One-Dimensional Transient Heat Conduction in Composite Living Perfuse Tissue

2013 ◽  
Vol 135 (7) ◽  
Author(s):  
S. M. Becker
Keyword(s):  
Heat Conduction ◽  
Initial Conditions ◽  
Separation Of Variables ◽  
Composite Layer ◽  
Circulatory System ◽  
Transient Heat Conduction ◽  
Bioheat Equation ◽  
Perfusion Rate ◽  
One Dimensional ◽  
Perfuse Tissue

Modeling the conduction of heat in living tissue requires the consideration of sudden spatial discontinuities in property values as well as the presence of the body's circulatory system. This paper presents a description of the separation of variables method that results in a remarkably simple solution of transient heat conduction in a perfuse composite slab for which at least one of the layers experiences a zero perfusion rate. The method uses the natural analytic approach and formats the description so that the constants of integration of each composite layer are expressed in terms of those of the previous layer's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. The method is presented in the context of the Pennes bioheat equation for which the solution is developed for a system composed of any number of N layers with arbitrary initial conditions.

Enhanced Explicit Scheme to Solve Transient Heat Conduction Problem

2007 ◽  
Vol 2 (1) ◽  
Author(s):  
Ganesh Hegde ◽  
Madhu Gattumane
Keyword(s):  
Heat Conduction ◽  
Correction Factor ◽  
Truncation Error ◽  
Transient Heat Conduction ◽  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Time Step ◽  
One Dimensional ◽  
Partial Derivatives

Improvement in accuracy without sacrificing stability and convergence of the solution to unsteady diffusion heat transfer problems by computational method of enhanced explicit scheme (EES), has been achieved and demonstrated, through transient one dimensional and two dimensional heat conduction. The truncation error induced in the explicit scheme using finite difference technique is eliminated by optimization of partial derivatives in the Taylor series expansion, by application of interface theory developed by the authors. This theory, in its simple terms gives the optimum values to the decision vectors in a redundant linear equation. The time derivatives and the spatial partial derivatives in the transient heat conduction, take the values depending on the time step chosen and grid size assumed. The time correction factor and the space correction factor defined by step sizes govern the accuracy, stability and convergence of EES. The comparison of the results of EES with analytical results, show decreased error as compared to the result of explicit scheme. The paper has an objective of reducing error in the explicit scheme by elimination of truncation error introduced by neglecting the higher order terms in the expansion of the governing function. As the pilot examples of the exercise, the implementation is aimed at solving one-dimensional and two-dimensional problems of transient heat conduction and compared with the results cited in the referred literature.

A Simple Model for Transient Heat Conduction in an Infinite Cylinder With Convective Boundary Conditions

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Messaoud Guellal ◽  
Hamou Sadat ◽  
Christian Prax
Keyword(s):  
Heat Conduction ◽  
Simple Model ◽  
Perturbation Method ◽  
Explicit Solution ◽  
Heat Conduction Problem ◽  
Transient Heat Conduction ◽  
Infinite Cylinder ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
One Dimensional

A perturbation method is used to solve an unsteady one-dimensional heat conduction problem in a cylinder. A simple second order explicit solution is obtained. It is shown that this solution is accurate even for high values of the Biot number in a region surrounding the center of the cylinder.

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