeled as a theoretical volumetric heat source profile dependent on both time and space, placed in the innermost layer. While a pure FIT approach may only be used to solve Neumann boundary conditions, the combined scheme applied here copes with any type of boundary condition. Hence, the integration of FIT with homogenization satisfies the Dirichlet condition requirement of the TP&A procedure. The methodology was verified through an equivalent Finite Volume Method (FVM) solution obtained using a commercial code. The results evidenced that within the 5 min duration of thermite reaction, the entire circumferential section of the production tube exceeds the steel melting temperature by at least 227 °C, thus fulfilling both milestones set. The research outcomes are part of a series of investigation steps to thoroughly analyze the new TP&A technology with regard to its compliance with the regulatory norms established to P&A operations.

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

The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical 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.

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.

Analytical solution of two-dimensional unsteady heat conduction in multi-layered cylindrical sectors applied to thermal plug and abandonment of oil wells

In this work, a hybrid analytical and numerical solution for transient heat conduction across a composite cylindrical sector is presented. A two-dimensional domain consisting of a multi-layer circular sector of angle φ was investigated (φ<2π). The Separation of Variables Method (SVM) was applied to solve the partial differential equation with non-homogeneous boundary conditions of the first, second and third kinds prescribed in the radial direction. Homogeneous boundary conditions of first and second kinds were arbitrated in the angular direction. A spatial time-independent source term gi(r,θ) was considered. The radial eigenvalues problem for the (r,θ) domain returns only real quantities and depends implicitly on the angular eigenvalues. Results for time dependent temperatures using the Separation of Variables Method were compared with numerical results, showing good accuracy. A second set of results was developed to investigate boundary conditions, material properties and the thermal source power required to rise temperature levels (mainly around the mid-angle φ/2) high enough to promote melting of certain layers of materials. These results might be useful for investigating a novel technology for the decommissioning of oil wells using thermal sources, often referred to in the literature as Thermal Plug and Abandonment (TP&A).

Transient Heat Conduction in a Planar Slab With Convection and Radiation Effects

Abstract This article presents new semi-analytical solutions for transient heat conduction in a slab, which exchanges heat at the surface with its surroundings via convection, radiation, or simultaneous convection–radiation mechanisms. The concept of thermal penetration depth together with the integral method of heat balances form the basis of the formulation. Explicit expressions are derived that eliminate the need for numerical integration of the ordinary differential equations (ODEs) obtained from space integration of the heat equation. Verification of the temperature relations with a convection boundary condition is performed using an exact solution. In the case of radiation and combined convection–radiation boundary conditions, the accuracy of the explicit solutions is assessed against a numerical model. In all three cases, the new relations predict the surface temperature with high precision. For the radiation boundary condition, the computed midpoint temperature is also in excellent agreement with the numerical solution. In the case of convection heat exchange at the surface, the temperature at internal locations has close agreement with the exact solution for Biot numbers up to Bi = 1. The model prediction deviates from the exact solution in the interior positions for higher values of Bi with the highest deviation occurring at the center. For example, at Bi = 3, the highest relative error is 7.5%. Likewise, for the case of combined convection–radiation boundary conditions, the predicted midpoint temperature is found to closely follow the numerical solution at early stages of the process only. The predictability of the explicit solution of the convection-only boundary condition is more accurate than other approximate solutions at a Fourier number less than 0.2. Additional relations are presented for determination of the total heat transfer as a function of the Fourier and Biot numbers.

Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

Two-dimensional laser-induced thermal ablation modeling with integrated melt flow and vapor dynamics

A new detailed, physics-based computational framework for the direct prediction of material ablation under nanosecond pulsed laser irradiation is presented. The framework, named directed energy illumination and visualization (DEIVI), integrates two-dimensional (axisymmetric) models for the following physical processes within a single framework for the first time: (i) coupling of the laser irradiation with the solid, (ii) transient heat conduction in the condensed (solid and melt) phases including melt front tracking, (iii) flow of the melt layer under the action of recoil pressure, (iv) vaporization of the melt and development of recoil pressure, and (v) dynamics of the vapor/gas. The solution for transient heat conduction and melt interface tracking using the enthalpy method is similar to the development by Ho et al. [J. Appl. Phys. 78, 4696–4709 (1995)]; however, they did not model the flow of the melt layer and modeled one-dimensional heat flow in the solid. To model the melt flow, DEIVI solves the incompressible, viscous Navier–Stokes equations that include a pressure correction solver, treatment of spatial derivatives using upwind or hybrid schemes, and a staggered grid. The compressive interface capturing scheme for arbitrary mesh volume of fluid method [O. Ubbink, Ph.D. thesis, University of London (1997)] is implemented for tracking the melt/vapor interfaces. The recoil pressure is applied by setting a fixed-pressure boundary condition in grid cells located at the melt-vapor interface. Surface tension is also included in order to ensure realistic surface morphology behavior at the melt-vapor interface. The vaporization process is treated using a quasiequilibrium convection-dominated approach combined with Knudsen-type jump conditions [S. I. Anisimov, Sov. Phys. JETP 27, 182–183 (1968) and C. J. Knight, AIAA J. 17, 519–523 (1979)] to obtain the input conditions for solving the two-dimensional (axisymmetric) gasdynamics equations for the vapor. An operator splitting procedure is used to solve the Euler equations using a Lax–Wendroff or Colella and Woodward [J. Comput. Phys. 54, 174–201 (1984).] piecewise parabolic scheme. This paper presents key aspects of the integrated framework as well as verification analyses performed to ensure the accuracy of individual modules.

Analytical solution of transient heat conduction in a bi-layered circular plate irradiated by laser pulse

In this study, the heat conduction behavior of a bi-layered circular plate during pulsed laser heating is investigated. A laser pulse with non-Gaussian temporal form is applied to the center of the top surface of the plate at the beginning of time. The spatial distribution of the heat energy of the laser pulse is also considered. The Green’s function for heat conduction equation of two-dimensional laminated structure is determined and is used to derive the analytical expression of temperature increment of the problem. The validation of this theoretical model is obtained through comparison with numerical results obtained by the ANSYS software package. The spatial and temporal variations of the temperature increment are illustrated. The influence of the duration time of the laser pulse on the temperature increment is also analyzed.

Closed-form analytical solutions of transient heat conduction in hollow composite cylinders with any number of layers

Conventional analytical methods for transient heat conduction solutions, due to complicated derivations and formidable calculations, have been limited to composite bodies with two or three layers. Developed in this work is a new analytical solution method for transient heat conduction in hollow composite cylinders with an arbitrary number of layers and subject to general boundary conditions. In this method, a distributed transfer function formulation gives an s-domain solution of a composite cylinder and inverse Laplace transform of the s-domain solution via a newly derived residue formula yields the transient solution of the cylinder in an explicit and closed form. Unlike conventional analytical methods, the proposed method does not require different derivations for different cylinder configurations (such as number of layers, boundary conditions and thermal resistance at layer interfaces). Because it only involves two-by-two matrices in the solution process, the proposed method is highly efficient in computation, as demonstrated in numerical examples.

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.

Temperature Solution for Transient Heat Conduction in a Thin Bilayer Between Semi-Infinite Media in Thermal Effusivity Measurement

A series expansion and an approximation have been derived for the temperature in the general case of transient heat conduction in two thin layers between two semi-infinite media at different uniform initial temperatures. The temperature accuracy and thermal effusivity measurement accuracy have been mapped to establish a window of operation for direct thermal effusivity measurement by the modified transient plane source (MTPS) method. The presented temperature series enables quick thermal analysis of thermally thick and semithick samples without factory correction/calibration.

Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform

Analytical solution of transient heat conduction through a semi-infinite fractal medium is developed. The solution focuses on application of a local fractional derivative operator to model the heat transfer process and a solution through the Yang-Laplace transform.

One-Dimensional Transient Heat Conduction in Composite Living Perfuse Tissue

Modeling the conduction of heat in living tissue requires the consideration of sudden spatial discontinuities in property values as well as the presence of the body's circulatory system. This paper presents a description of the separation of variables method that results in a remarkably simple solution of transient heat conduction in a perfuse composite slab for which at least one of the layers experiences a zero perfusion rate. The method uses the natural analytic approach and formats the description so that the constants of integration of each composite layer are expressed in terms of those of the previous layer's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. The method is presented in the context of the Pennes bioheat equation for which the solution is developed for a system composed of any number of N layers with arbitrary initial conditions.

One dimensional transient heat conduction in segmented fin-like geometries with distinct discrete peripheral convection

This study presents the solution of transient heat conduction in a composite extended surface whose periphery is exposed to convection and whose thermophysical properties experience discontinuities along the longitudinal direction. The solution development uses the natural analytic approach and formats the description so that the constants of integration of each of the composite segments are expressed in terms of the previous segment's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. A three-segment case study is also provided to show the simplicity and applicability of the implementation of this solution.

Analytic Solutions for Heat Conduction in Functionally Graded Circular Hollow Cylinders with Time-Dependent Boundary Conditions

An analytic solution method, without integral transformation, is developed to find the exact solutions for transient heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions. By introducing suitable shifting functions, the governing second-order regular singular differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. The exact solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analyses are given to illustrate the efficiency and the accuracy of the analysis.

Green’s function solution for transient heat conduction in annular fin during solidification of phase change material

The Green’s function method is applied for the transient temperature of an annular fin when a phase change material (PCM) solidifies on it. The solidification of the PCMs takes place in a cylindrical shell storage. The thickness of the solid PCM on the fin varies with time and is obtained by the Megerlin method. The models are found with the Bessel equation to form an analytical solution. Three different kinds of boundary conditions are investigated. The comparison between analytical and numerical solutions is given. The results demonstrate that the significant accuracy is obtained for the temperature distribution for the fin in all cases.

Experimental validation of a frequency domain BEM model to study 2D and 3D heat transfer by conduction

Abstract This article presents an experimental validation of 2D and 2.5D boundary element method (BEM) solutions for transient heat conduction in systems containing heterogeneities. The problem is formulated in the frequency domain. The responses in the time domain are obtained by means of an inverse Fourier transform. Complex frequencies with a small imaginary part are introduced to cope with aliasing. Their effect is taken into account by rescaling the results in the time domain.For validation purposes, the solutions provided by the proposed BEM formulation were first verified against analytical solutions and then compared with experimental results. In the laboratory tests a steel inclusion was embedded in a confined host medium and unsteady temperatures were applied to its boundary. Two host media were tested: molded expanded polystyrene and medium-density fiberboard. The systems were subjected to plane and point heat sources. The thermal properties of these materials have been previously defined experimentally. The results show that the BEM solutions agree well with the experimental results.

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.