scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

scholarly journals Explicit Solution for Cylindrical Heat Conduction

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Kaitlyn Parsons ◽  
Tyler Reichanadter ◽  
Andi Vicksman ◽  
Harvey Segur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Analytical Method ◽  
Explicit Solution ◽  
Separation Of Variables ◽  
Partial Differential ◽  
Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

Transient Temperatures and Thermal Stress Problems in Small Gas Turbines

1961 ◽  
Author(s):  
Robert R. van Nimwegen
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Gas Turbines ◽  
Heat Conduction Equation ◽  
Transient Heat Conduction ◽  
Analog Computer ◽  
Transient Temperature ◽  
Advantages And Disadvantages ◽  
Difference Form ◽  
Typical Temperature

A discussion is given of the steady and transient temperature problems associated with the design of small gas turbines and starter equipment. The transient heat-conduction equation is given in differential and finite-difference form. Boundary conditions of convective heat transfer, radiation and prescribed temperatures are discussed. Several methods for solution of these equations in irregularly shaped bodies are given. Examples of solutions of typical temperature problems in small gas turbines are shown. Solution by means of an electrolytic tank, analog computer and a digital computer are discussed, with the advantages and disadvantages of each method given.

The numerical solution of the heat conduction equation in one dimension

1964 ◽  
Vol 60 (4) ◽  
pp. 897-907 ◽  
Author(s):  
M. Wadsworth ◽  
A. Wragg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Heat Conduction Equation ◽  
Linear Equations ◽  
Conduction Equation ◽  
Partial Differential ◽  
Direct Numerical Method

AbstractThe replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.

Effect of Arbitrary Nonsteady Wall Temperature on Incompressible Heat Transfer

1962 ◽  
Vol 84 (4) ◽  
pp. 347-351 ◽  
Author(s):  
T. R. Goodman
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Wall Temperature ◽  
Integral Method ◽  
Nonsteady Heat ◽  
Uniform Heat ◽  
Special Cases ◽  
Title Problem

The title problem is solved using an integral method and ignoring viscous dissipation. A partial differential equation is derived which yields as special cases Lighthill’s non-uniform heat-transfer formula and the nonsteady heat conduction in a slab. The differential equation is then specialized to the nonsteady but uniform heat transfer on a flat plate. Comparisons with other solutions are made when available, and it is shown that the integral method produces accuracy of a few per cent in these limiting cases.

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 195-196 ◽  
pp. 712-717
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao ◽  
Niang Zhi Fan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated. Transient heat conduction analysises of infinite plate with uniform thickness and two dimensional rectangle region have been realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

Accurate analytical/numerical solution of the heat conduction equation

2014 ◽  
Vol 24 (7) ◽  
pp. 1519-1536 ◽  
Author(s):  
Antonio Campo ◽  
Abraham J. Salazar ◽  
Diego J. Celentano ◽  
Marcos Raydan
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Numerical Solution ◽  
Time Domain ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Method Of Lines ◽  
Conduction Equation ◽  
Content Type ◽  
Entire Time

Purpose – The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables. Design/methodology/approach – The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense. Findings – In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent. Originality/value – Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 510 ◽  
pp. 205-210
Author(s):  
Xiao Feng Xiao ◽  
Qiong Xue
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated.. Transient heat conduction analysis of infinite plate with uniform thickness and two dimensional rectangle region are realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

scholarly journals Modeling methodology of locally non-equilibrium heat conductivity processes

Vestnik IGEU ◽  
2020 ◽  
pp. 65-71
Author(s):  
A.V. Eremin
Keyword(s):  
Mathematical Modeling ◽  
Heat Transfer ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Equation ◽  
Exact Analytical Solution ◽  
Separation Of Variables ◽  
Infinite Plate ◽  
Transfer Processes ◽  
Non Equilibrium

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.

Peridynamics for Heat Conduction

2020 ◽  
Author(s):  
Yozo Mikata
Keyword(s):  
Heat Conduction ◽  
Constitutive Equation ◽  
Heat Equation ◽  
Governing Equation ◽  
Heat Conduction Equation ◽  
Transient Heat Conduction ◽  
Anisotropic Materials ◽  
Time Dependent ◽  
Heat Sources ◽  
Exact Analytical Solutions

Abstract Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches 0. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent 3D peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

Two-Dimensional Vibrating And Thermal Response of Skin Tissue To Different Types of Heat Sources Considering The Hyperbolic Heat Transfer Equation

2021 ◽  
Author(s):  
Mina Ghanbari ◽  
Ghader Rezazadeh
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Skin Tissue ◽  
Transient Temperature ◽  
Hyperbolic Heat Conduction ◽  
Two Dimensional ◽  
Thermomechanical Response ◽  
Different Types

Abstract Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation of tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of skin displacement and heat transfer with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along the depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

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