Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux

2007 ◽  
Vol 47 (1) ◽  
pp. 107-117 ◽  
Author(s):  
N. A. Kudryashov ◽  
M. A. Chmykhov
Keyword(s):  
Heat Conduction ◽  
Approximate Solutions ◽  
Nonlinear Heat Conduction ◽  
One Dimensional

Approximate Solutions of Canonical Heat Conduction Equations

1990 ◽  
Vol 112 (4) ◽  
pp. 836-842 ◽  
Author(s):  
B. D. Vujanovic´ ◽  
S. E. Jones
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Conduction ◽  
Integral Method ◽  
Approximate Solutions ◽  
Boundary Layer Theory ◽  
Variational Procedure ◽  
Nonlinear Heat Conduction ◽  
One Dimensional ◽  
The Third

We consider three analytical methods for finding the approximate solutions of one-dimensional, transient, and nonlinear heat conduction problems based upon the canonical equations of heat transfer. The first method can be considered as a generalization or refinement of the integral method. The second is an iterative method similar to that of Targ utilized in boundary layer theory. The third method is a variational procedure introduced in the spirit of Gauss’ variational principle of least constraint.

Simulation of Nonlinear Heat Conduction in Perforated Material by Improved Moving Particle Semi-Implicit Method

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
Jianqiang Wang ◽  
Xiaobing Zhang
Keyword(s):  
Heat Conduction ◽  
Laplacian Operator ◽  
Nonlinear Heat Conduction ◽  
Temperature Dependent ◽  
One Dimensional ◽  
Particle Distributions ◽  
Temperature Dependent Conductivity ◽  
Irregular Particle ◽  
Moving Particle ◽  
Temperature Dependent Thermal Conductivity

Abstract An improved moving particle semi-implicit (MPS) method is presented to simulate heat conduction with temperature-dependent thermal conductivity. Based on Taylor expansion, a modified Laplacian operator is proposed, and its accuracy in irregular particle distributions is verified. Two problems are considered: (1) heat conduction in a one-dimensional (1D) slab and (2) heat conduction in a perforated sector with different boundary conditions. Consistent results with a mesh-based method are obtained, and the feasibility of the proposed method for heat conduction simulation with temperature-dependent conductivity is demonstrated.

scholarly journals Approximate Solution of the Nonlinear Heat Conduction Equation in a Semi-Infinite Domain

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Jun Yu ◽  
Yi Yang ◽  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Energy Density ◽  
Approximation Method ◽  
Heat Conduction Equation ◽  
Boundary Energy ◽  
Approximate Solutions ◽  
Nonlinear Heat Conduction ◽  
Infinite Domain ◽  
Nonlinear Heat Conduction Equation ◽  
Self Similar

We use an approximation method to study the solution to a nonlinear heat conduction equation in a semi-infinite domain. By expanding an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain, we reduce the partial differential equation to a set of first-order ordinary differential equations in time. We describe a systematic approach to derive approximate solutions using Taylor polynomials of a different degree. For a special case, we derive an analytical solution and compare it with the result of a self-similar analysis. A comparison with the numerically integrated results demonstrates good accuracy of our approximate solutions. We also show that our approximation method can be applied to cases where boundary energy density and the corresponding effective conductivity are more general than those that are suitable for the self-similar method. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions.

Some Algebraically Explicit Analytical Solutions of Unsteady Nonlinear Heat Conduction

2001 ◽  
Vol 123 (6) ◽  
pp. 1189-1191 ◽  
Author(s):  
Ruixian Cai ◽  
Na Zhang
Keyword(s):  
Heat Conduction ◽  
Analytical Solutions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Conduction Equation ◽  
Nonlinear Heat Conduction ◽  
One Dimensional ◽  
Nonlinear Heat Conduction Equation ◽  
Conduction Theory ◽  
Unsteady Heat Conduction

The analytical solutions of nonlinear unsteady heat conduction equation are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. However, very few explicit analytical solutions have been known for the unsteady nonlinear heat conduction. In order to develop the heat conduction theory, some algebraically explicit analytical solutions of nonlinear heat conduction equation have been derived in this paper, which include one-dimensional and two-dimensional unsteady heat conduction solutions with thermal conductivity, density and specific heat being functions of temperature.

Improved Lumped Models for a One-Dimensional Nonlinear Heat Conduction Problem

2007 ◽  
Author(s):  
Ge Su ◽  
Zheng Tan ◽  
Jian Su
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Distributed Parameter ◽  
Nonlinear Heat Conduction ◽  
Lumped Model ◽  
One Dimensional ◽  
Average Temperature ◽  
Lumped Parameter ◽  
Lumped Parameter Models ◽  
Lumped Models

This work reports improved lumped-parameter models for a class of one-dimensional nonlinear heat conduction problems in a slab, cylinder or sphere with linearly temperature-dependent thermal conductivity and subject to combined convective and radiative boundary condition. The improved lumped models are obtained through two point Hermite approximations for integrals. It is shown by comparison with numerical solution of the original distributed parameter models that the higher order lumped models (H1, 1/H0, 0 approximation for slab and cylinder, H2, 1/H0, 0 for sphere) yield significant improvement of average temperature predictions over the classical lumped model.

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