scholarly journals Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

2022 ◽  
Vol 2022 ◽  
pp. 1-8
Author(s):  
Endalew Getnet Tsega
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Numerical Solution ◽  
Numerical Method ◽  
Heat Conduction Equation ◽  
Three Dimensional ◽  
Transient Heat Conduction ◽  
Cylindrical Coordinates ◽  
Central Difference ◽  
Central Difference Method

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

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.

Transient Thermoelastic Fields in a Transversely Isotropic Infinite Solid With a Penny-Shaped Crack

1987 ◽  
Vol 54 (4) ◽  
pp. 854-860 ◽  
Author(s):  
N. Noda ◽  
F. Ashida
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transient Heat Conduction ◽  
Thermoelastic Problem ◽  
Conduction Equation ◽  
Thermal Stress Field ◽  
Penny Shaped Crack ◽  
Shaped Crack

The present paper deals with a transient thermoelastic problem for an axisymmetric transversely isotropic infinite solid with a penny-shaped crack. A finite difference formulation based on the time variable alone is proposed to solve a three-dimensional transient heat conduction equation in an orthotropic medium. Using this formulation, the heat conduction equation reduces to a differential equation with respect to the spatial variables. This formulation is applied to attack the transient thermoelastic problem for an axisymmetric transversely isotropic infinite solid containing a penny-shaped crack subjected to heat absorption and heat exchange through the crack surface. Thus, the thermal stress field is analyzed by means of the transversely isotropic potential function method.

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.

scholarly journals A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

2020 ◽  
Vol 5 ◽  
pp. 36
Author(s):  
S. Abbas ◽  
A. A. Khan ◽  
B. Shakia
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Central Difference ◽  
Cyclic Reduction ◽  
Difference Formula ◽  
Reduction Methods ◽  
Crank Nicolson

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

Numerical Solution using SPH Method: An Application to Transient Heat Conduction

2019 ◽  
Author(s):  
Almério José Venâncio Pains Soares Pamplona ◽  
Karoliny Freitas Silva ◽  
Cláudio Bucar Filho ◽  
Joel Vasco
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Transient Heat Conduction ◽  
Sph Method

Effect of Three-Dimensional Electric Field and Heat Conduction to Electrodes on the Temperature Rise During Irreversible Electroporation

2011 ◽  
Author(s):  
Seiji Nomura ◽  
Kosaku Kurata ◽  
Hiroshi Takamatsu
Keyword(s):  
Heat Conduction ◽  
Electric Field ◽  
Temperature Rise ◽  
Heat Conduction Equation ◽  
Thermal Damage ◽  
Irreversible Electroporation ◽  
Three Dimensional ◽  
End Effects ◽  
Abnormal Cells ◽  
Novel Method

The irreversible electroporation (IRE) is a novel method to ablate abnormal cells by applying a high voltage between two electrodes that are stuck into abnormal tissues. One of the advantages of the IRE is that the extracellular matrix (ECM) may be kept intact, which is favorable for healing. For a successful IRE, it is therefore important to avoid thermal damage of ECM resulted from the Joule heating within the tissue. A three-dimensional (3-D) analysis was conducted in this study to predict temperature rise during the IRE. The equation of electric field and the heat conduction equation were solved numerically by a finite element method. It was clarified that the highest temperature rise occurred at the base of electrodes adjacent to the insulated surface. The result was significantly different from a two-dimensional (2-D) analysis due to end effects, suggesting that the 3-D analysis is required to determine the optimal condition.

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