Approximate solutions of the heat conduction equation for a two-layer plate with sublimation at the outer surface

1967 ◽  
Vol 12 (4) ◽  
pp. 236-241
Author(s):  
P. P. Smyshlyaev
Keyword(s):  
Heat Conduction ◽  
Outer Surface ◽  
Heat Conduction Equation ◽  
Approximate Solutions ◽  
Conduction Equation

Approximate solutions in the theory of thermal explosions for semi-infinite explosives

1968 ◽  
Vol 305 (1481) ◽  
pp. 205-217 ◽  
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Numerical Solutions ◽  
Surface Flux ◽  
Approximate Solutions ◽  
Conduction Equation ◽  
Inert Material ◽  
Zero Order Reaction ◽  
Thermal Explosions

If the solution, of the heat conduction equation θ τ ( 0 ) = θ ξ ξ ( 0 ) , ξ > 0 , τ > 0 of a chemically ‘inert’ material is known, then an approximate formula for the explosion time, ד expl. , of an explosive satisfying the heat conduction equation with zero order reaction, θ ד = θ ξξ +exp(-1/θ), ξ > 0, ד 0, and the same initial and boundary conditions as the ‘inert’, is given by the root of the equation, − ∂ θ ( 0 ) ( ξ , τ expt . ) / ∂ ξ | ξ − 0 = ∫ 0 ∞ exp ⁡ [ − 1 / θ ( 0 ) ( ξ , τ expl . ) ] d ξ provided 1/θ (0) (ξ, ד) is suitably expanded about the surface ξ = 0 such that the integrand vanishes as ξ→∞. Similar results hold for one-dimensional cylindrically and spherically symmetric problems. The derivation of the explosion criterion is based on observation of existing numerical solutions where it is seen that (i) almost to the onset of explosion, the solution θ(ξ, ד )does not differ appreciably from θ (0) (ξ, ד ) (ii) the onset of explosion is indicated by the appearance of a temperature maximum at the surface. Simple formulas for ד expl. readily obtainable for a wide variety of boundary conditions, are given for seven sample problems. Among these are included a semi-infinite explosive with constant surface flux, convective surface heat transfer, and constant surface temperature with and without subsurface melting. The derived values of ד expl. are in satisfactory agreement with those obtained from finite-difference solutions for the problems that can be compared.

A variational formulation applied to heat conduction equation

1970 ◽  
Vol 56 (1) ◽  
pp. 367-379
Author(s):  
Enrico Lorenzini
Keyword(s):  
Heat Conduction ◽  
Variational Formulation ◽  
Heat Conduction Equation ◽  
Conduction Equation

scholarly journals FORMULATION OF THE HEAT CONDUCTION EQUATION FOR HETEROGENEOUS MEDIA WITH MULTIPLE SPATIAL SCALES USING REITERATED HOMOGENIZATION

2016 ◽  
Vol 15 (1) ◽  
pp. 96
Author(s):  
E. Iglesias-Rodríguez ◽  
M. E. Cruz ◽  
J. Bravo-Castillero ◽  
R. Guinovart-Díaz ◽  
R. Rodríguez-Ramos ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Small Parameter ◽  
Heat Conduction Equation ◽  
Heterogeneous Media ◽  
Spatial Scales ◽  
Conduction Equation ◽  
Small Scale ◽  
Multiple Spatial Scales ◽  
Effective Coefficients ◽  
Reiterated Homogenization

Heterogeneous media with multiple spatial scales are finding increased importance in engineering. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. The objective in this paper is to formulate the strong-form Fourier heat conduction equation for such media using the method of reiterated homogenization. The phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter ε. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter ε . The technique leads to two pairs of local and homogenized equations, linked by effective coefficients. In this manner the medium behavior at the smallest scales is seen to affect the macroscale behavior, which is the main interest in engineering. To facilitate the physical understanding of the formulation, an analytical solution is obtained for the heat conduction equation in a functionally graded material (FGM). The approach presented here may serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

scholarly journals A Monte Carlo Method of Solving Heat Conduction Problems

1980 ◽  
Vol 102 (1) ◽  
pp. 121-125 ◽  
Author(s):  
S. K. Fraley ◽  
T. J. Hoffman ◽  
P. N. Stevens
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Monte Carlo Method ◽  
Transport Equation ◽  
Heat Conduction Equation ◽  
Analytical Techniques ◽  
Conduction Equation ◽  
Geometric Complexity ◽  
New Approach ◽  
Temperature Distributions

A new approach in the use of Monte Carlo to solve heat conduction problems is developed using a transport equation approximation to the heat conduction equation. A variety of problems is analyzed with this method and their solutions are compared to those obtained with analytical techniques. This Monte Carlo approach appears to be limited to the calculation of temperatures at specific points rather than temperature distributions. The method is applicable to the solution of multimedia problems with no inherent limitations as to the geometric complexity of the problem.

An Inverse Problem for the Heat Conduction Equation, III

1982 ◽  
Vol 108 (1) ◽  
pp. 73-78
Author(s):  
Gottfried Anger ◽  
Regine Czerner
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Conduction Equation

The iterative alternating decomposition explicit (iade) method to solve the heat conduction equation

1993 ◽  
Vol 47 (3-4) ◽  
pp. 219-229 ◽  
Author(s):  
M. S. Sahimi ◽  
A. Ahmad ◽  
A. A. Bakar
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Conduction Equation

On the stochastic inverse problem for the heat conduction equation

1988 ◽  
Vol 26 (2) ◽  
pp. 245-259 ◽  
Author(s):  
P. Kazimierczyk
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Conduction Equation
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