Analytical Solution for One-Dimensional Heat Conduction-Convection Equation

Through the present work the authors determined the analytical solution of a transient two-dimensional heat conduction problem using Green’s Functions (GF). This method is very useful for solving cases where heat conduction is transient and whose boundary conditions vary with time. Boundary conditions of the problem in question, with rectangular geometry, are of the prescribed temperature type - prescribed flow in the direction x and prescribed flow - prescribed flow in the direction y, implying in the corresponding GF given by GX21Y22. The initial temperature of the space domain is assumed to be different from the prescribed temperature occurring at one of the boundaries along x. The temperature field solution of the two-dimensional problem was determined. The intrinsic verification of this solution was made by comparing the solution of a 1D problem. This was to consider the incident heat fluxes at y = 0 and y = 2b tending to zero, thus making the problem one-dimensional, with corresponding GF given by GX21. When comparing the results obtained in both cases, for a time of t = 1 s, it was seen that the temperature field of both was very similar, which validates the solution obtained for the 2D problem.

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Heat conduction in one-dimensional functionally graded media based on the dual-phase-lag theory

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406212456651 ◽

2012 ◽

Vol 227 (4) ◽

pp. 744-759 ◽

Cited By ~ 19

Author(s):

AH Akbarzadeh ◽

ZT Chen

Keyword(s):

Heat Conduction ◽

Analytical Solution ◽

Functionally Graded ◽

Hyperbolic Heat Conduction ◽

Phase Lag ◽

Dual Phase ◽

Spherical Coordinates ◽

One Dimensional ◽

Phase Lags ◽

Functionally Graded Media

In this article, heat conduction in one-dimensional functionally graded media is investigated based on the dual-phase-lag theory to consider the microstructural interactions in the fast transient process of heat conduction. All material properties of the media are assumed to vary continuously according to a power-law formulation with arbitrary non-homogeneity indices except the phase lags which are taken constant for simplicity. The one-dimensional heat conduction equations based on the dual-phase-lag theory are derived in a unified form which can be used for Cartesian, cylindrical, and spherical coordinates. A semi-analytical solution for temperature and heat flux is presented using the Laplace transform to eliminate the time dependency of the problem. The results in the time domain are then given by employing a numerical Laplace inversion technique. The semi-analytical solution procedure leads to exact expressions for the thermal wave speed in one-dimensional functionally graded media with different geometries based on the dual-phase-lag and hyperbolic heat conduction theories. The transient temperature distributions have been found for various types of dynamic thermal loading. The numerical results are shown to reveal the effects of phase lags, non-homogeneity indices, and thermal boundary conditions on the thermal responses for different temporal disturbances. The results are verified with those reported in the literature for hyperbolic heat conduction in cylindrical and spherical coordinates.

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Electrocaloric devices part I: Analytical solution of one-dimensional transient heat conduction in a multilayer electrocaloric system

Journal of Advanced Dielectrics ◽

10.1142/s2010135x20500289 ◽

2020 ◽

Vol 10 (06) ◽

pp. 2050028

Author(s):

Farrukh Najmi ◽

Wenxian Shen ◽

Lorenzo Cremaschi ◽

Z.-Y. Cheng

Keyword(s):

Heat Conduction ◽

Analytical Solution ◽

Analytical Solutions ◽

Initial Temperature ◽

Transient Heat Conduction ◽

Typical Structure ◽

One Dimensional ◽

Conduction Heat Transfer ◽

Materials Used

The analytical solution is reported for one-dimensional (1D) dynamic conduction heat transfer within a multilayer system that is the typical structure of electrocaloric devices. Here, the multilayer structure of typical electrocaloric devices is simplified as four layers in which two layers of electrocaloric materials (ECMs) are sandwiched between two semi-infinite bodies representing the thermal sink and source. The temperature of electrocaloric layers can be instantaneously changed by external electric field to establish the initial temperature profile. The analytical solution includes the temperatures in four bodies as a function of both time and location and heat flux through each of the three interfaces as a function of time. Each of these analytical solutions includes five infinite series. It is proved that each of these series is convergent so that the sum of each series can be calculated using the first [Formula: see text] terms of the series. The formula for calculating the value of [Formula: see text] is presented so that the simulation of an electrocaloric device, such as the temperature distribution and heat transferred from one body to another can be performed. The value of [Formula: see text] is dependent on the thickness of electrocaloric material layers, the time of heat conduction, and thermal properties of the materials used. Based on a case study, it is concluded that the [Formula: see text] is mostly less than 20 and barely reaches more than 70. The application of the analytical solutions for the simulation of real electrocaloric devices is discussed.

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ANALITYCAL SOLUTION OF THE ONE DIMENSIONAL NONLINEAR TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN’S FUNCTIONS

Revista de Engenharia Térmica ◽

10.5380/reterm.v20i2.81788 ◽

2021 ◽

Vol 20 (2) ◽

pp. 55

Author(s):

S. S. Ribeiro ◽

G. C. Oliveira ◽

J. R. F. Oliveira ◽

G. Guimarães

Keyword(s):

Heat Conduction ◽

Analytical Solution ◽

Green's Functions ◽

Green’S Functions ◽

Heat Conduction Problem ◽

Transient Heat Conduction ◽

One Dimensional ◽

Transient Heat Conduction Problem ◽

Kirchhoff Transformation ◽

Nonlinear Transient

Analytical solutions showed to be an important and strong tool for understand thermal problems using mathematic tools. In this work we propose an approach about one dimensional analytical solution for a nonlinear transient heat conduction problem, were used mathematical elements such as Kirchhoff transformation, Green’s functions and the combination of them. The combination of this two methods showed that was possible to determinate an analytical solution for the nonlinear thermal problem, and showed a good approximation when compared with results from numerical methods.

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