Solution Of Heat Conduction Research Articles

Finite element solution of heat conduction in complex 3D geometries

This paper presents the use of the finite element method (FEM) to solve heat conduction problems in complex 3-dimensional geometries not amenable to analytical solutions. Heat conduction is important across engineering domains, but closed-form solutions only exist for basic shapes. For intricate real-world component geometries, numerical techniques like FEM must be applied. The paper outlines the mathematical formulation of FEM, starting from the heat conduction governing equations. The domain is discretized into a mesh of interconnected finite elements. Element equations are derived and assembled into a global matrix system relating nodal temperatures. Boundary conditions are imposed and the matrix equations solved to find the temperature distribution. An example problem analyzes steady state conduction in an L-shaped block with 90-degree corners and surface convection. Results show FEM can capture localized gradients and discontinuities difficult to model otherwise. Detailed temperature contours provide insight. FEM enables robust thermal simulation of complex 3D geometries with localized effects, expanding analysis capabilities beyond basic analytical shapes. Proper application of FEM is critical for accurate results.

Construction of Short-Time Heat Conduction Solutions in One-Dimensional Finite Rectangular Bodies

Abstract The concept of both penetration and deviation times for rectangular coordinates along with the principle of superposition for linear problems allows short-time solutions to be constructed for a one-dimensional (1D) rectangular finite body from the well-known solutions of a semi-infinite medium. Some adequate physical considerations due to thermal symmetries with respect to the middle plane of a slab to simulate homogeneous boundary conditions of the first and second kinds are also needed. These solutions can be applied at the level of accuracy desired (one part in 10A, with A = 2, 3, …, 15) with respect to the maximum temperature variation (that always occurs at the active surface and at the time of evaluation) in place of the exact analytical solution to the problem of interest consisting of an infinite number of terms and, hence, unapplicable.

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.

Finite integral transform solution of unsteady heat conduction applied to thermal plug and abandonment of oil wells

The end of the production phase of an offshore oil well represents a remarkable shift in field operations from extraction to plug and abandonment. The international normative requirements for permanent well plugging demand a series of technical maneuvers to avoid structural failures and the formation of leakage paths during or after the sealing process. The current closure technique requires a complete or partial removal of the production column before the borehole may be plugged with Portland cement. However, the tube removal process frequently results in an increase in the involved time and costs. Hence, the current research was aimed at investigating the prototype approach of Thermal Plug and Abandonment of wells by applying a thermite heat emitter device to melt the production column's steel. Here, the Finite Integral Transform analytical method was computationally implemented through an inhouse code to calculate the resultant transient temperature fields at the multilayered medium. The results were compared to a Finite Volume Method numerical solution to enhance the study’s reliability. The research outcomes provided insight that even in the event of a partial thermite reaction failure and a highly nonuniform heat pattern, the resultant molten azimuthal length of the production column may still allow enough room for cement flow. It was estimated through the temperatures achieved that the heat emitter is capable of melting at least nearly three-quarters of the production tube’s azimuthal length, thus eliminating the need for its removal and significantly reducing the sealing process operational costs.

Non-Fourier Heat Conduction of Nano-Films under Ultra-Fast Laser.

The ultra-fast laser heating process of nano-films is characterized by an ultra-short duration and ultra-small space size, in which the classical Fourier law based on the hypothesis of local equilibrium is no longer applicable. Based on the Cattaneo-Vernotte (CV) model and the dual-phase-lag (DPL) model, the two-dimensional analytical solutions of heat conduction in nano-films under ultra-fast laser are obtained using the integral transformation method. The results show that there is a thermal wave phenomenon inside the film, which becomes increasingly evident as the elapse of the lag time of the temperature gradient. Moreover, the wave amplitude in the vertical direction is much larger than that in the horizontal direction of the nano-film. By comparing the numerical result of the two models, it is found that the temperature distribution inside the nano-film based on the DPL model is gentler than that of the CV model. Additionally, the temperature distribution in the two-dimensional solution is lower than that in the one-dimensional solution under the same Knudsen number. In the comparison results of the CV model, the maximum peak difference in the thermal wave reaches 75.08 K when the Knudsen number is 1.0. This demonstrates that the horizontal energy carried by the laser source significantly impacts the temperature distribution within the film.

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.

The Effect of Biot Number on a Generalized Heat Conduction Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number values, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tabular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.

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.

Numerical Analysis of Airfoil Flow Control Using a Nanosecond Pulsed Plasma Actuator

The control mechanism of a nanosecond dielectric barrier discharge (NS-DBD) actuator for suppressing laminar separation on airfoils was investigated using two-dimensional scale-adaptive simulations. To model the control effect of the NS-DBD actuator, a surface heating approach based on the analytical solution for transient one-dimensional heat conduction has been used. The focus of the investigation lies in the study of vortex formation and evolution induced by the actuator. A NACA 0015 profile was simulated at 14° with 250,000 chord-based Reynolds number. Regarding energy deposition, a low-energy case and a high-energy case were simulated. Several actuation parameters were varied, including the electrode position and surface temperature. In addition, the influence of constant and temperature-dependent modeling of the kinematic viscosity was investigated. The results for the time-averaged pressure coefficients show excellent agreement with measurement results. High grid resolution in the boundary layer allowed a detailed investigation of the vortex formation process. The main finding is that vortices are generated by baroclinic torque and vortex dilatation and not by a Kelvin–Helmholtz instability. Initially, tiny vortices form at the beginning of the separation region. Subsequently, these vortices are carried downstream and grow in size, thus preventing full separation due to momentum transfer from the external flow.

The Application of Boundary Element Method for The Solution of The Inverse Heat Conduction

In this paper, the general and supplementary conditions are used to determine the inverse heat conduction in the parabolic equation. The measured conditions ensure that the inverse problem has a unique solution, but the solution is not stable, so the problem is ill-posed. Using the boundary element method to discretize the original problem, it is used to solve the problem. On this basis, the direct method, the least square method and the regular method are tried. Finally, the minimal energy method is used to solve the system. Numerical experiments are carried out and the results are discussed.

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).

Exact solutions of one-dimensional steady heat conduction analysis for multi-layered mediums by tri-diagonal matrix algorithm (TDMA) applicable to Excel program (Analytical solutions of steady heat conduction for multi-layered symmetrical plat plate, solid cylinder, sphere, unsymmetric plat plate, pipe, and spherical shell, and these applications to engineering problems)

Exact solutions of one-dimensional steady heat conduction analysis for multi-layered mediums by tri-diagonal matrix algorithm (TDMA) applicable to Excel program (Analytical solutions of steady heat conduction for multi-layered symmetrical plat plate, solid cylinder, sphere, unsymmetric plat plate, pipe, and spherical shell, and these applications to engineering problems)

GLOBAL WELL-POSEDNESS OF A CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH MEMORY

In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources.

Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

In the current research, a comprehensive analytical technique is presented to evaluate the non-Fourier thermal behavior of a 3-D hollow sphere subjected to arbitrarily-chosen space and time dependent boundary conditions. The transient hyperbolic thermal diffusion equation is driven based upon the Cattaneo-Vernotte (C-V) model and nondimensionalized using proper dimensionless parameters. The conventional procedure of separation of variables is applied for solving the 3-D hyperbolic heat conduction equation with general boundary conditions. In order to handle the time dependency of the boundary conditions, Duhamel's theorem is employed. Furthermore, for the purpose of demonstrating the applicability of the obtained general solution, two particular cases with different time-space varying boundary conditions are considered. Subsequently, their respective non-Fourier thermal characteristics are elaborately discussed and compared with the Fourier case. The quantitative analysis is carried out, including the profiles of the time-dependent temperature and 3-D distributions of temperature at different time frames. Eventually, the influences of the controlling factors such as Fourier number and Vernotte number on the temperature field distributions within a hollow sphere for both cases are assessed. The findings reveal that the lag time in the hyperbolic thermal propagation diminishes with a decrement of Vernotte number, and it asymptotically vanishes for the Fourier case. Also, the number and severity of the jump points that occurred in the non-Fourier cases decrease by increasing the Fourier number, and these points finally vanish at particular Fourier numbers.

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.

A semi-analytical solution technique for predicting circulating mud temperatures

Accurate prediction of circulating fluid temperature in wellbore is required for proper drilling operations and design of the closed-loop systems. This study involves the improvement of the traditional analytical solution based on the assumption of constant input values. The presented solutions are derived from the analytical solution of the transient radial heat conduction. Application of the superposition theorem and combination of the analytical solutions established for each segment constituting the wellbore to account for temporal and positional variations are the main features of the semi-analytical solution presented in this study. In this respect, it can be asserted that the solution is also valid for variable mud/water circulation in deviated and horizontal wells.The presented solution is validated using the pertinent results of the traditional analytical solution. The results of the semi-analytical solution for realistic wellbore design reveal the possible deviations from temperature profile corresponding to analytical solution. The temperature profile as a result of the circulation with variable rate in a designed horizontal well is examined to test the presented semi-analytical solution. The trials indicate that the semi-analytical solution which is able to be implemented with plausible computational effort can be an alternative improvement for the traditional analytical solution.

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.

Analytic investigation on error of heat flux measurement and data processing for large curvature models in hypersonic shock tunnels

Due to short test time, heat conduction was considered as transient in hypersonic shock tunnels. The heat flux measurement and data processing were operated basing on one-dimensional semi-infinite heat conduction theory. However, for models with local large curvature or small radius, it resulted in significant compression or expansion of space for heat transfer, or lateral heat conduction, which made the hypothesis of one-dimensional unsatisfied and errors. In this paper, approximate solutions for the unsteady heat conduction in cylindrically convex and concave shells were established, and were used for further analysis of the errors, with forms of heating load, location and curvature radius of heated surface taken into consideration.